Large language models (LLMs) have shown promise in simulating human-like social behaviors.Social graphs provide high-quality supervision signals that encode both local interactions and global network structure, yet they remain underutilized for LLM training.To address this gap, we propose Graphia, the first general LLM-based social graph simulation framework that leverages graph data as supervision for LLM post-training via reinforcement learning.With GNN-based structural rewards, Graphia trains specialized agents to predict whom to interact with (destination selection) and how to interact (edge generation), followed by designed graph generation pipelines.We evaluate Graphia under two settings: Transductive Dynamic Graph Generation (TDGG), a micro-level task with our proposed node-wise interaction alignment metrics; and Inductive Dynamic Graph Generation (IDGG), a macro-level task with our proposed metrics for aligning emergent network properties.On three real-world networks, Graphia improves micro-level alignment by 6.1% in the composite destination selection score, 12% in edge classification accuracy, and 27.9% in edge content BERTScore over the strongest baseline.For macro-level alignment, it achieves 35.98% higher structural similarity and 28.71% better replication of social phenomena such as power laws and echo chambers.Our results show that social graphs can serve as high-quality supervision signals for LLM post-training, closing the gap between agent behaviors and network dynamics for LLMbased simulation.Code is available at https: //github.com/Ji-Cather/Graphia.git.

We initiate a study of solving a row/column diagonally dominant (RDD/CDD) linear system $Mx=b$ in sublinear time, with the goal of estimating $t^{\top}x^*$ for a given vector $t\in R^n$ and a specific solution $x^*$. This setting naturally generalizes the study of sublinear-time solvers for symmetric diagonally dominant (SDD) systems [AKP19] to the asymmetric case. Our first contributions are characterizations of the problem's mathematical structure. We express a solution $x^*$ via a Neumann series, prove its convergence, and upper bound the truncation error on this series through a novel quantity of $M$, termed the maximum $p$-norm gap. This quantity generalizes the spectral gap of symmetric matrices and captures how the structure of $M$ governs the problem's computational difficulty. For systems with bounded maximum $p$-norm gap, we develop a collection of algorithmic results for locally approximating $t^{\top}x^*$ under various scenarios and error measures. We derive these results by adapting the techniques of random-walk sampling, local push, and their bidirectional combination, which have proved powerful for special cases of solving RDD/CDD systems, particularly estimating PageRank and effective resistance on graphs. Our general framework yields deeper insights, extended results, and improved complexity bounds for these problems. Notably, our perspective provides a unified understanding of Forward Push and Backward Push, two fundamental approaches for estimating random-walk probabilities on graphs. Our framework also inherits the hardness results for sublinear-time SDD solvers and local PageRank computation, establishing lower bounds on the maximum $p$-norm gap or the accuracy parameter. We hope that our work opens the door for further study into sublinear solvers, local graph algorithms, and directed spectral graph theory.

Antigen-antibody binding is a critical process in the immune response. Although recent progress has advanced antibody design, current methods lack a generative framework for end-to-end modeling of full-atom antibody structures and struggle to fully exploit antigen-specific geometric information for optimizing local binding interfaces and global structures. To overcome these limitations, we introduce AbFlow, a paratope-restricted one-step flow-matching framework for designing full-atom antibodies end-to-end. AbFlow incorporates an extended velocity field network featuring an equivariant Surface Multi-channel Encoder, which uses surface-level antigen interaction data to refine the antibody structure, particularly the CDR-H3 region. Extensive experiments in paratope-centric antibody design, multi-CDRs and full-atom antibody design, binding affinity optimization, and complex structure prediction show that AbFlow produces superior antigen-antibody complexes, especially at the contact interface, and markedly improves the binding affinity of generated antibodies.

Link prediction for directed graphs is a crucial task with diverse real-world applications. Recent advances in embedding methods and Graph Neural Networks (GNNs) have shown promising improvements. However, these methods often lack a thorough analysis of their expressiveness and suffer from the lack of effective benchmarks for fair evaluation. In this paper, we propose a unified framework to assess the expressiveness of existing methods, highlighting the impact of dual embeddings and decoder design on directed link prediction performance. To address limitations in current benchmark setups, we introduce DirLinkBench, a robust new benchmark with comprehensive coverage, standardized evaluation, and modular extensibility. The results on DirLinkBench show that current methods struggle to achieve strong performance, while DiGAE outperforms other baselines overall. We further revisit DiGAE theoretically, showing its graph convolution aligns with GCN on an undirected bipartite graph. Inspired by these insights, we propose a novel Spectral Directed Graph Auto-Encoder SDGAE that achieves state-of-the-art average performance on DirLinkBench. Finally, we analyze key factors influencing directed link prediction and highlight open challenges in this field.

Graph Neural Networks (GNNs) have become widely popular across various applications, with their vulner ability to adversarial attacks being a key concern. Among the different types of graph attacks, Restricted Black-box Attacks (RBAs) present the most strict constraints, as attackers have limited access only to node features and graph structure. Existing RBAs rely on homophily assumptions or shift-based losses as their objectives to conduct structural perturbations, but we demonstrate that all the approaches fail on heterophilic graphs. To address this challenge, we introduce node-wise distance metrics as the objective to fundamentally quantify the quality of the graph structure after perturbations. Our theoretical results show that the proposed objective allows RBAs to effectively handle graphs beyond homophily. Leveraging this objective, we propose HetAttack, a scalable method that significantly reduces the distinguishability of nodes on the victim graph. Experiments on both synthetic and real-world graphs confirm the efficacy of HetAttack across varying levels of homophily, achieving performance comparable to split-unknown white-box attacks without prior knowledge of labels or the target model.

Diffusion-based molecular graph generative models have achieved significant success in template-free, single-step retrosynthesis prediction. However, these models typically generate reactants from scratch, often overlooking the fact that the scaffold of a product molecule typically remains unchanged during chemical reactions. To leverage this useful observation, we introduce a retrieval-augmented molecular graph generation framework. Our framework comprises three key components: a retrieval component that identifies similar molecules for the given product, an integration component that learns valuable clues from these molecules about which part of the product should remain unchanged, and a base generative model that is prompted by these clues to generate the corresponding reactants. We explore various design choices for critical and under-explored aspects of this framework and instantiate it as the Retrieval-Augmented RetroBridge (RARB). RARB demonstrates state-of-the-art performance on standard benchmarks, achieving a 14.8% relative improvement in top-1 accuracy over its base generative model, highlighting the effectiveness of retrieval augmentation. Additionally, RARB excels in handling out-of-distribution molecules, and its advantages remain significant even with smaller models or fewer denoising steps. These strengths make RARB highly valuable for real-world retrosynthesis applications, where extrapolation to novel molecules and high-throughput prediction are essential.

Continual pre-training (CPT) has been an important approach for adapting language models to specific domains or tasks. In this paper, we comprehensively study its key designs to balance the new abilities while retaining the original abilities, and present an effective CPT method that can greatly improve the Chinese language ability and scientific reasoning ability of LLMs. To achieve it, we design specific data mixture and curriculum strategies based on existing datasets and synthetic high-quality data. Concretely, we synthesize multidisciplinary scientific QA pairs based on related web pages to guarantee the data quality, and also devise the performance tracking and data mixture adjustment strategy to ensure the training stability. For the detailed designs, we conduct preliminary studies on a relatively small model, and summarize the findings to help optimize our CPT method. Extensive experiments on a number of evaluation benchmarks show that our approach can largely improve the performance of Llama-3 (8B), including both the general abilities (+8.81 on C-Eval and +6.31 on CMMLU) and the scientific reasoning abilities (+12.00 on MATH and +4.13 on SciEval). Our model, data, and codes are available at https://github.com/RUC-GSAI/Llama-3-SynE.

Future link prediction is a fundamental challenge in various real-world dynamic systems. To address this, numerous temporal graph neural networks (temporal GNNs) and benchmark datasets have been developed. However, these datasets often feature excessive repeated edges and lack complex sequential dynamics, a key characteristic inherent in many real-world applications such as recommender systems and "Who-To-Follow" on social networks. This oversight has led existing methods to inadvertently downplay the importance of learning sequential dynamics, focusing primarily on predicting repeated edges. In this study, we demonstrate that existing methods, such as GraphMixer and DyGFormer, are inherently incapable of learning simple sequential dynamics, such as "a user who has followed OpenAI and Anthropic is more likely to follow AI at Meta next." Motivated by this issue, we introduce the Temporal Graph Benchmark with Sequential Dynamics (TGB-Seq), a new benchmark carefully curated to minimize repeated edges, challenging models to learn sequential dynamics and generalize to unseen edges. TGB-Seq comprises large real-world datasets spanning diverse domains, including e-commerce interactions, movie ratings, business reviews, social networks, citation networks and web link networks. Benchmarking experiments reveal that current methods usually suffer significant performance degradation and incur substantial training costs on TGB-Seq, posing new challenges and opportunities for future research. TGB-Seq datasets, leaderboards, and example codes are available at https://tgb-seq.github.io/.

Graph unlearning has emerged as a pivotal research area for ensuring privacy protection, given the widespread adoption of Graph Neural Networks (GNNs) in applications involving sensitive user data. Among existing studies, certified graph unlearning is distinguished by providing robust privacy guarantees. However, current certified graph unlearning methods are impractical for large-scale graphs because they necessitate the costly re-computation of graph propagation for each unlearning request. Although numerous scalable techniques have been developed to accelerate graph propagation for GNNs, their integration into certified graph unlearning remains uncertain as these scalable approaches introduce approximation errors into node embeddings. In contrast, certified graph unlearning demands bounded model error on exact node embeddings to maintain its certified guarantee. To address this challenge, we present ScaleGUN, the first approach to scale certified graph unlearning to billion-edge graphs. ScaleGUN integrates the approximate graph propagation technique into certified graph unlearning, offering certified guarantees for three unlearning scenarios: node feature, edge and node unlearning. Extensive experiments on real-world datasets demonstrate the efficiency and unlearning efficacy of ScaleGUN. Remarkably, ScaleGUN accomplishes $(\epsilon,\delta)=(1,10^{-4})$ certified unlearning on the billion-edge graph ogbn-papers100M in 20 seconds for a 5,000 random edge removal request -- of which only 5 seconds are required for updating the node embeddings -- compared to 1.91 hours for retraining and 1.89 hours for re-propagation. Our code is available at https://github.com/luyi256/ScaleGUN.

In this paper, we investigate the acceleration of adaptive subgradient methods through frequent directions (FD), a widely-used matrix sketching technique. The state-of-the-art regret bound exhibits a _linear_ dependence on the dimensionality $d$, leading to unsatisfactory guarantees for high-dimensional problems. Additionally, it suffers from an $O(\tau^2 d)$ time complexity per round, which scales quadratically with the sketching size $\tau$. To overcome these issues, we first propose an algorithm named FTSL, achieving a tighter regret bound that is independent of the dimensionality. The key idea is to integrate FD with adaptive subgradient methods under _the primal-dual framework_ and add the cumulative discarded information of FD back. To reduce its time complexity, we further utilize fast FD to expedite FTSL, yielding a better complexity of $O(\tau d)$ while maintaining the same regret bound. Moreover, to mitigate the computational cost for optimization problems involving matrix variables (e.g., training neural networks), we adapt FD to Shampoo, a popular optimization algorithm that accounts for the structure of decision, and give a novel analysis under _the primal-dual framework_. Our proposed method obtains an improved dimension-free regret bound. Experimental results have verified the efficiency and effectiveness of our approaches.

Second-order Online Kernel Learning (OKL) has attracted considerable research interest due to its promising predictive performance in streaming environments. However, existing second-order OKL approaches suffer from at least quadratic time complexity with respect to the pre-set budget, rendering them unsuitable for large-scale datasets. Moreover, the singular value decomposition required to obtain explicit feature mapping is computationally expensive due to the complete decomposition process. To address these issues, we propose FORKS, a fast incremental matrix sketching and decomposition approach tailored for second-order OKL. FORKS constructs an incremental maintenance paradigm for second-order kernelized gradient descent, which includes incremental matrix sketching for kernel approximation and incremental matrix decomposition for explicit feature mapping construction. Theoretical analysis demonstrates that FORKS achieves a logarithmic regret guarantee on par with other second-order approaches while maintaining a linear time complexity w.r.t. the budget, significantly enhancing efficiency over existing methods. We validate the performance of our method through extensive experiments conducted on real-world datasets, demonstrating its superior scalability and robustness against adversarial attacks.

Multivariate Time Series (MTS) forecasting is vital in various practical applications. Current research in this area is categorized into Spatial-Temporal Forecasting (STF) and Long-term Time Series Forecasting (LTSF). While these tasks share similarities, the methods and benchmarks used differ significantly. Spatio-Temporal Graph Neural Networks (STGNNs) excel at modeling interrelationships in STF tasks but face difficulties with long sequence inputs due to inefficient training. In contrast, LTSF models handle long sequences well but struggle with capturing complex variable interrelationships. This paper proposes the Spectral Spatio-Temporal Graph Neural Network (S2GNN) to address these challenges, proposing a framework capable of handling long-term spatiotemporal forecasting. S2GNN leverages a decoupled GNN along with an MLP architecture to ensure efficiency. Specifically, it employs spectral GNNs for global feature extraction on an adaptive graph structure; the visualization of the filters resembles a band-rejection shape, indicating the presence of both homophilic and heterophilic relationships between nodes. Additionally, we introduce scale-adaptive node embeddings and cross-correlation embeddings for better differentiation between similar temporal patterns. Extensive experiments on eight public datasets, including STF and LTSF, demonstrate that S2GNN consistently outperforms state-of-the-art models across diverse prediction tasks. Code is available at https://github.com/superarthurlx/S2GNN.

