Given a social network$G=(V, E)$, the unconstrained profit maximization problem aims to identify a subset$S \subseteq V$that maximizes the net profit, defined as the expected influence spread$\Gamma (S)$of set$S$minus the associated cost$c(S)$, i.e.,$\Gamma (S) - c(S)$. However, this problem presupposes an unlimited budget, which is often impractical in real scenarios. Motivated by this, we investigate the budgeted profit maximization (BPM) problem by adding a budget constraint. Unfortunately, addressing the BPM problem with a theoretical approximation guarantee remains relatively under-explored in the literature. In response, assuming$\Gamma (S)$is known for any$S \subseteq V$, we propose an algorithm that guarantees returning a set$S^{o}$such that$\Gamma (S^{o}) - c(S^{o}) \geq (1 - \frac{1}{e}) \frac{\Gamma (S^*)}{2} - \frac{c(S^*)}{2}$, where$S^*$denotes an optimal solution for BPM. Then, we develop a practical solution, which uses the reverse reachable set (RR-set) technique for influence estimation, without assuming knowledge of$\Gamma (S)$, while still maintaining a strong approximation guarantee. Additionally, similar to existing RR-set-based solutions for influence cascade-related problems, our RR-set-based solution relies on generating a large number of random RR-sets to accurately estimate$\Gamma (S)$. However, the existing RR-set generation method suffers from high memory stall rates due to its irregular memory access patterns, leaving room for further efficiency improvement. Therefore, we propose a new RR-set generation method that utilizes batch execution and cache prefetching. When a memory access is required, instead of stalling while waiting for data, the CPU first issues an asynchronous prefetch request to load the target data into the cache, and then switches to processing the generation of other RR-sets within the same batch, effectively hiding memory access latency. This method can be seamlessly integrated into existing RR-set-based solutions to improve their efficiency. Finally, we conduct extensive experiments on real, large-scale datasets to demonstrate the effectiveness and efficiency of our proposed solutions.

Graph clustering is a fundamental data mining task that clusters vertices into different groups. The structural graph clustering algorithm ($SCAN$) is a widely used graph clustering algorithm that derives not only clustering results, but also special roles of vertices like hubs and outliers. In this paper, we consider structural graph clustering on dynamic graphs under Jaccard similarity. The state-of-the-art index-based solution focuses on static graphs and incurs prohibitive update costs to maintain indices. Lately, an efficient approximate dynamic structural graph clustering algorithm DynStrClu under Jaccard similarity is proposed. However, their solution needs to fix input parameters while parameter settings of SCAN usually need to be fine-tuned to achieve good clustering results. Motivated by these limitations, we present a study on devising effective index structures for SCAN algorithm on dynamic graphs. Similar to the state-of-the-art dynamic scheme, our main idea to reduce the time complexity is still by bringing approximation to clustering results. However, our solution does not need to fix the input parameters. To achieve this, our solution includes two key components. The first is to maintain a bottom-$k$sketch for each vertex so that the similarities of affected vertices can be easily updated. The second key is a bucketing strategy that allows us to update clustering results and roles of vertices efficiently. Our theoretical analysis shows that our proposed algorithm achieves$O(\log \log {\frac{M+m}{p_{f}}}\cdot \log {\frac{M+m}{p_{f}}})$expected update cost and guarantees to return approximate clustering results with probability$1-p_{f}$after up to$M$updates. Extensive experiments show that our solution is up to two orders of magnitude faster than the state-of-the-art index-based solution while still achieving high-quality clustering results.

A$k$-clique is a dense graph, consisting of$k$fully-connected nodes, that finds numerous applications, such as community detection and network analysis. In this paper, we study a new problem, that finds a maximum set of disjoint$k$-cliques in a given large real-world graph with a user-defined fixed number$k$, which can contribute to a good performance of teaming collaborative events in online games. However, this problem is NP-hard when$k\geq 3$, making it difficult to solve. To address that, we propose an efficient lightweight method that avoids significant overheads and achieves a$k$-approximation to the optimal, which is equipped with several optimization techniques, including the ordering method, degree estimation in the clique graph, and a lightweight implementation. Besides, to handle dynamic graphs that are widely seen in real-world social networks, we devise an efficient indexing method with careful swapping operations, leading to the efficient maintenance of a near-optimal result with frequent updates in the graph. In various experiments on several large graphs, our proposed approaches significantly outperform the competitors by up to 2 orders of magnitude in running time and 13.3% in the number of computed disjoint$k$-cliques, which demonstrates the superiority of the proposed approaches in terms of efficiency and effectiveness.

A multi-view attributed graph (MVAG)$\mathcal{G}$captures the diverse relationships and properties of real-world entities through multiple graph views and attribute views. Effectively utilizing all views in$\mathcal{G}$is essential for MVAG clustering and embedding, which are important for applications like recommendation systems, anomaly detection, social network analysis, etc. Existing methods either achieve inferior result quality or incur significant computational costs to handle large-scale MVAGs. In this paper, we present a spectrum-guided Laplacian aggregation scheme with an effective objective formulation and two efficient algorithms SGLA and SGLA+, to cohesively integrate all views of$\mathcal{G}$into an MVAG Laplacian matrix, which readily enables classic graph algorithms to handle$\mathcal{G}$with superior performance in clustering and embedding tasks. We begin by conducting a theoretical analysis to design an integrated objective that consists of two components, the eigengap and connectivity objectives, aiming to link the spectral properties of the aggregated MVAG Laplacian with the underlying community and connectivity properties of$\mathcal{G}$. A constrained optimization problem is then formulated for the integration, which is computationally expensive to solve. Thus, we first develop the SGLA algorithm, which already achieves excellent performance compared with existing methods. To further enhance efficiency, we design SGLA+ to reduce the number of costly objective evaluations via sampling and approximation to quickly find an approximate optimum. Extensive experiments compare our methods against 12 baselines for clustering and 8 baselines for embedding on 8 multi-view attributed graphs, validating the superior performance of SGLA and SGLA+ in terms of result quality and efficiency. Compared with the most effective baselines, our methods are significantly faster, often by up to orders of magnitude. Our implementation is available at https://github.com/CyanideCentral/SGLA/.

This paper addresses the Poisson πps sampling problem, a topic of significant academic interest in various domains and with practical data mining applications, such as influence maximization. The problem includes a set S of n elements, where each element v is assigned a weight w(v) reflecting its importance. The goal is to generate a random subset X of S, where each element v ∈ S is included in X independently with probability c ⋅ w(v)over ∑v ∈ S w(v), where 0 < c≤ 1 is a constant. The subsets must be independent across different queries. While the Poisson πps sampling problem can be reduced to the well-studied subset sampling problem, updates in Poisson πps sampling, such as adding a new element or removing an element, would cause the probabilities of all n elements to change in the corresponding subset sampling problem, making this approach impractical for dynamic scenarios. To address this, we propose a dynamic index specifically tailored for the Poisson πps sampling problem, supporting optimal expected O (1) query time and O (1) index update time, with an optimal O (n) space cost. Our solution involves recursively partitioning the set by weights and ultimately using table lookup. The core of our solution lies in addressing the challenges posed by weight explosion and correlations between elements. Empirical evaluations demonstrate that our approach achieves significant speedups in update time while maintaining consistently competitive query time compared to the subset-sampling-based methods.

Graph Anomaly Detection (GAD) in heterogeneous networks presents unique challenges due to node and edge heterogeneity. Existing Graph Neural Network (GNN) methods primarily focus on homogeneous GAD and thus fail to address three key issues: (C1) Capturing abnormal signal and rich semantics across diverse meta-paths; (C2) Retaining high-frequency content in HIN dimension alignment; and (C3) Learning effectively from difficult anomaly samples with class imbalance. To overcome these, we propose ChiGAD, a spectral GNN framework based on a novel Chi-Square filter, inspired by the wavelet effectiveness in diverse domains. Specifically, ChiGAD consists of: (1) Multi-Graph Chi-Square Filter, which captures anomalous information via applying dedicated Chi-Square filters to each meta-path graph; (2) Interactive Meta-Graph Convolution, which aligns features while preserving high-frequency information and incorporates heterogeneous messages by a unified Chi-Square Filter; and (3) Contribution-Informed Cross-Entropy Loss, which prioritizes difficult anomalies to address class imbalance. Extensive experiments on public and industrial datasets show that ChiGAD outperforms state-of-the-art models on multiple metrics. Additionally, its homogeneous variant, ChiGNN, excels on seven GAD datasets, validating the effectiveness of Chi-Square filters. Our code is available at https://github.com/HsipingLi/ChiGAD.

We study structural clustering on graphs in dynamic scenarios, where graphs can be updated by arbitrary insertions or deletions of edges/vertices. Our goal is to efficiently compute structural clustering results under three conditions: 1) for any clustering parameters ε and μ provided on the fly, 2) for arbitrary graph update patterns, and 3) for all typical similarity measurements. To achieve this, we propose an algorithm named VD-STAR that is much simpler yet more efficient than state of the art. With a theoretical guarantee on clustering result's quality, VD-STAR can produce clustering results with up to 99.9% accuracy. Moreover, VD-STAR is easy to implement as it just needs to maintain sorted linked lists and hash tables, making it highly deployable in practice. Most importantly, VD-STAR improves the expected per-update time bound from state-of-the-art O(log2 n), which relies on specific assumption on update pattern, to O(log n) amortized in expectation without any assumption on update pattern. We further design two variants of VD-STAR to enhance its empirical performance. Experimental results show that our algorithms consistently outperform state-of-the-art competitors by up to 9,315 times in update time across nine real datasets, while maintaining similar update time and memory usage.

The smoothing issue in graph learning leads to indistinguishable node representations, posing significant challenges for graph-related tasks. However, our experiments reveal that this problem can uncover underlying properties of node anomaly detection (NAD) that previous research has missed. We introduce Individual Smoothing Patterns (ISP) and Neighborhood Smoothing Patterns (NSP), which indicate that the representations of anomalous nodes are harder to smooth than those of normal ones. In addition, we explore the theoretical implications of these patterns, demonstrating the potential benefits of ISP and NSP for NAD tasks. Motivated by these findings, we propose SmoothGNN, a novel unsupervised NAD framework. First, we design a learning component to explicitly capture ISP for detecting node anomalies. Second, we design a spectral graph neural network to implicitly learn ISP to enhance detection. Third, we design an effective coefficient based on our findings that NSP can serve as coefficients for node representations, aiding in the identification of anomalous nodes. Furthermore, we devise a novel anomaly measure to calculate loss functions and anomalous scores for nodes, reflecting the properties of NAD using ISP and NSP. Extensive experiments on 9 real datasets show that SmoothGNN outperforms the best rival by an average of 14.66% in AUC and 7.28% in Average Precision, with 75x running time speedup, validating the effectiveness and efficiency of our framework. Our code is available at https://github.com/xydong127/SmoothGNN.

Clustering over a graph seeks to partition the nodes therein into disjoint groups such that nodes within the same cluster are tightly-knit, while those across clusters are distant from each other. In practice, graphs are often attended with rich attributes, which are termed attributed graphs. By leveraging the complementary nature of graph topology and node attributes in such graphs, graph neural networks (GNNs) have obtained encouraging performance in graph clustering. However, existing GNN-based approaches strongly rely on the homophilic assumption of the input graph, and thus, largely fail on heterophilic graphs and others embodying numerous missing or noisy links, which are widely present in real life.