Graph Neural Networks (GNNs) play a pivotal role in graph-based tasks for their proficiency in representation learning. Among the various GNN methods, spectral GNNs employing polynomial filters have shown promising performance on tasks involving both homophilous and heterophilous graph structures. However, The scalability of spectral GNNs on large graphs is limited because they learn the polynomial coefficients through multiple forward propagation executions during forward propagation. Existing works have attempted to scale up spectral GNNs by eliminating the linear layers on the input node features, a change that can disrupt end-to-end training, potentially impact performance, and become impractical with high-dimensional input features. To address the above challenges, we propose ''Spectral Graph Neural Networks with Laplacian Sparsification (SGNN-LS)'', a novel graph spectral sparsification method to approximate the propagation patterns of spectral GNNs. We prove that our proposed method generates Laplacian sparsifiers that can approximate both fixed and learnable polynomial filters with theoretical guarantees. Our method allows the application of linear layers on the input node features, enabling end-to-end training as well as the handling of raw text features. We conduct an extensive experimental analysis on datasets spanning various graph scales and properties to demonstrate the superior efficiency and effectiveness of our method. The results show that our method yields superior results in comparison with the corresponding approximated base models, especially on dataset Ogbn-papers100M(111M nodes, 1.6B edges) and MAG-scholar-C (2.8M features).

Due to the proficiency of self-attention mechanisms (SAMs) in capturing dependencies in sequence modeling, several existing dynamic graph neural networks (DGNNs) utilize Transformer architectures with various encoding designs to capture sequential evolutions of dynamic graphs. However, the effectiveness and efficiency of these Transformer-based DGNNs vary significantly, highlighting the importance of properly defining the SAM on dynamic graphs and comprehensively encoding temporal and interactive dynamics without extra complex modules. In this work, we propose TIDFormer, a dynamic graph TransFormer that fully exploits Temporal and Interactive Dynamics in an efficient manner. We clarify and verify the interpretability of our proposed SAM, addressing the open problem of its uninterpretable definitions on dynamic graphs in previous works. To model the temporal and interactive dynamics, respectively, we utilize the calendar-based time partitioning information and extract informative interaction embeddings for both bipartite and non-bipartite graphs using merely the sampled first-order neighbors. In addition, we jointly model temporal and interactive features by capturing potential changes in historical interaction patterns through a simple decomposition. We conduct extensive experiments on several dynamic graph datasets to verify the effectiveness and efficiency of TIDFormer. The experimental results demonstrate that TIDFormer excels, outperforming state-of-the-art models across most datasets and experimental settings. Furthermore, TIDFormer exhibits significant efficiency advantages compared to previous Transformer-based methods.

We study the problem of efficiently approximating the effective resistance (ER) on undirected graphs, where ER is a widely used node proximity measure with applications in graph spectral sparsification, multi-class graph clustering, network robustness analysis, graph machine learning, and more. Specifically, given any nodes s and t in an undirected graph G, we aim to efficiently estimate the ER value R(s,t) between nodes s and t, ensuring a small absolute error ε. The previous best algorithm for this problem has a worst-case computational complexity of Õ(Lmax3/ε2d2), where the value of Lmax depends on the mixing time of random walks on G, d = mind(s), d(t), and d(s), d(t) denote the degrees of nodes s and t, respectively. We improve this complexity to Õ ( min Lmax7/3/ε2/3, Lmax3/ε2d2,mLmax ), achieving a theoretical improvement of Õ (maxLmax2/3/ε4/3d2,1, Lmax2/ε2d2m2) over previous results. Here, m denotes the number of edges. Given that Lmax is often very large in real-world networks (e.g., Lmax > 104), our improvement on Lmax is significant, especially for real-world networks. We also conduct extensive experiments on real-world and synthetic graph datasets to empirically demonstrate the superiority of our method. The experimental results show that our method achieves a 10× to 1000× speedup in running time while maintaining the same absolute error compared to baseline methods.

Future link prediction on temporal graphs is a fundamental task with wide applicability in real-world dynamic systems. These scenarios often involve both recurring (seen) and novel (unseen) interactions, requiring models to generalize effectively across both types of edges. However, existing methods typically rely on complex memory and aggregation modules, yet struggle to handle unseen edges. In this paper, we revisit the architecture of existing temporal graph models and identify two essential but overlooked modeling requirements for future link prediction: representing nodes with unique identifiers and performing target-aware matching between source and destination nodes. To this end, we propose Cross-Attention based Future Link Predictor on Temporal Graphs (CRAFT), a simple yet effective architecture that discards memory and aggregation modules and instead builds on two components: learnable node embeddings and cross-attention between the destination and the source's recent interactions. This design provides strong expressive power and enables target-aware modeling of the compatibility between candidate destinations and the source's interaction patterns. Extensive experiments on diverse datasets demonstrate that CRAFT consistently achieves superior performance with high efficiency, making it well-suited for large-scale real-world applications.

The past few years have witnessed a growing interest in LLM-based recommender systems (RSs), although their industrial deployment remains in a preliminary stage. Most existing deployments leverage LLMs offline as feature enhancers, generating augmented knowledge for downstream tasks. However, in recommendation scenarios with numerous users and items, even offline knowledge generation with LLMs demands significant time and computational resources. This inefficiency arises from the autoregressive nature of LLMs. A promising solution is speculative decoding, a Draft-Then-Verify approach that increases the number of tokens generated per decoding step. In this work, we first identify recommendation knowledge generation as a highly fitting use case for retrieval-based speculative decoding. Then, we discern its two characteristics: (1) the vast number of items and users in RSs leads to retrieval inefficiency, and (2) RSs exhibit high diversity tolerance for LLM-generated text. Building on these insights, we introduce Lossless Acceleration via Speculative Decoding for LLM-based Recommender Systems (LASER), which features a Customized Retrieval Pool to enhance retrieval efficiency and Relaxed Verification to improve the acceptance rate of draft tokens. LASER achieves a 3-5x speedup on public datasets and saves about 67% of computational resources during the online A/B test on a large-scale advertising scenario with lossless downstream recommendation performance. Our code is available at https://github.com/YunjiaXi/LASER

Abstract Geometric graphs are a special kind of graph with geometric features, which are vital to model many scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries of translations, rotations, and reflections, making them ineffectively processed by current Graph Neural Networks (GNNs). To address this issue, researchers proposed a variety of geometric GNNs equipped with invariant/equivariant properties to better characterize the geometry and topology of geometric graphs. Given the current progress in this field, it is imperative to conduct a comprehensive survey of data structures, models, and applications related to geometric GNNs. In this paper, based on the necessary but concise mathematical preliminaries, we formalize geometric graph as the data structure, on top of which we provide a unified view of existing models from the geometric message passing perspective. Additionally, we summarize the applications as well as the related datasets to facilitate later research for methodology development and experimental evaluation. We also discuss the challenges and future potential directions of geometric GNNs at the end of this survey.

Abstract Graph neural networks (GNNs) have emerged as a powerful tool for effectively mining and learning from graph-structured data, with applications spanning numerous domains. However, most research focuses on static graphs, neglecting the dynamic nature of real-world networks where topologies and attributes evolve over time. By integrating sequence modeling modules into traditional GNN architectures, dynamic GNNs aim to bridge this gap, capturing the inherent temporal dependencies of dynamic graphs for a more authentic depiction of complex networks. This paper provides a comprehensive review of the fundamental concepts, key techniques, and state-of-the-art dynamic GNN models. We present the mainstream dynamic GNN models in detail and categorize models based on how temporal information is incorporated. We also discuss large-scale dynamic GNNs and pre-training techniques. Although dynamic GNNs have shown superior performance, challenges remain in scalability, handling heterogeneous information, and lack of diverse graph datasets. The paper also discusses possible future directions, such as adaptive and memory-enhanced models, inductive learning, and theoretical analysis.

Graph neural networks (GNNs) have emerged as a promising method for learning from graph data, but large-scale GNN training requires extensive memory and computation resources. To address this, researchers have proposed using multi-GPU processing, which partitions graph data across GPUs for parallel training. However, vertex dependencies in multi-GPU GNN training lead to significant neighbor replications across GPUs, increasing memory consumption. The substantial intermediate data generated during training further exacerbates this issue. Neighbor replication and intermediate data constitute the primary memory consumption in GNN training (i.e., typically accounting for over 80%). In this work, we propose GNN task parallelism for multi-GPU GNN training, which reduces neighbor replication by partitioning training tasks in each layer across different GPUs rather than partitioning the graph structure. This approach only partitions the graph data within individual GPUs, reducing the memory requirements of single tasks while overlapping subgraph computation across different GPUs. Shared neighbor embeddings among different subgraphs can be efficiently reused within a single GPU. Additionally, we employ a task-decoupled GNN training framework, which decouples different training tasks to manage their associated intermediate data independently and release it as early as possible to reduce memory usage. By integrating these techniques, we propose a multi-GPU GNN training system, NeutronTask. Experimental results on a 4×A5000 GPU server show that NeutronTask effectively supports billion-scale full-graph GNN training. For small graphs where the training data fits into the GPUs, NeutronTask achieves 1.27× - 5.47× speedup compared to state-of-the-art GNN systems including NeutronStar and Sancus.

Graph Neural Networks (GNNs) have demonstrated remarkable effectiveness in recommendation systems. However, conventional graph-based recommenders, such as LightGCN, require maintaining embeddings of size d for each node, resulting in a parameter complexity of O ( n X d ), where n represents the total number of users and items. This scaling pattern poses significant challenges for deployment on large-scale graphs encountered in real-world applications. To address this scalability limitation, we propose Lighter-X , an efficient and modular framework that can be seamlessly integrated with existing GNN-based recommender architectures. Our approach substantially reduces both parameter size and computational complexity while preserving the theoretical guarantees and empirical performance of the base models, thereby enabling practical deployment at scale. Specifically, we analyze the original structure and inherent redundancy in their parameters, identifying opportunities for optimization. Based on this insight, we propose an efficient compression scheme for the sparse adjacency structure and high-dimensional embedding matrices, achieving a parameter complexity of O ( h X d ), where h >> n. Furthermore, the model is optimized through a decoupled framework, reducing computational complexity during the training process and enhancing scalability. Extensive experiments demonstrate that Lighter-X achieves comparable performance to baseline models with significantly fewer parameters. In particular, on large-scale interaction graphs with millions of edges, we are able to attain even better results with only 1% of the parameter over LightGCN.

Graph Neural Networks (GNNs) have aroused increasing research attention for their effectiveness on graph mining tasks. However, full-batch training methods based on stochastic gradient descent (SGD) require substantial resources since all gradient-required computational processes are stored in the acceleration device. The bottleneck of storage challenges the training of classic GNNs on large-scale datasets within one acceleration device. Meanwhile, message-passing based (spatial) GNN designs usually necessitate the homophily hypothesis of the graph, which easily fails on heterophilous graphs. In this paper, we propose the random walk extension for those message-passing based GNNs, enriching them with spectral powers. We prove that our random walk sampling with appropriate correction coefficients generates an unbiased approximation of the$K$-order polynomial filter matrix, thus promoting the neighborhood aggregation of the central nodes. Node-wise sampling strategy and historical embedding allow the classic models to be trained with mini-batches, which extends the scalability of the basic models. To show the effectiveness of our method, we conduct a thorough experimental analysis on some frequently-used benchmarks with diverse homophily and scale. The empirical results show that our model achieves significant performance improvements in comparison with the corresponding base GNNs and some state-of-the-art baselines in node classification tasks.

With the increasing application of deep learning to solve scientific problems in biochemistry, molecular federated learning has become popular due to its ability to offer distributed privacy-preserving solutions. However, most existing molecular federated learning methods rely on joint training with public datasets, which are difficult to obtain in practice. These methods also fail to leverage multi-modal molecular representations effectively. To address the above issues, we propose a novel framework, Federated Heterogeneous Contrastive Distillation (FedHCD), which enables to jointly train global models from clients with heterogeneous data modalities, learning tasks, and molecular models. To aggregate data representations of different modalities in a data-free manner, we design a global multi-modal contrastive strategy to align the representation of clients without public dataset. Utilizing intrinsic characteristics of molecular data in different modalities, we tackle the exacerbation of local model drift and data Non-IIDness caused by multi-modal clients. We introduce a multi-view contrastive knowledge transfer to extract features from atoms, substructures, and molecules, solving the issue of information distillation failure due to dimensional biases in different data modalities. Our evaluations on eight real-world molecular datasets and ablation experiments show that FedHCD outperforms other state-of-the-art FL methods, irrespective of whether or not they use public datasets.

The drastic performance degradation of Graph Neural Networks (GNNs) as the depth of the graph propagation layers exceeds 8-10 is widely attributed to a phenomenon of Over-smoothing. Although recent research suggests that Over-smoothing may not be the dominant reason for such a performance degradation, they have not provided rigorous analysis from a theoretical view, which warrants further investigation In this paper, we systematically analyze the real dominant problem in deep GNNs and identify the issues that these GNNs towards addressing Over-smoothing essentially work on via empirical experiments and theoretical gradient analysis. We theoretically prove that the difficult training problem of deep MLPs is actually the main challenge, and various existing methods that supposedly tackle Over-smoothing actually improve the trainability of MLPs, which is the main reason for their performance gains. Our further investigation into trainability issues reveals that properly constrained smaller upper bounds of gradient flow notably enhance the trainability of GNNs. Experimental results on diverse datasets demonstrate consistency between our theoretical findings and empirical evidence. Our analysis provides new insights in constructing deep graph models.