The spread of rumors on social media, particularly during significant events like the US elections and the COVID-19 pandemic, poses a serious threat to social stability and public health. Current rumor detection methods primarily rely on propagation graphs to improve the model performance. However, the effectiveness of these methods is often compromised by noisy and irrelevant structures in the propagation process. To tackle this issue, techniques such as weight adjustment and data augmentation have been proposed. However, they depend heavily on rich original propagation structures, limiting their effectiveness in handling rumors that lack sufficient propagation information, especially in the early stages of dissemination. In this work, we introduce the Key Propagation Graph Generator (KPG), a novel reinforcement learning-based framework, that generates contextually coherent and informative propagation patterns for events with insufficient topology information and identifies significant substructures in events with redundant and noisy propagation structures. KPG comprises two key components: the Candidate Response Generator (CRG) and the Ending Node Selector (ENS). CRG learns latent variable distributions from refined propagation patterns to eliminate noise and generate new candidates for ENS, while ENS identifies the most influential substructures in propagation graphs and provides training data for CRG. Furthermore, we develop an end-to-end framework that utilizes rewards derived from a pre-trained graph neural network to guide the training process. The resulting key propagation graphs are then employed in downstream rumor detection tasks. Extensive experiments conducted on four datasets demonstrate that KPG outperforms current state-of-the-art methods.

Graphs are a ubiquitous data structure in diverse domains such as machine learning, social networks, and data mining. As real-world graphs continue to grow beyond the memory capacity of single machines, out-of-core graph processing systems have emerged as a viable solution. Yet, existing systems that rely on strictly synchronous, iteration-by-iteration execution incur significant overheads. In particular, their scheduling mechanisms lead to I/O inefficiencies, stemming from read and work amplification, and induce costly synchronization stalls hindering sustained disk utilization. To overcome these limitations, we present ACGraph, a novel asynchronous graph processing system optimized for SSD-based environments with constrained memory resources. ACGraph employs a dynamic, block-centric priority scheduler that adjusts in real time based on workload, along with an online asynchronous worklist that minimizes redundant disk accesses by efficiently reusing active blocks in memory. Moreover, ACGraph unifies asynchronous I/O with computation in a pipelined execution model that maintains sustained I/O activation, and leverages a highly optimized hybrid storage format to expedite access to low-degree vertices. We implement popular graph algorithms, such as Breadth-First Search (BFS), Weakly Connected Components (WCC), personalized PageRank (PPR), PageRank (PR), and k -core on ACGraph and demonstrate that ACGraph substantially outperforms state-of-the-art out-of-core graph processing systems in both runtime and I/O efficiency.

Recent advancements in AI have enabled models to map real-world entities, such as product images, into high-dimensional vectors, making approximate nearest neighbor search (ANNS) crucial for various applications. Often, these vectors are associated with additional attributes like price, prompting the need for range-filtered ANNS where users seek similar items within specific attribute ranges. Naive solutions like pre-filtering and post-filtering are straightforward but inefficient. Specialized indexes, such as SeRF, SuperPostFiltering, and iRangeGraph, have been developed to address these queries effectively. However, these solutions do not support dynamic updates, limiting their practicality in real-world scenarios where datasets frequently change. To address these challenges, we propose DIGRA, a novel dynamic graph index for range-filtered ANNS. DIGRA supports efficient dynamic updates while maintaining a balance among query efficiency, update efficiency, indexing cost, and result quality. Our approach introduces a dynamic multi-way tree structure combined with carefully integrated ANNS indices to handle range filtered ANNS efficiently. We employ a lazy weight-based update mechanism to significantly reduce update costs and adopt optimized choice of ANNS index to lower construction and update overhead. Experimental results demonstrate that DIGRA achieves superior trade-offs, making it suitable for large-scale dynamic datasets in real-world applications.

Given a set O of objects consisting of n high-dimensional vectors, the problem of approximate nearest neighbor (ANN) search for a query vector q is crucial in many applications where objects are represented as feature vectors in high-dimensional spaces. Each object in O often has attributes like popularity or price, which influence the search. Practically, searching for the nearest neighbor to q might include a range filter specifying the desired attribute values, e.g., within a specific price range. Existing solutions for range filtered ANN search often face trade-offs among excessive storage, poor query performance, and limited support for updates. To address this challenge, we propose RangePQ, a novel indexing scheme that supports efficient range filtered ANN searches and updates, requiring only linear space. Our scheme integrates seamlessly with existing PQ-based index---a widely recognized, scalable index type for ANN searches---to enhance range-filtered ANN queries and update capabilities. Our indexing method, supporting arbitrary range filters, has a space complexity of (O(n log K)), where K is a parameter of the PQ-based index and log K scales with O(log n). To reduce the space cost, we further present a hybrid two-layer structure to reduce space usage to O(n), preserving query efficiency without additional update costs. Experimental results demonstrate that our indexing scheme significantly improves query performance while maintaining competitive update performance and space efficiency.

In the era of big data, the demand for historical data analytics is growing across various applications. Simultaneously, range queries have been extensively explored within the domain of databases. Binary search trees are a classic type of in-memory index for facilitating range queries. Persistent binary search trees provide read-only snapshots of these trees, allowing range queries to be processed during updates while ensuring consistency. Additionally, multiple versions of snapshots support queries related to historical moments to meet the demands of numerous applications. However, existing implementations do not support both highspeed updates and efficient, accurate historical queries on multi-core platforms. Motivated by this gap, we propose a novel concurrent update strategy to balance update and query performance. For a binary search tree containing n elements, our approach completes m updates in O (log n + m ) time using O (log n ) threads. We further implement a hybrid concurrent strategy to improve the scalability and practical performance of our solution. The experimental results demonstrate that our proposal strikes a good balance between update and query performance. In particular, our proposal outperforms existing solutions under workloads with different data distributions and varying update-query ratios.

In the era of big data, the landscape of data management and analytics has significantly transformed, presenting diverse challenges for cloud platforms. Modern data warehouses face increasing challenges in handling hybrid transactional and analytical processing (HTAP) workloads efficiently in cloud environments. Traditional shared-nothing architectures provide high-performance query execution but suffer from high storage costs and limited elasticity, while shared-storage approaches improve scalability but often struggle with query efficiency due to increased data movement and indexing overhead. Furthermore, existing execution engines lack optimized support for vectorized processing and real-time analytics, limiting their ability to handle large-scale workloads efficiently. To address these limitations, we introduce AnalyticDB-PG (ADB-PG), a cloud-native, high-performance data warehouse designed for modern analytical workloads. It integrates a unified architecture supporting both Shared-Nothing and Shared-Storage modes, allowing flexible deployment and seamless elasticity. In ADB-PG, we introduce Beam, a hybrid storage engine that efficiently balances row-based and columnar storage for real-time analytics, and Laser, an optimized execution engine leveraging vectorized execution and Just-In-Time compilation to accelerate query processing. The system further incorporates advanced indexing mechanisms, adaptive runtime filtering, and dictionary encoding to enhance performance. Extensive evaluations on TPC-H and TPC-DS benchmarks demonstrate that ADB-PG achieves significant performance improvements while reducing storage and operational costs, making it a compelling solution for modern cloud-based data analytics.

We study the problem of Quality-of-Service (QoS)-Aware Personalized PageRank (PPR) computation. Existing studies mostly focus on improving the PPR query processing time. However, the query processing time alone may not reflect the service quality in real-world PPR-based systems. The query response time can be a more service-relevant measure in many applications such as the online game service of Tencent and the related-pin recommendation module of Pinterest. We make the first attempt at studying QoS-Aware PPR computation and present Quota, a system that adapts the state-of-the-art PPR algorithms to a given environment for minimizing query response time. Equipped with mathematical tools including queuing theory, algorithmic complexity analysis, and constrained optimization, Quota is designed to adapt itself to a wide spectrum of workloads. We conduct extensive experiments on real datasets and show that Quota can reduce the query response time compared with state-of-the-art PPR algorithms, often by a significant margin.

Node embedding learns low-dimensional vectors for nodes in the graph. Recent state-of-the-art embedding approaches take Personalized PageRank (PPR) as the proximity measure and factorize the PPR matrix or its adaptation to generate embeddings. However, little previous work analyzes what information is encoded by these approaches, and how the information correlates with their superb performance in downstream tasks. In this work, we first show that state-of-the-art embedding approaches that factorize a PPR-related matrix can be unified into a closed-form framework. Then, we study whether the embeddings generated by this strategy can be inverted to better recover the graph topology information than random-walk based embeddings. To achieve this, we propose two methods for recovering graph topology via PPR-based embeddings, including the analytical method and the optimization method. Extensive experimental results demonstrate that the embeddings generated by factorizing a PPR-related matrix maintain more topological information, such as common edges and community structures, than that generated by random walks, paving a new way to systematically comprehend why PPR-based node embedding approaches outperform random walk-based alternatives in various downstream tasks. To the best of our knowledge, this is the first work that focuses on the interpretability of PPR-based node embedding approaches.

Given a social network G , a cost associated with each user, and an influence threshold η, the minimum cost seed selection problem (MCSS) aims to find a set of seeds that minimizes the total cost to reach η users. Existing works are mainly devoted to providing an expected coverage guarantee on reaching η, classified as MCSS-ECG, where their solutions either rely on an impractical influence oracle or cannot attain the expected influence threshold. More importantly, due to the expected coverage guarantee, the actual influence in a campaign may drift from the threshold evidently. Thus, the advertisers would like to request for a probability guarantee of reaching η. This motivates us to further solve the MCSS problem with a probabilistic coverage guarantee, termed MCSS-PCG. In this paper, we first propose our algorithm CLEAR to solve MCSS-ECG, which reaches the expected influence threshold without any influence oracle or influence shortfall but a practical approximation ratio. However, the ratio involves an unknown term (i.e., the optimal cost). Thus, we further devise the STAR method to derive a lower bound of the optimal cost and then obtain the first explicit approximation ratio for MCSS-ECG. In MCSS-PCG, it is necessary to estimate the probability that the current seeds reach η, to decide when to stop seed selection. To achieve this, we design a new technique named MRR, which provides efficient probability estimation with a theoretical guarantee. With MRR in hand, we propose our algorithm SCORE for MCSS-PCG, whose performance guarantee is derived by measuring the gap between MCSS-ECG and MCSS-PCG, and applying the theoretical results in MCSS-ECG. Finally, extensive experiments demonstrate that our algorithms achieve up to two orders of magnitude speed-up compared to alternatives while meeting the requirement of MCSS-PCG with the smallest cost.

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.