Personalized PageRank (PPR) is an extensively studied and applied node proximity measure in graphs. For a pair of nodes $s$ and $t$ on a graph $G=(V,E)$, the PPR value $π(s,t)$ is defined as the probability that an $α$-discounted random walk from $s$ terminates at $t$, where the walk terminates with probability $α$ at each step. We study the classic Single-Source PPR query, which asks for PPR approximations from a given source node $s$ to all nodes in the graph. Specifically, we aim to provide approximations with absolute error guarantees, ensuring that the resultant PPR estimates $\hatπ(s,t)$ satisfy $\max_{t\in V}\big|\hatπ(s,t)-π(s,t)\big|\le\varepsilon$ for a given error bound $\varepsilon$. We propose an algorithm that achieves this with high probability, with an expected running time of - $\widetilde{O}\big(\sqrt{m}/\varepsilon\big)$ for directed graphs, where $m=|E|$; - $\widetilde{O}\big(\sqrt{d_{\mathrm{max}}}/\varepsilon\big)$ for undirected graphs, where $d_{\mathrm{max}}$ is the maximum node degree in the graph; - $\widetilde{O}\left(n^{γ-1/2}/\varepsilon\right)$ for power-law graphs, where $n=|V|$ and $γ\in\left(\frac{1}{2},1\right)$ is the extent of the power law. These sublinear bounds improve upon existing results. We also study the case when degree-normalized absolute error guarantees are desired, requiring $\max_{t\in V}\big|\hatπ(s,t)/d(t)-π(s,t)/d(t)\big|\le\varepsilon_d$ for a given error bound $\varepsilon_d$, where the graph is undirected and $d(t)$ is the degree of node $t$. We give an algorithm that provides this error guarantee with high probability, achieving an expected complexity of $\widetilde{O}\left(\sqrt{\sum_{t\in V}π(s,t)/d(t)}\big/\varepsilon_d\right)$. This improves over the previously known $O(1/\varepsilon_d)$ complexity.

Recently, Graph Contrastive Learning (GCL) has achieved significantly superior performance in self-supervised graph representation learning. However, the existing GCL technique has inherent smooth characteristics because of its low-pass GNN encoder and objective based on homophily assumption, which poses a challenge when applying it to heterophilic graphs. In supervised learning tasks, spectral GNNs with polynomial approximation excel in both homophilic and heterophilic settings by adaptively fitting graph filters of arbitrary shapes. Yet, their applications in unsupervised learning are rarely explored. Based on the above analysis, a natural question arises: Can we incorporate the excellent properties of spectral polynomial filters into graph contrastive learning? In this paper, we address the question by studying the necessity of introducing high-pass information for heterophily from a spectral perspective. We propose PolyGCL, a GCL pipeline that utilizes polynomial filters to achieve contrastive learning between the low-pass and high-pass views. Specifically, PolyGCL utilizes polynomials with learnable filter functions to generate different spectral views and an objective that incorporates high-pass information through a linear combination. We theoretically prove that PolyGCL outperforms previous GCL paradigms when applied to graphs with varying levels of homophily. We conduct extensive experiments on both synthetic and real-world datasets, which demonstrate the promising performance of PolyGCL on homophilic and heterophilic graphs.

Predicting the binding sites of target proteins plays a fundamental role in drug discovery. Most existing deep-learning methods consider a protein as a 3D image by spatially clustering its atoms into voxels and then feed the voxelized protein into a 3D CNN for prediction. However, the CNN-based methods encounter several critical issues: 1) defective in representing irregular protein structures; 2) sensitive to rotations; 3) insufficient to characterize the protein surface; 4) unaware of protein size shift. To address the above issues, this work proposes EquiPocket, an E(3)-equivariant Graph Neural Network (GNN) for binding site prediction, which comprises three modules: the first one to extract local geometric information for each surface atom, the second one to model both the chemical and spatial structure of protein and the last one to capture the geometry of the surface via equivariant message passing over the surface atoms. We further propose a dense attention output layer to alleviate the effect incurred by variable protein size. Extensive experiments on several representative benchmarks demonstrate the superiority of our framework to the state-of-the-art methods.

Spectral Graph Neural Networks have demonstrated superior performance in graph representation learning. However, many current methods focus on employing shared polynomial coefficients for all nodes, i.e., learning node-unified filters, which limits the filters' flexibility for node-level tasks. The recent DSF attempts to overcome this limitation by learning node-wise coefficients based on positional encoding. However, the initialization and updating process of the positional encoding are burdensome, hindering scalability on large-scale graphs. In this work, we propose a scalable node-wise filter, PolyAttn. Leveraging the attention mechanism, PolyAttn can directly learn node-wise filters in an efficient manner, offering powerful representation capabilities. Building on PolyAttn, we introduce the whole model, named PolyFormer. In the lens of Graph Transformer models, PolyFormer, which calculates attention scores within nodes, shows great scalability. Moreover, the model captures spectral information, enhancing expressiveness while maintaining efficiency. With these advantages, PolyFormer offers a desirable balance between scalability and expressiveness for node-level tasks. Extensive experiments demonstrate that our proposed methods excel at learning arbitrary node-wise filters, showing superior performance on both homophilic and heterophilic graphs, and handling graphs containing up to 100 million nodes. The code is available at https://github.com/air029/PolyFormer.

Graph Neural Networks (GNNs) excel across various applications but remain vulnerable to adversarial attacks, particularly Graph Injection Attacks (GIAs), which inject malicious nodes into the original graph and pose realistic threats. Text-attributed graphs (TAGs), where nodes are associated with textual features, are crucial due to their prevalence in real-world applications and are commonly used to evaluate these vulnerabilities. However, existing research only focuses on embedding-level GIAs, which inject node embeddings rather than actual textual content, limiting their applicability and simplifying detection. In this paper, we pioneer the exploration of GIAs at the text level, presenting three novel attack designs that inject textual content into the graph. Through theoretical and empirical analysis, we demonstrate that text interpretability, a factor previously overlooked at the embedding level, plays a crucial role in attack strength. Among the designs we investigate, the Word-frequency-based Text-level GIA (WTGIA) is particularly notable for its balance between performance and interpretability. Despite the success of WTGIA, we discover that defenders can easily enhance their defenses with customized text embedding methods or large language model (LLM)--based predictors. These insights underscore the necessity for further research into the potential and practical significance of text-level GIAs.

Virtual Screening is an essential technique in the early phases of drug discovery, aimed at identifying promising drug candidates from vast molecular libraries. Recently, ligand-based virtual screening has garnered significant attention due to its efficacy in conducting extensive database screenings without relying on specific protein-binding site information. Obtaining binding affinity data for complexes is highly expensive, resulting in a limited amount of available data that covers a relatively small chemical space. Moreover, these datasets contain a significant amount of inconsistent noise. It is challenging to identify an inductive bias that consistently maintains the integrity of molecular activity during data augmentation. To tackle these challenges, we propose S-MolSearch, the first framework to our knowledge, that leverages molecular 3D information and affinity information in semi-supervised contrastive learning for ligand-based virtual screening. % S-MolSearch processes both labeled and unlabeled data, trains molecular structural encoders, and generates soft labels for unlabeled data, drawing on the principles of inverse optimal transport. Drawing on the principles of inverse optimal transport, S-MolSearch efficiently processes both labeled and unlabeled data, training molecular structural encoders while generating soft labels for the unlabeled data. This design allows S-MolSearch to adaptively utilize unlabeled data within the learning process. Empirically, S-MolSearch demonstrates superior performance on widely-used benchmarks LIT-PCBA and DUD-E. It surpasses both structure-based and ligand-based virtual screening methods for AUROC, BEDROC and EF.

We revisit ApproxContributions, the classic local graph exploration algorithm proposed by Andersen, Borgs, Chayes, Hopcroft, Mirrokni, and Teng (WAW ’07, Internet Math. ’08) for computing an є-approximation of the PageRank contribution vector for a target node t on a graph with n nodes and m edges. We give a worst-case complexity bound of it as O(nπ(t)/є·min(Δin,Δout,√m)), where π(t) is the PageRank score of t, and Δin and Δout are the maximum in-degree and out-degree of the graph, resp. We also give a lower bound of Ω(min(Δin/δ,Δout/δ,√m/δ,m)) for detecting t’s δ-contributing set, showing that the simple ApproxContributions algorithm is already optimal.

Vision Transformers (ViTs) have revolutionized the field of computer vision, yet their deployments on resource-constrained devices remain challenging due to high computational demands. To expedite pre-trained ViTs, token pruning and token merging approaches have been developed, which aim at reducing the number of tokens involved in the computation. However, these methods still have some limitations, such as image information loss from pruned tokens and inefficiency in the token-matching process. In this paper, we introduce a novel Graph-based Token Propagation (GTP) method to resolve the challenge of balancing model efficiency and information preservation for efficient ViTs. Inspired by graph summarization algorithms, GTP meticulously propagates less significant tokens’ information to spatially and semantically connected tokens that are of greater importance. Consequently, the remaining few tokens serve as a summarization of the entire token graph, allowing the method to reduce computational complexity while preserving essential information of eliminated tokens. Combined with an innovative token selection strategy, GTP can efficiently identify image tokens to be propagated. Extensive experiments have validated GTP’s effectiveness, demonstrating both efficiency and performance improvements. Specifically, GTP decreases the computational complexity of both DeiT-S and DeiT-B by up to 26% with only a minimal 0.3% accuracy drop on ImageNet-1K without finetuning, and remarkably surpasses the state-of-the-art token merging method on various backbones at an even faster inference speed. The source code is available at https://github.com/Ackesnal/GTP-ViT.

Heterogeneous Graph Neural Networks (HGNNs) have gained significant popularity in various heterogeneous graph learning tasks. However, most existing HGNNs rely on spatial domain-based methods to aggregate information, i.e., manually selected meta-paths or some heuristic modules, lacking theoretical guarantees. Furthermore, these methods cannot learn arbitrary valid heterogeneous graph filters within the spectral domain, which have limited expressiveness. To tackle these issues, we present a positive spectral heterogeneous graph convolution via positive noncommutative polynomials. Then, using this convolution, we propose PSHGCN, a novel Positive Spectral Heterogeneous Graph Convolutional Network. PSHGCN offers a simple yet effective method for learning valid heterogeneous graph filters. Moreover, we demonstrate the rationale of PSHGCN in the graph optimization framework. We conducted an extensive experimental study to show that PSHGCN can learn diverse heterogeneous graph filters and outperform all baselines on open benchmarks. Notably, PSHGCN exhibits remarkable scalability, efficiently handling large real-world graphs comprising millions of nodes and edges. Our codes are available at https://github.com/ivam-he/PSHGCN.

Recently, how the model performance scales with the training sample size has been extensively studied for large models on vision and language related domains. Nevertheless, the ubiquitous node classification tasks on web-scale graphs were ignored, where the traits of these tasks, such as non-IIDness and transductive setting, are likely to cause different scaling laws and motivate novel techniques to beat the law. Therefore, we first explore the neural scaling law for node classification tasks on three large-scale graphs. Then, we benchmark several state-of-the-art data pruning methods on these tasks, not only validating the possibility of improving the original unsatisfactory power law but also gaining insights into a hard-and-representative principle on picking an effective subset of training nodes. Moreover, we leverage the transductive setting to propose a novel data pruning method, which instantiates our principle in a test set-targeted manner. Our method consistently outperforms related methods on all three datasets. Meanwhile, we utilize a PAC-Bayesian framework to analyze our method, extending prior results to account for both hardness and representativeness. In addition to a promising way to ease GNN training on web-scale graphs, our study offers knowledge of the relationship between training nodes and GNN generalization.

Abstract Autonomous agents have long been a research focus in academic and industry communities. Previous research often focuses on training agents with limited knowledge within isolated environments, which diverges significantly from human learning processes, and makes the agents hard to achieve human-like decisions. Recently, through the acquisition of vast amounts of Web knowledge, large language models (LLMs) have shown potential in human-level intelligence, leading to a surge in research on LLM-based autonomous agents. In this paper, we present a comprehensive survey of these studies, delivering a systematic review of LLM-based autonomous agents from a holistic perspective. We first discuss the construction of LLM-based autonomous agents, proposing a unified framework that encompasses much of previous work. Then, we present a overview of the diverse applications of LLM-based autonomous agents in social science, natural science, and engineering. Finally, we delve into the evaluation strategies commonly used for LLM-based autonomous agents. Based on the previous studies, we also present several challenges and future directions in this field.

The problem of estimating data properties using sampling frequency histograms has attracted extensive interest in the area of databases. The properties include the number of distinct values (NDV), entropy, and so on. In the field of databases, property estimation is fundamental to complex applications. For example, NDV estimation is the foundation of query optimization, and entropy estimation is the foundation of data compression. Among them, methods originating from statistics exhibit desirable theoretical guarantees but rely on specific assumptions about the distribution of data, resulting in poor performance in real-world applications. Learning-based methods, which use information from training data, are adaptable in the real world but often lack theoretical guarantees or explainability. In addition, a unified framework for estimating these frequency-based estimators with machine learning is lacking. Given the aforementioned challenges, it is natural to wonder if a unified framework with theoretical guarantees can be established for property estimation. The recent literature has presented theoretical studies that propose estimation frameworks based on polynomials. These studies also prove estimation errors with respect to the sample size. Motivated by the above polynomial estimation framework, we propose a learning-based estimation framework with polynomial approximation, which aims to learn the coefficients of the polynomial, providing theoretical guarantees to the learning framework. Through comprehensive experiments on both synthetic and real-world datasets for estimating various data properties like NDV, entropy, and power sum, our results show the superiority of our algorithms over previous estimators.

Matrix sketching, aimed at approximating a matrix A ∈ R N×d consisting of vector streams of length N with a smaller sketching matrix B ∈ R ℓ×d , ℓ ≪ N , has garnered increasing attention in fields such as large-scale data analytics and machine learning. A well-known deterministic matrix sketching method is the FreqentDirections algorithm, which achieves the optimal [EQUATION] space bound and provides a covariance error guarantee of ε = || A ⊤ A - B ⊤ B || 2 /|| A || 2 F. The matrix sketching problem becomes particularly interesting in the context of sliding windows, where the goal is to approximate the matrix A W , formed by input vectors over the most recent N time units. However, despite recent efforts, whether achieving the optimal [EQUATION] space bound on sliding windows is possible has remained an open question. In this paper, we introduce the DS-FD algorithm, which achieves the optimal [EQUATION] space bound for matrix sketching over row-normalized, sequence-based sliding windows. We also present matching upper and lower space bounds for time-based and unnormalized sliding windows, demonstrating the generality and optimality of DS-FD across various sliding window models. This conclusively answers the open question regarding the optimal space bound for matrix sketching over sliding windows. We conduct extensive experiments with both synthetic and real-world datasets, validating our theoretical claims and thus confirming the correctness and effectiveness of our algorithm, both theoretically and empirically.

Cardinality estimation (CardEst) is essential for optimizing query execution plans. Recent ML-based CardEst methods achieve high accuracy but face deployment challenges due to high preparation costs and lack of transferability across databases. In this paper, we propose PRICE, a PRetrained multI-table CardEst model, which addresses these limitations. PRICE takes low-level but transferable features w.r.t. data distributions and query information and elegantly applies self-attention models to learn meta-knowledge to compute cardinality in any database. It is generally and adaptively applicable to any unseen new database to attain high estimation accuracy, while its preparation cost is as little as the basic one-dimensional histogram-based CardEst methods. Moreover, PRICE can be finetuned to further enhance its performance on any specific database. We pretrained PRICE using 30 diverse datasets, completing the process in about 5 hours with a resulting model size of only about 40MB. Evaluations show that PRICE consistently outperforms existing methods, achieving the highest estimation accuracy on several unseen databases and generating faster execution plans with lower overhead. After finetuning with a small volume of database-specific queries, PRICE could even find plans that were very close to the optimal ones. Meanwhile, PRICE is generally applicable to different settings such as data updates, data scaling, and query workload shifts.

The successes of applying Transformer to graphs have been witnessed on small graphs (e.g., molecular graphs), yet two barriers prevent its adoption on large graphs (e.g., citation networks). First, despite the benefit of the global receptive field, enormous distant nodes might distract the necessary attention of each target node from its neighborhood. Second, training a Transformer model on large graphs is costly due to the node-to-node attention mechanism's quadratic computational complexity. To break down these barriers, we propose a two-view architectureCoarformer, wherein a GNN-based module captures fine-grained local information from the original graph, and a Transformer-based module captures coarse yet long-range information on the coarse graph. We further design a cross-view propagation scheme so that these two views can enhance each other. Our graph isomorphism analysis shows the complementary natures of GNN and Transformer, justifying the motivation and design ofCoarformer. We conduct extensive experiments on real-world datasets, whereCoarformersurpasses any single-view method that solely applies a GNN or Transformer. As an ablation,Coarformeroutperforms straightforward combinations of a GNN model and a Transformer-based model, verifying the effectiveness of our coarse global view and the cross-view propagation scheme. Meanwhile,Coarformerconsumes the least runtime and GPU memory than those combinations.

Personalized PageRank (PPR) is a traditional measure for node proximity on large graphs. For a pair of nodes$\boldsymbol{s}$and$\boldsymbol{t}$, the PPR value$\boldsymbol{\pi_{s}(t)}$equals the probability that an$\boldsymbol{\alpha }$-discounted random walk from$\boldsymbol{s}$terminates at$\boldsymbol{t}$and reflects the importance between$\boldsymbol{s}$and$\boldsymbol{t}$in a bidirectional way. As a generalization of Google's celebrated PageRank centrality, PPR has been extensively studied and has found multifaceted applications in many fields, such as network analysis, graph mining, and graph machine learning. Despite numerous studies devoted to PPR over the decades, efficient computation of PPR remains a challenging problem, and there is a dearth of systematic summaries and comparisons of existing algorithms. In this paper, we recap several frequently used techniques for PPR computation and conduct a comprehensive survey of various recent PPR algorithms from an algorithmic perspective. We classify these approaches based on the types of queries they address and review their methodologies and contributions. We also discuss some representative algorithms for computing PPR on dynamic graphs and in parallel or distributed environments.

We study the query version of the approximate heavy hitter and quantile problems. In the former problem, the input is a parameter $\varepsilon$ and a set $P$ of $n$ points in $\mathbb{R}^d$ where each point is assigned a color from a set $C$, and we want to build a structure s.t. given any geometric range $γ$, we can efficiently find a list of approximate heavy hitters in $γ\cap P$, i.e., colors that appear at least $\varepsilon |γ\cap P|$ times in $γ\cap P$, as well as their frequencies with an additive error of $\varepsilon |γ\cap P|$. In the latter problem, each point is assigned a weight from a totally ordered universe and the query must output a sequence $S$ of $1+1/\varepsilon$ weights s.t. the $i$-th weight in $S$ has approximate rank $i\varepsilon|γ\cap P|$, meaning, rank $i\varepsilon|γ\cap P|$ up to an additive error of $\varepsilon|γ\cap P|$. Previously, optimal results were only known in 1D [WY11] but a few sub-optimal methods were available in higher dimensions [AW17, ACH+12]. We study the problems for 3D halfspace and dominance queries. We consider the real RAM model with integer registers of size $w=Θ(\log n)$ bits. For dominance queries, we show optimal solutions for both heavy hitter and quantile problems: using linear space, we can answer both queries in time $O(\log n + 1/\varepsilon)$. Note that as the output size is $\frac{1}{\varepsilon}$, after investing the initial $O(\log n)$ searching time, our structure takes on average $O(1)$ time to find a heavy hitter or a quantile! For more general halfspace heavy hitter queries, the same optimal query time can be achieved by increasing the space by an extra $\log_w\frac{1}{\varepsilon}$ (resp. $\log\log_w\frac{1}{\varepsilon}$) factor in 3D (resp. 2D). By spending extra $\log^{O(1)}\frac{1}{\varepsilon}$ factors in time and space, we can also support quantile queries.

In long-term time series forecasting, most Transformer-based methods adopt the standard point-wise attention mechanism, which not only has high complexity but also cannot explicitly capture the predictive dependencies from contexts since the corresponding key and value are transformed from the same point. This paper proposes a predictive Transformer-based model called Preformer. Preformer introduces a novel efficient Multi-Scale Segment-Correlation mechanism that divides time series into segments and utilizes segment-wise correlation-based attention to replace point-wise attention. A multi-scale structure is developed to aggregate dependencies at different temporal scales and facilitate the selection of segment length. Preformer further designs a predictive paradigm for decoding, where the key and value come from two successive segments rather than the same segment. Experiments demonstrate that Preformer outperforms other Transformer-based models. The codes are available at https://github.com/ddz16/Preformer.

Polynomial filters, a kind of Graph Neural Networks, typically use a predetermined polynomial basis and learn the coefficients from the training data. It has been observed that the effectiveness of the model is highly dependent on the property of the polynomial basis. Consequently, two natural and fundamental questions arise: Can we learn a suitable polynomial basis from the training data? Can we determine the optimal polynomial basis for a given graph and node features? In this paper, we propose two spectral GNN models that provide positive answers to the questions posed above. First, inspired by Favard's Theorem, we propose the FavardGNN model, which learns a polynomial basis from the space of all possible orthonormal bases. Second, we examine the supposedly unsolvable definition of optimal polynomial basis from Wang et al. (2022) and propose a simple model, OptBasisGNN, which computes the optimal basis for a given graph structure and graph signal. Extensive experiments are conducted to demonstrate the effectiveness of our proposed models. Our code is available at https://github.com/yuziGuo/FarOptBasis.

Graph Neural Networks (GNNs) have emerged as a prominent research topic in the field of machine learning. Existing GNN models are commonly categorized into two types: spectral GNNs, which are designed based on polynomial graph filters, and spatial GNNs, which utilize a message-passing scheme as the foundation of the model. For the expressive power and universality of spectral GNNs, a natural approach is to improve the design of basis functions for better approximation ability. As for spatial GNNs, models like Graph Isomorphism Networks (GIN) analyze their expressive power based on Graph Isomorphism Tests. Recently, there have been attempts to establish connections between spatial GNNs and geometric concepts like curvature and cellular sheaves, as well as physical phenomena like oscillators. However, despite the recent progress, there is still a lack of comprehensive analysis regarding the universality of spatial GNNs from the perspectives of geometry and physics.

Graph Convolutional Networks (GCNs), which use a message-passing paradigm with stacked convolution layers, are foundational spatial methods for learning graph representations. Polynomial filters, which have an advantage on heterophilous graphs, are motivated differently from the spectral perspective of graph convolutions. Recent spatial GCN models use various residual connection techniques to alleviate the model degradation problem such as over-smoothing and gradient vanishing. However, current residual connections do not effectively harness the full potential of polynomial filters, which are commonly employed in the spectral domain of GNNs.

We study the fundamental problem of sampling independent events, called subset sampling. Specifically, consider a set of n distinct events S=x1, …, xn, in which each event xi has an associated probability p(xi). The subset sampling problem aims to sample a subset T ⊆ S, such that every xi is independently included in T with probability p(xi). A naive solution is to flip a coin for each event, which takes O(n) time. However, an ideal solution is a data structure that allows drawing a subset sample in time proportional to the expected output size μ=∑i=1n p(xi), which can be significantly smaller than n in many applications. The subset sampling problem serves as an important building block in many tasks and has been the subject of various research for more than a decade.

\em Personalized PageRank (PPR) stands as a fundamental proximity measure in graph mining. Given an input graph G with the probability of decay α, a source node s and a target node t, the PPR score π(s,t) of target t with respect to source s is the probability that an α-decay random walk starting from s stops at t. A \em single-source PPR (SSPPR) query takes an input graph G with decay probability α and a source s, and then returns the PPR π(s,v) for each node v ∈ V. Since computing an exact SSPPR query answer is prohibitive, most existing solutions turn to approximate queries with guarantees. The state-of-the-art solutions for approximate SSPPR queries are index-based and mainly focus on static graphs, while real-world graphs are usually dynamically changing. However, existing index-update schemes can not achieve a sub-linear update time. Motivated by this, we present an efficient indexing scheme for single-source PPR queries on evolving graphs. Our proposed solution is based on a classic framework that combines the forward-push technique with a random walk index for approximate PPR queries. Thus, our indexing scheme is similar to existing solutions in the sense that we store pre-sampled random walks for efficient query processing. One of our main contributions is an incremental updating scheme to maintain indexed random walks in expected O(1) time after each graph update. To achieve O(1) update cost, we need to maintain auxiliary data structures for both vertices and edges. To reduce the space consumption, we further revisit the sampling methods and propose a new sampling scheme to remove the auxiliary data structure for vertices while still supporting O(1) index update cost on evolving graphs. Extensive experiments show that our update scheme achieves orders of magnitude speed-up on update performance over existing index-based dynamic schemes without sacrificing the query efficiency.

Real-world graphs, such as social networks, financial transactions, and recommendation systems, often demonstrate dynamic behavior. This phenomenon, known as graph stream, involves the dynamic changes of nodes and the emergence and disappearance of edges. To effectively capture both the structural and temporal aspects of these dynamic graphs, dynamic graph neural networks have been developed. However, existing methods are usually tailored to process either continuous-time or discrete-time dynamic graphs, and cannot be generalized from one to the other. In this paper, we propose a decoupled graph neural network for large dynamic graphs, including a unified dynamic propagation that supports efficient computation for both continuous and discrete dynamic graphs. Since graph structure-related computations are only performed during the propagation process, the prediction process for the downstream task can be trained separately without expensive graph computations, and therefore any sequence model can be plugged-in and used. As a result, our algorithm achieves exceptional scalability and expressiveness. We evaluate our algorithm on seven real-world datasets of both continuous-time and discrete-time dynamic graphs. The experimental results demonstrate that our algorithm achieves state-of-the-art performance in both kinds of dynamic graphs. Most notably, the scalability of our algorithm is well illustrated by its successful application to large graphs with up to over a billion temporal edges and over a hundred million nodes.

The tremendous computing capacity of GPU offers significant potential in processing hierarchically compressed text data without decompression. However, current GPU techniques offer only traversal-based text data analytics; random access is exceedingly inefficient, limiting their utility significantly. To address this issue, we develop a novel and widely applicable solution that prompts random access to hierarchically compressed text data without decompression in GPU memory. We address three main challenges for enabling efficient random access to compressed text data on GPUs. The first challenge is designing GPU data structures that facilitate random access. The second challenge is efficiently generating data structures on GPU. The CPU is inefficient when generating data structures for random access, and this inefficiency increases considerably when PCIe transmission is incorporated. The third challenge is query processing on compressed text data in GPU memory. Random accesses, such as data updates, cause massive conflicts among countless threads. In order to address the first challenge, we develop several compressed GPU data structures, including indexing within the intricate GPU memory hierarchy. To handle the second challenge, we propose a two-phase process for producing these data structures on GPU. For the third challenge, a double-parsing design is proposed as a solution to avoid conflicts. We evaluate our solution on three platforms, two server-grade GPU platforms and one edge-grade GPU platform, using five real-world datasets. Experimental results show that random access operations on GPU achieve an average speedup of 52.98× compared to the state-of-the-art solution.