Real-world directed graphs are dynamically changing, and it is important to identify and maintain the strong connectivity information between nodes, which is useful in numerous applications. Given an input graphG, we study a new problem,minimum strongly connected subgraph collection(MSCSC), which asks for a complete collection of subgraphs, each of which contains amaximalset of nodes that are strongly connected to each other viaminimumnumber of edges inG. MSCSC is NP-hard, and its computation and maintenance are challenging, especially on large-scale dynamic graphs. Thus, we resort to approximate MSCSC with theoretical guarantees. We develop a series of approximate MSCSC methods for both static and dynamic graphs. Specifically, we first develop a static MSCSC method MSC that only needs one scan of the graphG, runs in linear timew.r.t., the number of edges, and provides rigorous approximation guarantees. Then, based on MSC, we leverage a reduced directed acyclic graph ofGto design incremental MSCSC method MSCiwith two variants to handle edge insertions efficiently. We further develop MSCdthat updates MSCSC under edge deletions by efficiently scanning only locally affected subgraphs. Moreover, to demonstrate the high utility, we conduct two use case studies to apply our MSCSC methods to boost the efficiency of dynamic strongly connected component (SCC) maintenance and dynamic SCC-based reachability index maintenance. Extensive experiments on 8 large graphs, including 3 billion-edge graphs, validate the superior efficiency of our methods.

Given a source node$s$and a target node$t$, the hitting probability tells us how likely an$\alpha$-terminating random walk (which stops with probability$\alpha$at each step) starting from$s$can hit$t$before it stops. This concept originates from the hitting time, a classic concept in random walks. In this paper, we focus on the group hitting probability (GHP) where the target is a set of nodes, measuring the node-to-group structural proximity. For this group version of the hitting probability, we present efficient algorithms for two types of GHP queries: the pairwise query which returns the GHP value of a target set$T$with respect to (w.r.t.) a source node$s$, and the top-$k$query which returns the top-$k$target sets with the largest GHP value w.r.t. a source node$s$. We first develop an efficient algorithm named SAMBA for the pairwise query, which is built on a group local push algorithm tailored for GHP, with rigorous analysis for correctness. Next, we show how to speed up SAMBA by combining the group local push algorithm with the Monte Carlo approach, where GHP brings new challenges as it might need to consider every hop of the random walk. We tackle this issue with a new formulation of the GHP and show how to provide approximation guarantees with a detailed theoretical analysis. With SAMBA as the backbone, we develop an iterative algorithm for top-$k$queries, which adaptively refines the bounds for the candidate target sets, and terminates as soon as it meets the stopping condition, thus saving unnecessary computational costs. We further present an optimization technique to accelerate the top-$k$query, improving its practical performance. Extensive experiments show that our solutions are orders of magnitude faster than their competitors.

Given a graph$\mathcal {G}$with a set of attributes, theattributed network embedding (ANE)aims to learn low-dimensional representations of nodes that preserve both graph topology and node attribute proximity. ANE is shown to be more effective than plain network embedding methods (using only graph topology) on many graph mining tasks. However, existing ANE solutions still provide inferior performance on tasks like node classification and link prediction, as will be shown in our experiments. The key issue is that when combining graph topology and attribute information, most existing solutions take attributes with equal importance, while in real scenarios, different attribute exerts distinct influence over the network due to the heterogeneous nature among attributes. Motivated by this, we presentLATAM, a learning-based framework for ANE via trainable proximity matrix factorization. To capture the node-attribute relationships, we first construct the attribute-augmented graph by adding attribute nodes (resp. edges) to the original graph. Then, we define the attribute-augmented random walk and proximity on the attribute-augmented graph, where the weights of different attributes can be learned automatically by our designed loss functions so that more indicative attributes tend to have higher weights, imposing a higher impact on the node connectivity. To achieve this, we incorporate a differentiable SVD to back-propagate gradients of attribute weights in an end-to-end process. To scale our LATAM to large graphs, we further propose sampling techniques to learn attribute weights and an efficient attribute-augmented push algorithm to compute the proximity matrix. Extensive experiments on 8 public attributed networks against 11 existing methods show the effectiveness of our LATAM.

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.

Abstract Active learning for graph neural networks (GNNs) aims to select B nodes to label for the best possible GNN performance. Carefully selected labeled nodes can help improve GNN performance and hence motivates a line of research works. Unfortunately, existing methods still provide inferior GNN performance or cannot scale to large networks.Motivated by these limitations, in this paper, we present FICOM, an effective and scalable GNN active learning framework. Firstly, we formulate the node selection as an optimization problem where we consider the importance of a node from (i) the importance of a node during the feature propagation with a connection to the personalized PageRank (PPR), and (ii) the diversity of a node brings in the embedding space generated by feature propagation. We show that the defined problem is submodular, and a greedy solution can provide a $$(1-1/e)$$ ( 1 - 1 / e ) -approximate solution.However, a standard greedy solution requires getting the node with the maximum marginal gain of the objective score in each iteration, which incurs a prohibitive running cost and cannot scale to large datasets. As our main contribution, we present FICOM, an efficient and scalable solution that provides $$(1-1/e)$$ ( 1 - 1 / e ) -approximation guarantee and scales to graphs with millions of nodes on a single machine. The main idea is that we adaptively maintain the lower- and upper-bound of the marginal gain for each node v. In each iteration, we can first derive a small subset of candidate nodes and then compute the exact score for this subset of candidate nodes so that we can find the node with the maximum marginal gain efficiently. Extensive experiments on six benchmark datasets using four GNNs, including GCN, SGC, APPNP, and GCNII, show that our FICOM consistently outperforms existing active learning approaches on semi-supervised node classification tasks using different GNNs. Moreover, our solution can finish within 5 h on a million-node graph.

In graph analysis area, the shortest path is vital, and recent research shows that the shortest paths counting is crucial in applications like potential friend recommendation and betweenness analysis. Nevertheless, the existing works mainly focus on how to speed up it in a single machine with single core, which do not consider the scalability of it. It limits applications and wastes potential performance. The main bottleneck is dependency between their index is no considered. To fill this research gap, we provide a parallel method. The main approach is to release the dependency between the index as well as optimizations during the process. Moreover, our method could achieve a nearly linear speedup with the number of threads increase in terms of index time. The experimental results demonstrate the effectiveness and efficiency of our method than the baselines.