Masked autoencoder (MAE), as an effective self-supervised learner for computer vision and natural language processing, has been recently applied to molecule representation learning. In this paper, we identify two issues in applying MAE to pre-train Transformer-based models on molecular graphs that existing works have ignored. (1) As only atoms are abstracted as tokens and then reconstructed, the chemical bonds are not decided in the decoded molecule, making molecules with different arrangements of the same atoms indistinguishable. (2) Although a high mask ratio that corresponds to a challenging reconstruction task has been proved beneficial in the vision domain, it cannot be trivially leveraged on molecular graphs as there is less redundancy of information in graph data. To resolve these issues, we propose a novel framework, Molecular Graph Mask AutoEncoder (MGMAE). As the first step in MGMAE, we transform each molecular graph into a heterogeneous atom-bond graph to fully use the bond attributes and design unidirectional position encoding for such graphs. Then we propose a hybrid masking mechanism that exploits the complementary nature between atoms' attributive and spatial features. Meanwhile, we compensate for the mask embedding by a dynamic aggregation representation that exploits the correlations between topologically adjacent tokens. As a result, MGMAE can reconstruct the masked atoms, the masked bonds, and the relative distance among atoms simultaneously, with a high mask ratio. We compare MGMAE with the state-of-the-art methods on various molecular benchmarks and show the competitiveness of MGMAE in both regression and classification tasks.

Real-world graphs exhibit many interesting properties that differentiate them from random graphs, which have been extensively studied for the past decades. For various proposed generative models, a majority of them build the graph by sequentially adding each node and the attached edges. However, the growth of many real-world graphs, such as social networks, is naturally modeled by the sequential insertion of edges. Unfortunately, to the best of our knowledge, no generative model has been proposed to reveal this process. We propose the first sequence-of-edges model, denoted as temporal preferential attachment (TPA). It relies on preferential attachment (PA), one of the most influential mechanisms to generate scale-free graphs, and takes time-decay effect and node fitness into consideration. Empirical analysis demonstrates that our model preserves several key properties of the real-world graphs, including both the properties observed from the snapshot graphs (e.g., power-law distribution) and temporal properties observed from the graph generation process (e.g., shrinking diameter). Meanwhile, our model is sufficiently general to accommodate several forms of time decay and fitness distributions. Then, we design two efficient algorithms that generate TPA graphs with billions of edges in several minutes.

The prediction of protein-ligand binding affinity is of great significance for discovering lead compounds in drug research. Facing this challenging task, most existing prediction methods rely on the topological and/or spatial structure of molecules and the local interactions while ignoring the multi-level inter-molecular interactions between proteins and ligands, which often lead to sub-optimal performance. To solve this issue, we propose a novel global-local interaction (GLI) framework to predict protein-ligand binding affinity. In particular, our GLI framework considers the inter-molecular interactions between proteins and ligands, which involve not only the high-energy short-range interactions between closed atoms but also the low-energy long-range interactions between non-bonded atoms. For each pair of protein and ligand, our GLI embeds the long-range interactions globally and aggregates local short-range interactions, respectively. Such a joint global-local interaction modeling strategy helps to improve prediction accuracy, and the whole framework is compatible with various neural network-based modules. Experiments demonstrate that our GLI framework outperforms state-of-the-art methods with simple neural network architectures and moderate computational costs.

In data mining, estimating the number of distinct values (NDV) is a fundamental problem with various applications. Existing methods for estimating NDV can be broadly classified into two categories: i) scanning-based methods, which scan the entire data and maintain a sketch to approximate NDV; and ii) sampling-based methods, which estimate NDV using sampling data rather than accessing the entire data warehouse. Scanning-based methods achieve a lower approximation error at the cost of higher I/O and more time. Sampling-based estimation is preferable in applications with a large data volume and a permissible error restriction due to its higher scalability. However, while the sampling-based method is more effective on a single machine, it is less practical in a distributed environment with massive data volumes. For obtaining the final NDV estimators, the entire sample must be transferred throughout the distributed system, incurring a prohibitive communication cost when the sample rate is significant. This paper proposes a novel sketch-based distributed method that achieves sub-linear communication costs for distributed sampling-based NDV estimation under mild assumptions. Our method leverages a sketch-based algorithm to estimate the sample's frequency of frequency in the distributed streaming model, which is compatible with most classical sampling-based NDV estimators. Additionally, we provide theoretical evidence for our method's ability to minimize communication costs in the worst-case scenario. Extensive experiments show that our method saves orders of magnitude in communication costs compared to existing sampling- and sketch-based methods.

Recently, Neural Architecture Search (NAS) for GNN has received increasing popularity as it can seek an optimal architecture for a given new graph. However, the optimal architecture is applied to all the instances (i.e., nodes, in the context of graph) equally, which might be insufficient to handle the diverse local patterns ingrained in a graph, as shown in this paper and some very recent studies. Thus, we argue the necessity of node-wise architecture search for GNN. Nevertheless, node-wise architecture cannot be realized by trivially applying NAS methods node by node due to the scalability issue and the need for determining test nodes' architectures. To tackle these challenges, we propose a framework wherein the parametric controllers decide the GNN architecture for each node based on its local patterns. We instantiate our framework with depth, aggregator and resolution controllers, and then elaborate on learning the backbone GNN model and the controllers to encourage their cooperation. Empirically, we justify the effects of node-wise architecture through the performance improvements introduced by the three controllers, respectively. Moreover, our proposed framework significantly outperforms state-of-the-art methods on five of the ten real-world datasets, where the diversity of these datasets has hindered any graph convolution-based method to lead on them simultaneously. This result further confirms that node-wise architecture can help GNNs become versatile models.

Graph Neural Networks (GNNs) have been widely used for modeling graph-structured data. Recent breakthroughs have been made in improving the scalability of GNNs to work on graphs with millions of nodes. However, how to instantly represent continuous changes of large-scale dynamic graphs with GNNs is still an open problem. Existing dynamic GNNs focus on modeling the periodic evolution of graphs, often on a snapshot basis. Such methods suffer from two drawbacks: first, there is a substantial delay for the changes in the graph to be reflected in the graph representations, resulting in losses on the model's accuracy; second, repeatedly calculating the representation matrix on the entire graph in each snapshot is predominantly time-consuming and severely limits the scalability. In this paper, we propose Instant Graph Neural Network (InstantGNN), an incremental computation approach for the graph representation matrix of dynamic graphs. Set to work with dynamic graphs with the edge-arrival model, our method avoids time-consuming, repetitive computations and allows instant updates on the representation and instant predictions. Graphs with dynamic structures and dynamic attributes are both supported. The upper bounds of time complexity of those updates are also provided. Furthermore, our method provides an adaptive training strategy, which guides the model to retrain at moments when it can make the greatest performance gains. We conduct extensive experiments on several real-world and synthetic datasets. Empirical results demonstrate that our model achieves state-of-the-art accuracy while having orders-of-magnitude higher efficiency than existing methods.

Designing spectral convolutional networks is a challenging problem in graph learning. ChebNet, one of the early attempts, approximates the spectral graph convolutions using Chebyshev polynomials. GCN simplifies ChebNet by utilizing only the first two Chebyshev polynomials while still outperforming it on real-world datasets. GPR-GNN and BernNet demonstrate that the Monomial and Bernstein bases also outperform the Chebyshev basis in terms of learning the spectral graph convolutions. Such conclusions are counter-intuitive in the field of approximation theory, where it is established that the Chebyshev polynomial achieves the optimum convergent rate for approximating a function. In this paper, we revisit the problem of approximating the spectral graph convolutions with Chebyshev polynomials. We show that ChebNet's inferior performance is primarily due to illegal coefficients learnt by ChebNet approximating analytic filter functions, which leads to over-fitting. We then propose ChebNetII, a new GNN model based on Chebyshev interpolation, which enhances the original Chebyshev polynomial approximation while reducing the Runge phenomenon. We conducted an extensive experimental study to demonstrate that ChebNetII can learn arbitrary graph convolutions and achieve superior performance in both full- and semi-supervised node classification tasks. Most notably, we scale ChebNetII to a billion graph ogbn-papers100M, showing that spectral-based GNNs have superior performance. Our code is available at https://github.com/ivam-he/ChebNetII.

Graph Neural Networks (GNNs) have received extensive research attention for their promising performance in graph machine learning. Despite their extraordinary predictive accuracy, existing approaches, such as GCN and GPRGNN, are not robust in the face of homophily changes on test graphs, rendering these models vulnerable to graph structural attacks and with limited capacity in generalizing to graphs of varied homophily levels. Although many methods have been proposed to improve the robustness of GNN models, most of these techniques are restricted to the spatial domain and employ complicated defense mechanisms, such as learning new graph structures or calculating edge attentions. In this paper, we study the problem of designing simple and robust GNN models in the spectral domain. We propose EvenNet, a spectral GNN corresponding to an even-polynomial graph filter. Based on our theoretical analysis in both spatial and spectral domains, we demonstrate that EvenNet outperforms full-order models in generalizing across homophilic and heterophilic graphs, implying that ignoring odd-hop neighbors improves the robustness of GNNs. We conduct experiments on both synthetic and real-world datasets to demonstrate the effectiveness of EvenNet. Notably, EvenNet outperforms existing defense models against structural attacks without introducing additional computational costs and maintains competitiveness in traditional node classification tasks on homophilic and heterophilic graphs.

GPU's powerful computational capacity holds great potentials for processing hierarchically-compressed data without decompression in data science domain. Unfortunately, existing GPU approaches offer only traversal-based data analytics; random access is extremely inefficient, substantially limiting their utility. To solve this problem, we develop a novel and broadly applicable optimization that enables efficient random access to hierarchically-compressed data without decompression in GPU memory. We address three major challenges for enabling efficient random access to compressed data on GPUs. The first challenge is designing GPU data structures that support random access. The second challenge is efficiently generating data structures on GPU. Generating data structures for random access is costly on the CPU, and the inefficiency increases dramatically when PCIe data transmission is incorporated. The third challenge is query processing on compressed data in GPU memory. Random accesses, including data updates, result in significant conflicts between massive threads. To solve the first challenge, we propose and modify a number of compressed data structures, including indexing within the complicated GPU memory hierarchy. To address the second challenge, we develop a two-phase process for generating these data structures on the GPU. To handle the third challenge, we propose a double-parsing design to avoid data conflicts. We evaluate our solution on two GPU platforms using five real-world datasets. Experiments show that the random access operations on GPU can achieve 65.04x average speedup compared to the state-of-the-art method.

Personalized PageRank (PPR) is a popular node proximity metric in graph mining and network research. A single-source PPR (SSPPR) query asks for the PPR value of each node on the graph. Due to its importance and wide applications, decades of efforts have been devoted to the efficient processing of SSPPR queries. Among existing algorithms, LocalPush is a fundamental method for SSPPR queries and serves as a cornerstone for subsequent algorithms. In LocalPush , a push operation is a crucial primitive operation, which distributes the probability at a node u to ALL u 's neighbors via the corresponding edges. Although this push operation works well on unweighted graphs, unfortunately, it can be rather inefficient on weighted graphs. In particular, on unbalanced weighted graphs where only a few of these edges take the majority of the total weight among them, the push operation would have to distribute "insignificant" probabilities along those edges which just take the minor weights, resulting in expensive overhead. To resolve this issue, in this paper, we propose the EdgePush algorithm, a novel method for computing SSPPR queries on weighted graphs. EdgePush decomposes the aforementioned push operations in edge-based push , allowing the algorithm to operate at the edge level granularity. As a result, it can flexibly distribute the probabilities according to edge weights. Furthermore, our EdgePush allows a fine-grained termination threshold for each individual edge, leading to a superior complexity over LocalPush. Notably, we prove that EdgePush improves the theoretical query cost of LocalPush by an order of up to O ( n ) when the graph's weights are unbalanced. Our experimental results demonstrate that EdgePush significantly outperforms state-of-the-art baselines in terms of query efficiency on large motif-based and real-world weighted graphs.

Consider an edge-weighted graph, and a number of properties of interests (PoIs). Each vertex has a probability of exhibiting each PoI. The joint probability that a set of vertices exhibits a PoI is the probability that this set contains at least one vertex that exhibits this PoI. The probabilistic group Steiner tree problem is to find a tree such that (i) for each PoI, the joint probability that the set of vertices in this tree exhibits this PoI is no smaller than a threshold value, e.g. , 0.97; and (ii) the total weight of edges in this tree is the minimum. Solving this problem is useful for mining various graphs with uncertain vertex properties, but is NP-hard. The existing work focuses on certain cases, and cannot perform this task. To meet this challenge, we propose 3 approximation algorithms for solving the above problem. Let |Γ| be the number of PoIs, and ξ be an upper bound of the number of vertices for satisfying the threshold value of exhibiting each PoI. Algorithms 1 and 2 have tight approximation guarantees proportional to |Γ| and ξ, and exponential time complexities with respect to ξ and |Γ|, respectively. In comparison, Algorithm 3 has a looser approximation guarantee proportional to, and a polynomial time complexity with respect to, both |Γ| and ξ. Experiments on real and large datasets show that the proposed algorithms considerably outperform the state-of-the-art related work for finding probabilistic group Steiner trees in various cases.