In online social networks, users with similar interests tend to come together, forming social communities. Nowadays, user-defined communities become a prominent part of online social platforms as people who have joined such communities tend to be more active in social networks. Therefore, recommending explicit communities to users provides great potential to advance online services.

Trend analysis is a fundamental type of analytical query in online analytical processing (OLAP) systems. In trend analysis, a key step is to identify k valuable attributes whose distributions in two subsets under different predicates significantly differ for further investigation, where the difference is measured by metric functions. However, the exact solution that involves scanning all records is prohibitively expensive, particularly when handling large datasets in the era of big data. To minimize unnecessary data access, the existing state-of-the-art solution TopKAttr adopts sampling to avoid the expensive data scan. However, their solution still has two main drawbacks. Firstly, their solution is tailored only for two limited metric functions: the Earth Mover distance and Euclidean distance, and cannot be generalized to more complicated metric functions. Besides, their solution still aims to return the exact top-k answers via the sampling method, which still causes high running costs as shown in our experiment. Motivated by these limitations, we propose a general approximation framework for attribute recommendation that efficiently returns the top-k attributes with theoretical guarantees while supporting an extensive range of metric functions, such as the Kolmogorov-Smirnov test (KS-test), Chebyshev distance, the Earth Mover distance, Euclidean distance, and with the potential to more metrics. The key to our framework is a new bound estimation strategy that can be applied to a wide spectrum of metrics, as we listed above. Based on our estimation framework, we further devise an efficient approximation algorithm with theoretical guarantees to answer the top-k queries, which is widely used in attribute recommendation. Extensive experiments on four real large datasets show that our framework gains up to an order of magnitude speed-up and consistently high accuracy compared to TopKAttr, providing a promising alternative for attribute recommendation in OLAP systems.

Subset embedding is the task to learn low-dimensional representations for a subset of nodes according to the graph topology. It has applications when we focus on a subset of users, e.g., young adults, and aim to make better recommendations for these target users. In real-world scenarios, graphs are dynamically changing. Thus, it is more desirable to dynamically maintain the subset embeddings to reflect graph updates. The state-of-the-art methods, e.g., DynPPE, still adopt a hashing-based method, while hashing-based solutions are shown to be less effective than matrix factorization (MF)-based methods in existing studies. At the same time, MF-based methods in the literature are too expensive to update the embedding when the graph changes, making them inapplicable on dynamic graphs. Motivated by this, we present Tree-SVD, an efficient and effective MF-based method for dynamic subset embedding. If we simply maintain the whole proximity matrix, then we need to re-do the MF, e.g., truncated Singular Value Decomposition (SVD), on the whole matrix after graph updates, which is prohibitive. To tackle this issue, our main idea is to do hierarchical SVD (HSVD) on the proximity matrix of the given subset, which vertically divides the proximity matrix into multiple sub-matrices, and then repeatedly do SVD on sub-matrices and merge the intermediate results to obtain the final embedding. We first present Tree-SVD, which combines a sparse randomized SVD with an HSVD. Our theoretical analysis shows that our Tree-SVD gains the efficiency of sparse randomized SVD and the flexibility of the HSVD with theoretical guarantees. To further reduce update costs, we present a lazy-update strategy. In this strategy, we only update sub-matrices that changes remarkably in terms of the Frobenius norm. We present theoretical analysis to show the guarantees with our lazy-update strategy. Extensive experiments show the efficiency and effectiveness of Tree-SVD on node classification and link prediction tasks.

Given a graph G, a cost associated with each node, and a budget B, the budgeted influence maximization (BIM) aims to find the optimal set S of seed nodes that maximizes the influence among all possible sets such that the total cost of nodes in S is no larger than B. Existing solutions mainly follow the non-adaptive idea, i.e., determining all the seeds before observing any actual diffusion. Due to the absence of actual diffusion information, they may result in unsatisfactory influence spread. Motivated by the limitation of existing solutions, in this paper, we make the first attempt to solve the BIM problem under the adaptive setting, where seed nodes are iteratively selected after observing the diffusion result of the previous seeds. We design the first practical algorithm which achieves an expected approximation guarantee by probabilistically adopting a cost-aware greedy idea or a single influential node. Further, we develop an optimized version to improve its practical performance in terms of influence spread. Besides, the scalability issues of the adaptive IM-related problems still remain open. It is because they usually involve multiple rounds (e.g., equal to the number of seeds) and in each round, they have to construct sufficient new reverse-reachable set (RR-set) samples such that the claimed approximation guarantee can actually hold. However, this incurs prohibitive computation, imposing limitations on real applications. To solve this dilemma, we propose an incremental update approach. Specifically, it maintains extra construction information when building RR-sets, and then it can quickly correct a problematic RR-set from the very step where it is first affected. As a result, we recycle the RR-sets at a small computational cost, while still providing correctness guarantee. Finally, extensive experiments on large-scale real graphs demonstrate the superiority of our algorithms over baselines in terms of both influence spread and running time.

\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.