Real-world graphs exhibit many interesting properties that differentiate them from random graphs, which have been extensively studied for the past decades. For various proposed generative models, a majority of them build the graph by sequentially adding each node and the attached edges. However, the growth of many real-world graphs, such as social networks, is naturally modeled by the sequential insertion of edges. Unfortunately, to the best of our knowledge, no generative model has been proposed to reveal this process. We propose the first sequence-of-edges model, denoted astemporal preferential attachment (TPA). It relies onpreferential attachment (PA), one of the most influential mechanisms to generate scale-free graphs, and takes time-decay effect and node fitness into consideration. Empirical analysis demonstrates that our model preserves several key properties of the real-world graphs, including both the properties observed from the snapshot graphs (e.g., power-law distribution) and temporal properties observed from the graph generation process (e.g., shrinking diameter). Meanwhile, our model is sufficiently general to accommodate several forms of time decay and fitness distributions. Then, we design two efficient algorithms that generate TPA graphs with billions of edges in several minutes.

Given a social network G with n nodes and m edges, a positive integer k , and a cascade model C , the influence maximization (IM) problem asks for k nodes in G such that the expected number of nodes influenced by the k nodes under cascade model C is maximized. The state-of-the-art approximate solutions run in O(k(n+m) log n/ ε 2 ) expected time while returning a (1 - 1/ e - ε) approximate solution with at least 1 - 1/ n probability. A key phase of these IM algorithms is the random reverse reachable (RR) set generation, and this phase significantly affects the efficiency and scalability of the state-of-the-art IM algorithms. In this article, we present a study on this key phase and propose an efficient random RR set generation algorithm under IC model. With the new algorithm, we show that the expected running time of existing IM algorithms under IC model can be improved to O(k ċ n log n ċ 2 ), when for any node v , the total weight of its incoming edges is no larger than a constant. For the general IC model where the weights are skewed, we present a sampling algorithm SKIP. To the best of our knowledge, it is the first index-free algorithm that achieves the optimal time complexity of the sorted subset sampling problem. Moreover, existing approximate IM algorithms suffer from scalability issues in high influence networks where the size of random RR sets is usually quite large. We tackle this challenging issue by reducing the average size of random RR sets without sacrificing the approximation guarantee. The proposed solution is orders of magnitude faster than states of the art as shown in our experiment. Besides, we investigate the issues of forward propagation and derive its time complexity with our proposed subset sampling techniques. We also present a heuristic condition to indicate when the forward propagation approach should be utilized to estimate the expected influence of a given seed set.

A persistent data structure , also known as a multiversion data structure in the database literature, is a data structure that preserves all its previous versions as it is updated over time. Every update (inserting, deleting, or changing a data record) to the data structure creates a new version, while all the versions are kept in the data structure so that any previous version can still be queried. Persistent data structures aim at recording all versions accurately, which results in a space requirement that is at least linear to the number of updates. In many of today’s big data applications, in particular, for high-speed streaming data, the volume and velocity of the data are so high that we cannot afford to store everything. Therefore, streaming algorithms have received a lot of attention in the research community, which uses only sublinear space by sacrificing slightly on accuracy. All streaming algorithms work by maintaining a small data structure in memory, which is usually called a sketch , summary , or synopsis . The summary is updated upon the arrival of every element in the stream, thus it is ephemeral , meaning that it can only answer queries about the current status of the stream. In this article, we aim at designing persistent summaries, thereby giving streaming algorithms the ability to answer queries about the stream at any prior time.

Despite the recent success of graph neural networks (GNN), common architectures often exhibit significant limitations, including sensitivity to oversmoothing, long-range dependencies, and spurious edges, e.g., as can occur as a result of graph heterophily or adversarial attacks. To at least partially address these issues within a simple transparent framework, we consider a new family of GNN layers designed to mimic and integrate the update rules of two classical iterative algorithms, namely, proximal gradient descent and iterative reweighted least squares (IRLS). The former defines an extensible base GNN architecture that is immune to oversmoothing while nonetheless capturing long-range dependencies by allowing arbitrary propagation steps. In contrast, the latter produces a novel attention mechanism that is explicitly anchored to an underlying end-to-end energy function, contributing stability with respect to edge uncertainty. When combined we obtain an extremely simple yet robust model that we evaluate across disparate scenarios including standardized benchmarks, adversarially-perturbated graphs, graphs with heterophily, and graphs involving long-range dependencies. In doing so, we compare against SOTA GNN approaches that have been explicitly designed for the respective task, achieving competitive or superior node classification accuracy. Our code is available at https://github.com/FFTYYY/TWIRLS. And for an extended version of this work, please see https://arxiv.org/abs/2103.06064.

Efficient computation of node proximity queries such as transition probabilities, Personalized PageRank, and Katz are of fundamental importance in various graph mining and learning tasks. In particular, several recent works leverage fast node proximity computation to improve the scalability of Graph Neural Networks (GNN). However, prior studies on proximity computation and GNN feature propagation are on a case-by-case basis, with each paper focusing on a particular proximity measure.

Many representative graph neural networks, $e.g.$, GPR-GNN and ChebNet, approximate graph convolutions with graph spectral filters. However, existing work either applies predefined filter weights or learns them without necessary constraints, which may lead to oversimplified or ill-posed filters. To overcome these issues, we propose $\textit{BernNet}$, a novel graph neural network with theoretical support that provides a simple but effective scheme for designing and learning arbitrary graph spectral filters. In particular, for any filter over the normalized Laplacian spectrum of a graph, our BernNet estimates it by an order-$K$ Bernstein polynomial approximation and designs its spectral property by setting the coefficients of the Bernstein basis. Moreover, we can learn the coefficients (and the corresponding filter weights) based on observed graphs and their associated signals and thus achieve the BernNet specialized for the data. Our experiments demonstrate that BernNet can learn arbitrary spectral filters, including complicated band-rejection and comb filters, and it achieves superior performance in real-world graph modeling tasks. Code is available at https://github.com/ivam-he/BernNet.

Personalized PageRank (PPR) is a critical measure of the importance of a node t to a source node s in a graph. The Single-Source PPR (SSPPR) query computes the PPR's of all the nodes with respect to s on a directed graph G with n nodes and m edges; and it is an essential operation widely used in graph applications. In this paper, we propose novel algorithms for answering two variants of SSPPR queries: (i) high-precision queries and (ii) approximate queries.

Personalized PageRank (PPR) has wide applications in search engines, social recommendations, community detection, and so on. Nowadays, graphs are becoming massive and many IT companies need to deal with large graphs that cannot be fitted into the memory of most commodity servers. However, most existing state-of-the-art solutions for PPR computation only work for single-machines and are inefficient for the distributed framework since such solutions either (i) result in an excessively large number of communication rounds, or (ii) incur high communication costs in each round. Motivated by this, we present Delta-Push , an efficient framework for single-source and top- k PPR queries in distributed settings. Our goal is to reduce the number of rounds while guaranteeing that the load, i.e., the maximum number of messages an executor sends or receives in a round, can be bounded by the capacity of each executor. We first present a non-trivial combination of a redesigned parallel push algorithm and the Monte-Carlo method to answer single-source PPR queries. The solution uses pre-sampled random walks to reduce the number of rounds for the push al6gorithm. Theoretical analysis under the Massively Parallel Computing (MPC) model shows that our proposed solution bounds the communication rounds to [EQUATION] under a load of O ( m/p ), where m is the number of edges of the input graph, p is the number of executors, and ϵ is a user-defined error parameter. In the meantime, as the number of executors increases to p' = γ · p , the load constraint can be relaxed since each executor can hold O (γ · m/p' ) messages with invariant local memory. In such scenarios, multiple queries can be processed in batches simultaneously. We show that with a load of O (γ · m/p' ), our Delta-Push can process γ queries in a batch with [EQUATION] rounds, while other baseline solutions still keep the same round cost for each batch. We further present a new top- k algorithm that is friendly to the distributed framework and reduces the number of rounds required in practice. Extensive experiments show that our proposed solution is more efficient than alternatives.

Estimating the number of distinct values (NDV) in a column is useful for many tasks in database systems, such as columnstore compression and data profiling. In this work, we focus on how to derive accurate NDV estimations from random (online/offline) samples. Such efficient estimation is critical for tasks where it is prohibitive to scan the data even once. Existing sample-based estimators typically rely on heuristics or assumptions and do not have robust performance across different datasets as the assumptions on data can easily break. On the other hand, deriving an estimator from a principled formulation such as maximum likelihood estimation is very challenging due to the complex structure of the formulation. We propose to formulate the NDV estimation task in a supervised learning framework, and aim to learn a model as the estimator. To this end, we need to answer several questions: i) how to make the learned model workload agnostic; ii) how to obtain training data; iii) how to perform model training. We derive conditions of the learning framework under which the learned model isworkload agnostic, in the sense that the model/estimator can be trained with synthetically generated training data, and then deployed into any data warehouse simply as,e.g., user-defined functions (UDFs), to offer efficient (within microseconds on CPU) and accurate NDV estimations forunseen tables and workloads.We compare the learned estimator with the state-of-the-art sample-based estimators on nine real-world datasets to demonstrate its superior estimation accuracy. We publish our code for training data generation, model training, and the learned estimator online for reproducibility.

Interactive response time is important in analytical pipelines for users to explore a sufficient number of possibilities and make informed business decisions. We consider a forecasting pipeline with large volumes of high-dimensional time series data. Real-time forecasting can be conducted in two steps. First, we specify the part of data to be focused on and the measure to be predicted by slicing, dicing, and aggregating the data. Second, a forecasting model is trained on the aggregated results to predict the trend of the specified measure. While there are a number of forecasting models available, the first step is the performance bottleneck. A natural idea is to utilize sampling to obtain approximate aggregations in real time as the input to train the forecasting model. Our scalable real-time forecasting system FlashP (Flash Prediction) is built based on this idea, with two major challenges to be resolved in this paper: first, we need to figure out how approximate aggregations affect the fitting of forecasting models, and forecasting results; and second, accordingly, what sampling algorithms we should use to obtain these approximate aggregations and how large the samples are. We introduce a new sampling scheme, called GSW sampling, and analyze error bounds for estimating aggregations using GSW samples. We introduce how to construct compact GSW samples with the existence of multiple measures to be analyzed. We conduct experiments to evaluate our solution its alternatives on real data.

Graph convolutional networks (GCNs) are a powerful deep learning approach for graph-structured data. Recently, GCNs and subsequent variants have shown superior performance in various application areas on real-world datasets. Despite their success, most of the current GCN models are shallow, due to the \emph{over-smoothing} problem. In this paper, we study the problem of designing and analyzing deep graph convolutional networks. We propose the GCNII, an extension of the vanilla GCN model with two simple yet effective techniques: \emph{Initial residual} and \emph{Identity mapping}. We provide theoretical and empirical evidence that the two techniques effectively relieves the problem of over-smoothing. Our experiments show that the deep GCNII model outperforms the state-of-the-art methods on various semi- and full-supervised tasks.

Personalized PageRank (PPR) is a widely used node proximity measure in graph mining and network analysis. Given a source node s and a target node t, the PPR value π(s,t) represents the probability that a random walk from s terminates at t, and thus indicates the bidirectional importance between s and t. The majority of the existing work focuses on the single-source queries, which asks for the PPR value of a given source node s and every node t ∈ V. However, the single-source query only reflects the importance of each node t with respect to s. In this paper, we consider the single-target PPR query, which measures the opposite direction of importance for PPR. Given a target node t, the single-target PPR query asks for the PPR value of every node $s\in V$ to a given target node t. We propose RBS, a novel algorithm that answers approximate single-target queries with optimal computational complexity. We show that RBS improves three concrete applications: heavy hitters PPR query, single-source SimRank computation, and scalable graph neural networks. We conduct experiments to demonstrate that RBS outperforms the state-of-the-art algorithms in terms of both efficiency and precision on real-world benchmark datasets.

Graph Neural Networks (GNN) are an emerging field for learning on non-Euclidean data. Recently, there has been increased interest in designing GNN that scales to large graphs. Most existing methods use "graph sampling" or "layer-wise sampling" techniques to reduce training time; However, these methods still suffer from degrading performance and scalability problems when applying to graphs with billions of edges. In this paper, we present GBP, a scalable GNN that utilizes a localized bidirectional propagation process from both the feature vector and the training/testing nodes. Theoretical analysis shows that GBP is the first method that achieves sub-linear time complexity for both the precomputation and the training phases. An extensive empirical study demonstrates that GBP achieves state-of-the-art performance with significantly less training/testing time. Most notably, GBP is able to deliver superior performance on a graph with over 60 million nodes and 1.8 billion edges in less than 2,000 seconds on a single machine.

Given a social network G with n nodes and m edges, a positive integer k, and a cascade model C, the influence maximization (IM) problem asks for k nodes in G such that the expected number of nodes influenced by the k nodes under cascade model C is maximized. The state-of-the-art approximate solutions run in O(k(n+m)log(n)/ε2) expected time while returning a (1-1/e -ε) approximate solution with at least 1-1/n probability. A key phase of these IM algorithms is the random reverse reachable (RR) set generation, and this phase significantly affects the efficiency and scalability of the state-of-the-art IM algorithms. In this paper, we present a study on this key phase and propose an efficient random RR set generation algorithm under IC model. With the new algorithm, we show that the expected running time of existing IM algorithms under IC model can be improved to O(k· n log(n)/ε2), when for any node v, the total weight of its incoming edges is no larger than a constant. Moreover, existing approximate IM algorithms suffer from scalability issues in high influence networks where the size of random RR sets is usually quite large. We tackle this challenging issue by reducing the average size of random RR sets without sacrificing the approximation guarantee. The proposed solution is orders of magnitude faster than states of the art as shown in our experiment.