Structural graph clustering (SCAN) is a classic graph clustering algorithm. In SCAN, a key step is to compute the structural similarity between vertices according to the overlap ratio of one-hop neighborhoods. Given two vertices u and v, existing studies only consider the case when u and v are neighbors. However, the structural similarity between non-neighboring vertices in SCAN is always zero, and using only one-hop neighbors on weighted graphs discards the weights on each edge. Both may not reflect the true closeness of two vertices and may fail to return high-quality clustering results. To tackle this issue, we define and study the distance-based structural graph clustering problem. Given a distance threshold d and two vertices u and v, the structural similarity between u and v is defined as the ratio of their respective neighbors within a distance of no more than d. We show that the newly defined distance-based SCAN achieves better clustering results compared to the vanilla version of SCAN. However, the new definition brings challenges in the computation of final clustering results. To tackle this efficiency issue, we propose DistanceSCAN, an efficient approximate algorithm for solving the distance-based SCAN problem. The main idea of DistanceSCAN is to use all-distances bottom-k sketches (ADS) to speed up the computation of similarities. Given the ADS, we can derive the similarity between two vertices with a bounded cost of O(k). However, to ensure that the estimated similarity has an approximation guarantee, the value of k still needs to be set to as large as thousands. This brings high computational costs when computing the similarities between neighboring vertices. To tackle this issue, we further construct histograms to prune the structural similarity computations of vertices pairs. Extensive experiments on real datasets validate the effectiveness and efficiency of DistanceSCAN.

A bipartite graph is a graph that consists of two disjoint sets of vertices and only edges between vertices from different vertex sets. In this paper, we study the counting problems of two common types of em motifs in bipartite graphs: (i) butterflies (2x2 bicliques) and (ii) bi-triangles (length-6 cycles). Unlike most of the existing algorithms that aim to obtain exact counts, our goal is to obtain precise enough estimations of these counts in bipartite graphs, as such estimations are already sufficient and of great usefulness in various applications. While there exist approximate algorithms for butterfly counting, these algorithms are mainly based on the techniques designed for general graphs, and hence, they are less effective on bipartite graphs. Not to mention that there is still a lack of study on approximate bi-triangle counting. Motivated by this, we first propose a novel butterfly counting algorithm, called one-sided weighted sampling, which is tailored for bipartite graphs. The basic idea of this algorithm is to estimate the total butterfly count with the number of butterflies containing two randomly sampled vertices from the same side of the two vertex sets. We prove that our estimation is unbiased, and our technique can be further extended (non-trivially) for bi-triangle count estimation. Theoretical analyses under a power-law random bipartite graph model and extensive experiments on multiple large real datasets demonstrate that our proposed approximate counting algorithms can reach high accuracy, yet achieve up to three orders (resp. four orders) of magnitude speed-up over the state-of-the-art exact butterfly (resp. bi-triangle) counting algorithms. Additionally, we present an approximate clustering coefficient estimation framework for bipartite graphs, which shows a similar speed-up over the exact solutions with less than 1% relative error.

Given a weighted setSofnelements,weighted set sampling (WSS)samples an element inSso that each elementai; is sampled with a probability proportional to its weightw(ai). The classic alias method pre-processes an index inO(n) time withO(n) space and handles WSS withO(1) time. Yet, the alias method does not support dynamic updates. By minor modifications of existing dynamic WSS schemes, it is possible to achieve an expectedO(1) update time and drawtindependent samples in expectedO(t) time with linear space, which is theoretically optimal. But such a method is impractical and even slower than a binary search tree-based solution. How to support both efficient sampling and updates in practice is still challenging. Motivated by this, we designBUS, an efficient scheme that handles an update inO(1) amortized time and drawstindependent samples inO(logn + t)time with linear space. A natural extension of WSS is theweighted independent range sampling (WIRS), where each element inSis a data point from R. Given an arbitrary rangeQ= [ℓ,r] at query time, WIRS aims to do weighted set sampling on the setSQof data points falling into rangeQ.We show that by integrating the theoretically optimal dynamic WSS scheme mentioned above, it can handle an update inO(logn) time and can drawtindependent samples for WIRS inO(logn + t) time, the same as the state-of-the-art static algorithm. Again, such a solution by integrating the optimal dynamic WSS scheme is still impractical to handle WIRS queries. We further propose WIRS-BUS to integrate BUS to handle WIRS queries, which handles each update inO(logn) time and drawstindependent samples inO(log2n + t) time with linear space. Extensive experiments show that our BUS and WIRS-BUS are efficient for both sampling and updates.

With rapidly growing sizes of online social networks, computational challenges arise in analyzing the diffusion process over networks. Sampling methods are commonly used to study the cascade effect and estimate users' influence. In this paper, we propose a brand-new sampling method, called 2-hop+ sampling for quickly and accurately estimating the cascade size generated by a set of seed users under the independent cascade model. Our method generates only samples with at least one 2-hop live path from the source to reduce the number of samples. We further enhance the sampling efficiency of our method by a SkipEdge technique. Moreover, we improve the generalized stopping rule algorithm to obtain an (,)-estimate of the mean of random variables with fewer samples needed. Extensive experiments with real-world datasets show that our techniques can significantly improve the estimation efficiency compared to the state-of-the-art methods.

In this paper, we study the problem of constrained path search with submodular function maximization (CPS-SM). We aim to find the path with the best submodular function score under a given constraint (e.g., a length limit), where the submodular function score is computed over the set of nodes in this path. This problem can be used in many applications. For example, tourists may want to search the most diversified path (e.g., a path passing by the most diverse facilities such as parks and museums) given that the traveling time is less than 6 hours. We show that the CPS-SM problem is NP-hard. We first propose a concept called “submodular$\alpha$-dominance” by utilizing the submodular function properties, and we develop an algorithm with a guaranteed error bound based on this concept. By relaxing the submodular$\alpha$-dominance conditions, we design another more efficient algorithm that has the same error bound. We also utilize the way of bi-directional path search to further improve the efficiency of the algorithms. We finally propose a heuristic algorithm that is efficient yet effective in practice. The experiments conducted on several real datasets show that our proposed algorithms can achieve high accuracy and are faster than one state-of-the-art method by orders of magnitude.