SimRank is a popular measurement for evaluating the node-to-node similarities based on the graph topology. In recent years, single-source and top-k SimRank queries have received increasing attention due to their applications in web mining, social network analysis, and spam detection. However, a fundamental obstacle in studying SimRank has been the lack of ground truths. The only exact algorithm, Power Method, is computationally infeasible on graphs with more than 106 nodes. Consequently, no existing work has evaluated the actual trade-offs between query time and accuracy on large real-world graphs. In this paper, we present ExSim, the first algorithm that computes the exact single-source and top-k SimRank results on large graphs. With high probability, this algorithm produces ground truths with a rigorous theoretical guarantee. We conduct extensive experiments on real-world datasets to demonstrate the efficiency of ExactSim. The results show that ExactSim provides the ground truth for any single-source SimRank query with a precision up to 7 decimal places within a reasonable query time.

Graph embedding learns low-dimensional representations for nodes in a graph and effectively preserves the graph structure. Recently, a significant amount of progress has been made toward this emerging research area. However, there are several fundamental problems that remain open. First, existing methods fail to preserve the out-degree distributions on directed graphs. Second, many existing methods employ random walk based proximities and thus suffer from conflicting optimization goals on undirected graphs. Finally, existing factorization methods are unable to achieve scalability and non-linearity simultaneously. This paper presents an in-depth study on graph embedding techniques on both directed and undirected graphs. We analyze the fundamental reasons that lead to the distortion of out-degree distributions and to the conflicting optimization goals. We propose transpose proximity, a unified approach that solves both problems. Based on the concept of transpose proximity, we design STRAP, a factorization based graph embedding algorithm that achieves scalability and non-linearity simultaneously. STRAP makes use of the backward push algorithm to efficiently compute the sparse Personalized PageRank (PPR) as its transpose proximities. By imposing the sparsity constraint, we are able to apply non-linear operations to the proximity matrix and perform efficient matrix factorization to derive the embedding vectors. Finally, we present an extensive experimental study that evaluates the effectiveness of various graph embedding algorithms, and we show that \strap outperforms the state-of-the-art methods in terms of effectiveness and scalability.

Large-scale data labeling has become a major bottleneck for many applications, such as machine learning and data integration. This paper presents CrowdGame, a crowdsourcing system that harnesses the crowd to gather data labels in a cost-effective way. CrowdGame focuses on generating high-quality labeling rules to largely reduce the labeling cost while preserving quality. It first generates candidate rules, and then devises a game-based crowdsourcing approach to select rules with high coverage and accuracy. CrowdGame applies the generated rules for effective data labeling. We have implemented CrowdGame and provided a user-friendly interface for users to deploy their labeling applications. We will demonstrate CrowdGame in two representative data labeling scenarios, entity matching and relation extraction.

SimRank is a classic measure of the similarities of nodes in a graph. Given a node u in graph $G =(V, E)$, a \em single-source SimRank query returns the SimRank similarities $s(u, v)$ between node u and each node $v \in V$. This type of queries has numerous applications in web search and social networks analysis, such as link prediction, web mining, and spam detection. Existing methods for single-source SimRank queries, however, incur query cost at least linear to the number of nodes n, which renders them inapplicable for real-time and interactive analysis. This paper proposes \prsim, an algorithm that exploits the structure of graphs to efficiently answer single-source SimRank queries. \prsim uses an index of size $O(m)$, where m is the number of edges in the graph, and guarantees a query time that depends on the \em reverse PageRank distribution of the input graph. In particular, we prove that \prsim runs in sub-linear time if the degree distribution of the input graph follows the power-law distribution, a property possessed by many real-world graphs. Based on the theoretical analysis, we show that the empirical query time of all existing SimRank algorithms also depends on the reverse PageRank distribution of the graph. Finally, we present the first experimental study that evaluates the absolute errors of various SimRank algorithms on large graphs, and we show that \prsim outperforms the state of the art in terms of query time, accuracy, index size, and scalability.

Given an undirected graph G and a seed node s, the local clustering problem aims to identify a high-quality cluster containing s in time roughly proportional to the size of the cluster, regardless of the size of G. This problem finds numerous applications on large-scale graphs. Recently, heat kernel PageRank (HKPR), which is a measure of the proximity of nodes in graphs, is applied to this problem and found to be more efficient compared with prior methods. However, existing solutions for computing HKPR either are prohibitively expensive or provide unsatisfactory error approximation on HKPR values, rendering them impractical especially on billion-edge graphs. In this paper, we present TEA and TEA+, two novel local graph clustering algorithms based on HKPR, to address the aforementioned limitations. Specifically, these algorithms provide non-trivial theoretical guarantees in relative error of HKPR values and the time complexity. The basic idea is to utilize deterministic graph traversal to produce a rough estimation of exact HKPR vector, and then exploit Monte-Carlo random walks to refine the results in an optimized and non-trivial way. In particular, TEA+ offers practical efficiency and effectiveness due to non-trivial optimizations. Extensive experiments on real-world datasets demonstrate that TEA+ outperforms the state-of-the-art algorithm by more than four times on most benchmark datasets in terms of computational time when achieving the same clustering quality, and in particular, is an order of magnitude faster on large graphs including the widely studied Twitter and Friendster datasets.

The problem of crowdsourced entity collection solicits people (a.k.a. workers) to complete missing data in a database and has witnessed many applications in knowledge base completion and enterprise data collection. Although previous studies have attempted to address the “open world” challenge of crowdsourced entity collection, they do not pay much attention to the “distribution” of the collected entities. Evidently, in many real applications, users may have distribution requirements on the collected entities, e.g., even spatial distribution when collecting points-of-interest. In this paper, we study a new research problem, distribution-aware crowdsourced entity collection (CrowdDEC): Given an expected distribution w.r.t. an attribute (e.g., region or year), it aims to collect a set of entities via crowdsourcing and minimize the difference of the entity distribution from the expected distribution. Due to the openness of crowdsourcing, the CrowdDEC problem calls for effective crowdsourcing quality control. We propose an adaptive worker selection approach to address this problem. The approach estimates underlying entity distribution of workers on-the-fly based on the collected entities. Then, it adaptively selects the best set of workers that minimizes the difference from the expected distribution. Once workers submit their answers, it adjusts the estimation of workers' underlying distributions for subsequent adaptive worker selections. We prove the hardness of the problem, and develop effective estimation techniques as well as efficient worker selection algorithms to support this approach. We deployed the proposed approach on Amazon Mechanical Turk and the experimental results on two real datasets show that the approach achieves superiority on both effectiveness and efficiency.

The matching between trajectories and locations, called Trajectory-to-Location join (TL-Join), is fundamental functionality in spatiotemporal data management. Given a set of trajectories, a set of locations, and a threshold 8, the TL-Join finds all (trajectory, location) pairs from the two sets with spatiotemporal correlation above 8. This join targets diverse applications, including location recommendation, event tracking, and trajectory activity analyses. We address three challenges in relation to the TL-Join: how to define the spatiotemporal correlation between trajectories and locations, how to prune the search space effectively when computing the join, and how to perform the computation in parallel. Specifically, we define new metrics to measure the spatiotemporal correlation between trajectories and locations. We develop a novel parallel collaborative (PCol) search method based on a divide-and-conquer strategy. For each location o, we retrieve the trajectories with high spatiotemporal correlation to o, and then we merge the results. An upper bound on the spatiotemporal correlation and a heuristic scheduling strategy are developed to prune the search space. The trajectory searches from different locations are independent and are performed in parallel, and the result merging cost is independent of the degree of parallelism. Studies of the performance of the developed algorithms using large spatiotemporal data sets are reported.

Given a graph G , a source node s, and a target node t , the personalized PageRank ( PPR ) of t with respect to s is the probability that a random walk starting from s terminates at t . An important variant of the PPR query is single-source PPR ( SSPPR ), which enumerates all nodes in G and returns the top- k nodes with the highest PPR values with respect to a given source s . PPR in general and SSPPR in particular have important applications in web search and social networks, e.g., in Twitter’s Who-To-Follow recommendation service. However, PPR computation is known to be expensive on large graphs and resistant to indexing. Consequently, previous solutions either use heuristics, which do not guarantee result quality, or rely on the strong computing power of modern data centers, which is costly. Motivated by this, we propose effective index-free and index-based algorithms for approximate PPR processing, with rigorous guarantees on result quality. We first present FORA, an approximate SSPPR solution that combines two existing methods—Forward Push (which is fast but does not guarantee quality) and Monte Carlo Random Walk (accurate but slow)—in a simple and yet non-trivial way, leading to both high accuracy and efficiency. Further, FORA includes a simple and effective indexing scheme, as well as a module for top- k selection with high pruning power. Extensive experiments demonstrate that the proposed solutions are orders of magnitude more efficient than their respective competitors. Notably, on a billion-edge Twitter dataset, FORA answers a top-500 approximate SSPPR query within 1s, using a single commodity server.

Personalized PageRank (PPR) is a classic metric that measures the relevance of graph nodes with respect to a source node. Given a graph G, a source node s, and a parameter k, a top-k PPR query returns a set of k nodes with the highest PPR values with respect to s. This type of queries serves as an important building block for numerous applications in web search and social networks, such as Twitter's Who-To-Follow recommendation service. Existing techniques for top-k PPR, however, suffer from two major deficiencies. First, they either incur prohibitive space and time overheads on large graphs, or fail to provide any guarantee on the precision of top-k results (i.e., the results returned might miss a number of actual top-k answers). Second, most of them require significant pre-computation on the input graph G, which renders them unsuitable for graphs with frequent updates (e.g., Twitter's social graph).

Large-scale data annotation is indispensable for many applications, such as machine learning and data integration. However, existing annotation solutions either incur expensive cost for large datasets or produce noisy results. This paper introduces a cost-effective annotation approach, and focuses on the labeling rule generation problem that aims to generate high-quality rules to largely reduce the labeling cost while preserving quality. To address the problem, we first generate candidate rules, and then devise a game-based crowdsourcing approach C ROWD G AME to select high-quality rules by considering coverage and precision. C ROWD G AME employs two groups of crowd workers: one group answers rule validation tasks (whether a rule is valid) to play a role of rule generator, while the other group answers tuple checking tasks (whether the annotated label of a data tuple is correct) to play a role of rule refuter. We let the two groups play a two-player game: rule generator identifies high-quality rules with large coverage and precision, while rule refuter tries to refute its opponent rule generator by checking some tuples that provide enough evidence to reject rules covering the tuples. This paper studies the challenges in C ROWD G AME . The first is to balance the trade-off between coverage and precision. We define the loss of a rule by considering the two factors. The second is rule precision estimation. We utilize Bayesian estimation to combine both rule validation and tuple checking tasks. The third is to select crowdsourcing tasks to fulfill the game-based framework for minimizing the loss. We introduce a minimax strategy and develop efficient task selection algorithms. We conduct experiments on entity matching and relation extraction, and the results show that our method outperforms state-of-the-art solutions.

We study the problem of two-dimensional orthogonal range counting with additive error. Given a set P of n points drawn from an n × n grid and an error parameter ε, the goal is to build a data structure, such that for any orthogonal range R , it can return the number of points in P ∩ R with additive error ε n . A well-known solution for this problem is obtained by using ε-approximation , which is a subset A ⊆ P that can estimate the number of points in P ∩ R with the number of points in A ∩ R . It is known that an ε-approximation of size O (1/ε log 2.5 1/ε) exists for any P with respect to orthogonal ranges, and the best lower bound is Ω(1/ε log 1/ε). The ε-approximation is a rather restricted data structure, as we are not allowed to store any information other than the coordinates of the points. In this article, we explore what can be achieved without any restriction on the data structure. We first describe a simple data structure that uses O (1/ε(log 2 1/ε + log n )) bits and answers queries with error ε n . We then prove a lower bound that any data structure that answers queries with error ε n will have to use Ω(1/ε (log 2 1/ε + log n )) bits. Our lower bound is information-theoretic: We show that there is a collection of 2 Ω ( n log n ) point sets with large union combinatorial discrepancy and thus are hard to distinguish unless we use Ω( n log n ) bits.

In the independent range sampling (IRS) problem, given an input set P of n points in R^d, the task is to build a data structure, such that given a range R and an integer t >= 1, it returns t points that are uniformly and independently drawn from P cap R. The samples must satisfy inter-query independence, that is, the samples returned by every query must be independent of the samples returned by all the previous queries. This problem was first tackled by Hu, Qiao and Tao in 2014, who proposed optimal structures for one-dimensional dynamic IRS problem in internal memory and one-dimensional static IRS problem in external memory. In this paper, we study two natural extensions of the independent range sampling problem. In the first extension, we consider the static IRS problem in two and three dimensions in internal memory. We obtain data structures with optimal space-query tradeoffs for 3D halfspace, 3D dominance, and 2D three-sided queries. The second extension considers weighted IRS problem. Each point is associated with a real-valued weight, and given a query range R, a sample is drawn independently such that each point in P cap R is selected with probability proportional to its weight. Walker's alias method is a classic solution to this problem when no query range is specified. We obtain optimal data structure for one dimensional weighted range sampling problem, thereby extending the alias method to allow range queries.

We propose and investigate a novel query, the Collective Travel Planning (CTP) query, that finds the lowest-cost route connecting multiple query sources and a destination via at most k meeting points. This type of query is useful in organizing large events, and it can bring significant benefits to society and the environment: it can help optimize the allocation of transportation resources, reduce resource consumption, and enable smarter and greener transportation; and it can help reduce greenhouse-gas emissions and traffic congestion.

In many modern applications, input data is represented as matrices and often arrives continuously. The ability to summarize and approximate data matrices in streaming fashion has become a common requirement in many emerging environments. In these applications, input data is usually generated at multiple distributed sites and simply centralizing all data is often infeasible. Therefore, novel algorithmic techniques are required. Furthermore, in most of these applications, queries must be answered solely based on the recently observed data points (e.g., data collected over the last hour/day/month), which makes the problem even more challenging. In this paper, we propose to study the problem of tracking matrix approximations over distributed sliding windows. In this problem, there are m distributed sites each observing a stream of d-dimensional data points. The goal is to continuously track a small matrix B as an approximation to Aw, the matrix consists of data points in the union of the streams which arrived during the last W time units. The quality of the approximation is measured by the covariance error kAT wAw? BTBk = kAk2 F [1], and the primary goal is to minimize communication, while providing provable error guarantee. We propose novel communication-efficient algorithms for this problem. Our sampling-based algorithms continuously track a weighted sample of rows according to their squared norms, which generalize and simplify the sampling techniques in [2]. We also propose deterministic tracking algorithms that require only one-way communication and provide better error guarantee. All algorithms have provable guarantees, and extensive experimental studies on real and synthetic datasets validate our theoretical claims and demonstrate the efficiency of these algorithms.

Given a graph G, a source node s and a target node t, the personalized PageRank (PPR) of t with respect to s is the probability that a random walk starting from s terminates at t. A single-source PPR (SSPPR) query enumerates all nodes in G, and returns the top-k nodes with the highest PPR values with respect to a given source node s. SSPPR has important applications in web search and social networks, e.g., in Twitter's Who-To-Follow recommendation service. However, SSPPR computation is immensely expensive, and at the same time resistant to indexing and materialization. So far, existing solutions either use heuristics, which do not guarantee result quality, or rely on the strong computing power of modern data centers, which is costly.

Single-source and top- k SimRank queries are two important types of similarity search in graphs with numerous applications in web mining, social network analysis, spam detection, etc. A plethora of techniques have been proposed for these two types of queries, but very few can efficiently support similarity search over large dynamic graphs, due to either significant preprocessing time or large space overheads. This paper presents ProbeSim , an index-free algorithm for single-source and top- k SimRank queries that provides a non-trivial theoretical guarantee in the absolute error of query results. ProbeSim estimates SimRank similarities without precomputing any indexing structures, and thus can naturally support real-time SimRank queries on dynamic graphs. Besides the theoretical guarantee, ProbeSim also offers satisfying practical efficiency and effectiveness due to non-trivial optimizations. We conduct extensive experiments on a number of benchmark datasets, which demonstrate that our solutions outperform the existing methods in terms of efficiency and effectiveness. Notably, our experiments include the first empirical study that evaluates the effectiveness of SimRank algorithms on graphs with billion edges, using the idea of pooling.

The matching of similar pairs of objects, called similarity join, is fundamental functionality in data management. We consider the case of trajectory similarity join (TS-Join), where the objects are trajectories of vehicles moving in road networks. Thus, given two sets of trajectories and a threshold θ , the TS-Join returns all pairs of trajectories from the two sets with similarity above θ . This join targets applications such as trajectory near-duplicate detection, data cleaning, ridesharing recommendation, and traffic congestion prediction. With these applications in mind, we provide a purposeful definition of similarity. To enable efficient TS-Join processing on large sets of trajectories, we develop search space pruning techniques and take into account the parallel processing capabilities of modern processors. Specifically, we present a two-phase divide-and-conquer algorithm. For each trajectory, the algorithm first finds similar trajectories. Then it merges the results to achieve a final result. The algorithm exploits an upper bound on the spatiotemporal similarity and a heuristic scheduling strategy for search space pruning. The algorithm's per-trajectory searches are independent of each other and can be performed in parallel, and the merging has constant cost. An empirical study with real data offers insight in the performance of the algorithm and demonstrates that is capable of outperforming a well-designed baseline algorithm by an order of magnitude.

Large-scale matrix computation becomes essential for many data data applications, and hence the problem of sketching matrix with small space and high precision has received extensive study for the past few years. This problem is often considered in the row-update streaming model, where the data set is a matrix A -- Rn x d, and the processor receives a row (1 x d) of A at each timestamp. The goal is to maintain a smaller matrix (termed approximation matrix, or simply approximation) B -- Rl x d as an approximation to A, such that the covariance error |AT A - BTB| is small and l ll n.

Travel planning and recommendation are important aspects of transportation. We propose and investigate a novel Collective Travel Planning (CTP) query that finds the lowest-cost route connecting multiple sources and a destination, via at most$k$meeting points. When multiple travelers target the same destination (e.g., a stadium or a theater), they may want to assemble at meeting points and then go together to the destination by public transport to reduce their global travel cost (e.g., energy, money, or greenhouse-gas emissions). This type of functionality holds the potential to bring significant benefits to society and the environment, such as reducing energy consumption and greenhouse-gas emissions, enabling smarter and greener transportation, and reducing traffic congestions. The CTP query is Max SNP-hard. To compute the query efficiently, we develop two algorithms, including an exact algorithm and an approximation algorithm. The exact algorithm is capable finding the optimal result for small values of$k$(e.g.,$k = 2$) in interactive time, while the approximation algorithm, which has a$5$-approximation ratio, is suitable for other situations. The performance of the CTP query is studied experimentally with real and synthetic spatial data.

A persistent data structure, also known as a multiversion data structure in the database literature, is a data structure that preserves all its previous versions as it is updated over time. Every update (inserting, deleting, or changing a data record) to the data structure creates a new version, while all the versions are kept in the data structure so that any previous version can still be queried.

Maximum independent set (MIS) is a fundamental problem in graph theory and it has important applications in many areas such as social network analysis, graphical information systems and coding theory. The problem is NP-hard, and there has been numerous studies on its approximate solutions. While successful to a certain degree, the existing methods require memory space at least linear in the size of the input graph. This has become a serious concern in view of the massive volume of today's fast-growing graphs. In this paper, we study the MIS problem under the semi-external setting, which assumes that the main memory can accommodate all vertices of the graph but not all edges. We present a greedy algorithm and a general vertex-swap framework, which swaps vertices to incrementally increase the size of independent sets. Our solutions require only few sequential scans of graphs on the disk file, thus enabling in-memory computation without costly random disk accesses. Experiments on large-scale datasets show that our solutions are able to compute a large independent set for a massive graph with 59 million vertices and 151 million edges using a commodity machine, with a memory cost of 469MB and a time cost of three minutes, while yielding an approximation ratio that is around 99% of the theoretical optimum.

Database queries can be broadly classified into two categories: reporting queries and aggregation queries. The former retrieves a collection of records from the database that match the query's conditions, while the latter returns an aggregate, such as count, sum, average, or max (min), of a particular attribute of these records. Aggregation queries are especially useful in business intelligence and data analysis applications where users are interested not in the actual records, but some statistics of them. They can also be executed much more efficiently than reporting queries, by embedding properly precomputed aggregates into an index. However, reporting and aggregation queries provide only two extremes for exploring the data. Data analysts often need more insight into the data distribution than what those simple aggregates provide, and yet certainly do not want the sheer volume of data returned by reporting queries. In this article, we design indexing techniques that allow for extracting a statistical summary of all the records in the query. The summaries we support include frequent items, quantiles, and various sketches, all of which are of central importance in massive data analysis. Our indexes require linear space and extract a summary with the optimal or near-optimal query cost. We illustrate the efficiency and usefulness of our designs through extensive experiments and a system demonstration.

We study the problem of 2-dimensional orthogonal range counting with additive error. Given a set P of n points drawn from an n × n grid and an error parameter ε, the goal is to build a data structure, such that for any orthogonal range R, the data structure can return the number of points in P ∩ R with additive error εn. A well-known solution for this problem is the ε-approximation. Informally speaking, an ε-approximation of P is a subset A ⊆ P that allows us to estimate the number of points in P ∩ R by counting the number of points in A ∩ R. It is known that an ε-approximation of size exists for any P with respect to orthogonal ranges, and the best lower bound is . The ε-approximation is a rather restricted data structure, as we are not allowed to store any information other than the coordinates of a subset of points in P. In this paper, we explore what can be achieved without any restriction on the data structure. We first describe a data structure that uses bits that answers queries with error εn. We then prove a lower bound that any data structure that answers queries with error O(log n) must use Ω(n log n) bits. This lower bound has two consequences: 1) answering queries with error O(log n) is as hard as answering the queries exactly; and 2) our upper bound cannot be improved in general by more than an factor.

We study the mergeability of data summaries. Informally speaking, mergeability requires that, given two summaries on two datasets, there is a way to merge the two summaries into a single summary on the two datasets combined together, while preserving the error and size guarantees. This property means that the summaries can be merged in a way akin to other algebraic operators such as sum and max, which is especially useful for computing summaries on massive distributed data. Several data summaries are trivially mergeable by construction, most notably all the sketches that are linear functions of the datasets. But some other fundamental ones, like those for heavy hitters and quantiles, are not (known to be) mergeable. In this article, we demonstrate that these summaries are indeed mergeable or can be made mergeable after appropriate modifications. Specifically, we show that for ε-approximate heavy hitters, there is a deterministic mergeable summary of size O (1/ε); for ε-approximate quantiles, there is a deterministic summary of size O ((1/ε) log(ε n )) that has a restricted form of mergeability, and a randomized one of size O ((1/ε) log 3/2 (1/ε)) with full mergeability. We also extend our results to geometric summaries such as ε-approximations which permit approximate multidimensional range counting queries. While most of the results in this article are theoretical in nature, some of the algorithms are actually very simple and even perform better than the previously best known algorithms, which we demonstrate through experiments in a simulated sensor network. We also achieve two results of independent interest: (1) we provide the best known randomized streaming bound for ε-approximate quantiles that depends only on ε, of size O ((1/ε) log 3/2 (1/ε)), and (2) we demonstrate that the MG and the SpaceSaving summaries for heavy hitters are isomorphic.

We study the mergeability of data summaries. Informally speaking, mergeability requires that, given two summaries on two data sets, there is a way to merge the two summaries into a single summary on the union of the two data sets, while preserving the error and size guarantees. This property means that the summaries can be merged in a way like other algebraic operators such as sum and max, which is especially useful for computing summaries on massive distributed data. Several data summaries are trivially mergeable by construction, most notably all the sketches that are linear functions of the data sets. But some other fundamental ones like those for heavy hitters and quantiles, are not (known to be) mergeable. In this paper, we demonstrate that these summaries are indeed mergeable or can be made mergeable after appropriate modifications. Specifically, we show that for ε-approximate heavy hitters, there is a deterministic mergeable summary of size O(1/ε) for ε-approximate quantiles, there is a deterministic summary of size O(1 over ε log(εn))that has a restricted form of mergeability, and a randomized one of size O(1 over ε log 3/21 over ε) with full mergeability. We also extend our results to geometric summaries such as ε-approximations and εkernels.

Database queries can be broadly classified into two categories: reporting queries and aggregation queries. The former retrieves a collection of records from the database that match the query's conditions, while the latter returns an aggregate, such as count, sum, average, or max (min), of a particular attribute of these records. Aggregation queries are especially useful in business intelligence and data analysis applications where users are interested not in the actual records, but some statistics of them. They can also be executed much more efficiently than reporting queries, by embedding properly precomputed aggregates into an index. However, reporting and aggregation queries provide only two extremes for exploring the data. Data analysts often need more insight into the data distribution than what those simple aggregates provide, and yet certainly do not want the sheer volume of data returned by reporting queries. In this paper, we design indexing techniques that allow for extracting a statistical summary of all the records in the query. The summaries we support include frequent items, quantiles, various sketches, and wavelets, all of which are of central importance in massive data analysis. Our indexes require linear space and extract a summary with the optimal or near-optimal query cost.

The hash table, especially its external memory version, is one of the most important index structures in large databases. Assuming a truly random hash function, it is known that in a standard external hash table with block size b, searching for a particular key only takes expected average t_q=1+1/2Ω(b) disk accesses for any load factor α bounded away from $1$. However, such near-perfect performance is achieved only when b is known and the hash table is particularly tuned for working with such a blocking. In this paper we study if it is possible to build a cache-oblivious hash table that works well with any blocking. Such a hash table will automatically perform well across all levels of the memory hierarchy and does not need any hardware-specific tuning, an important feature in autonomous databases.

Hash tables are one of the most fundamental data structures in computer science, in both theory and practice. They are especially useful in external memory, where their query per-formance approaches the ideal cost of just one disk access. Knuth [16] gave an elegant analysis showing that with some simple collision resolution strategies such as linear probing or chaining, the expected average number of disk I/Os of a lookup is merely 1 + 1/2Ω(b) , where each I/O can read and/or write a disk block containing b items. Inserting a new item into the hash table also costs 1 + 1/2Ω(b) I/Os, which is again almost the best one can do if the hash ta-ble is entirely stored on disk. However, this requirement is unrealistic since any algorithm operating on an external hash table must have some internal memory (at least Ω(1) blocks) to work with. The availability of a small internal memory buffer can dramatically reduce the amortized insertion cost to o(1) I/Os for many external memory data structures. In this paper we study the inherent query-insertion tradeoff of external hash tables in the presence of a memory buffer. In particular, we show that for any constant c > 1, if the expected average successful query cost is targeted at 1+O(1/bc) I/Os, then it is not possible to support insertions in less than 1-O(1/bc-1/6) I/Os amortized, which means that the memory buffer is essentially useless. While if the query cost is relaxed to 1+O(1/bc) I/Os for any constant c < 1, there is a simple dynamic hash table with o(1) insertion cost.