Community search is a problem that seeks cohesive and connected subgraphs in a graph that satisfy certain topology constraints, e.g., degree constraints. The majority of existing works focus exclusively on the topology and ignore the nodes' influence in the communities. To tackle this deficiency, influential community search is further proposed to include the node's influence. Each node has a weight, namely influence value, in the influential community search problem to represent its network influence. The influence value of a community is produced by an aggregated function, e.g., max, min, avg, and sum, over the influence values of the nodes in the same community. The objective of the influential community search problem is to locate the top-r communities with the highest influence values while satisfying the topology constraints. Existing studies on influential community search have several limitations: (i) they focus exclusively on simple aggregation functions such as min, which may fall short of certain requirements in many real-world scenarios, and (ii) they impose no limitation on the size of the community, whereas most real-world scenarios do. This motivates us to conduct a new study to fill this gap. We consider the problem of identifying the top-r influential communities with/without size constraints while using more complicated aggregation functions such as sum or avg. We give a theoretical analysis demonstrating the hardness of the problems and propose efficient and effective heuristic solutions for our top-r influential community search problems. Extensive experiments on real large graphs demonstrate that our proposed solution is significantly more efficient than baseline solutions.

Empirical variance is a fundamental concept widely used in data management and data analytics, e.g., query optimization, approximate query processing, and feature selection. A direct solution to derive the empirical variance is scanning the whole data table, which is expensive when the data size is huge. Hence, most current works focus on approximate answers by sampling. For results with approximation guarantees, the samples usually need to be uniformly independent random, incurring high cache miss rates especially in compact columnar style layouts. An alternative uses block sampling to avoid this issue, which directly samples a block of consecutive records fitting page sizes instead of sampling one record each time. However, this provides no theoretical guarantee. Existing studies show that the practical estimations can be inaccurate as the records within a block can be correlated.

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.

Many real-world graphs, e.g., social networks, biological networks, knowledge graphs, naturally come with edge-labels, with different labels representing different relationships between nodes. On such edge-labeled graphs, an important query is the label-constrained reachability (LCR) query, where we are given a source s , a target t , a label set ψ, and the goal is to check if there exists any path P from s to t such that labels of edges on P all belong to ψ. Existing indexing schemes for LCR queries still focus on static graphs, despite the fact that many edge-labeled graphs are dynamic in nature. Motivated by the limitations of existing solutions, we present a study on how to effectively maintain the indexing scheme on dynamic graphs. Our proposed approach is based on the state-of-the-art 2-hop index for LCR queries. In this paper, we present efficient algorithms for updating the index structure in response to dynamic edge insertions/deletions and demonstrate the correctness of our update algorithms. Following that, we present that adopting a query-friendly but update-unfriendly indexing scheme results in surprisingly superb query/update efficiency and outperforms those update-friendly ones. We analyze and demonstrate that the query-friendly indexing scheme actually achieves the same time complexity as those of update-friendly ones. Finally, we present the batched update algorithms where the updates may include multiple edge insertions/deletions. Extensive experiments show the effectiveness of the proposed update algorithms, query-friendly indexing scheme, and batched update algorithms.

Graph clustering is a fundamental data mining task that clusters vertices into different groups. The structural graph clustering algorithm ( SCAN ) is a widely used graph clustering algorithm that derives not only clustering results, but also special roles of vertices like hubs and outliers. In this paper, we consider structural graph clustering on dynamic graphs under Jaccard similarity. The state-of-the-art index-based solution focuses on static graphs and incurs prohibitive update costs to maintain indices. Lately, an efficient approximate dynamic structural graph clustering algorithm Dyn-StrClu under Jaccard similarity is proposed. However, their solution needs to fix input parameters while parameter settings of SCAN usually need to be fine-tuned to achieve good clustering results. Motivated by these limitations, we present a study on devising effective index structures for SCAN algorithm on dynamic graphs. Similar to the state-of-the-art dynamic scheme, our main idea to reduce the time complexity is still by bringing approximation to clustering results. However, our solution does not need to fix the input parameters. To achieve this, our solution includes two key components. The first is to maintain a bottom- k sketch for each vertex so that the similarities of affected vertices can be easily updated. The second key is a bucketing strategy that allows us to update clustering results and roles of vertices efficiently. Our theoretical analysis shows that our proposed algorithm achieves O (log n · log M + m / pf ) expected update cost and guarantees to return approximate clustering results with probability 1 - pf after up to M updates. Extensive experiments show that our solution is up to two orders of magnitude faster than the state-of-the-art index-based solution while still achieving high-quality clustering results.

Finding the fastest path in the time-dependent road network is time consuming because its problem complexity is$\Omega (T(|V|\log |V|+|E|))$, where$T$is the size of the result's time-dependent function,$|V|$and$|E|$are the number of vertices and edges. There are three kinds of fastest path problems:SSFP (Single-Staring Time Fastest Path)that has a fixed departure time,ISFP (Interval-Staring Time Fastest Path)that selects the best departure time from an interval, andFPP (Fastest Path Profile)that returns the travel time of the entire time domain. In this paper, we aim to answer these three queries in time-dependent road network faster by extending the2-hop labelingapproach, which is fast in answering shortest distance query in the static graph. However, it is hard to construct index forSSFPandISFPbecause there are$|\mathcal {T}|$and$|\mathcal {T}|^2$possible time points and intervals, where$\mathcal {T}$is the time domain. Therefore, we first propose thetime-dependent hop-labelingforFPP, then provide the specific optimizations forSSFPandISFPquery answering. Moreover, it is both time and space consuming to build an index in a large time-dependent graph, so we partition road network into smaller sub-graphs and build indexes within and between the partitions. Furthermore, we propose an online approximation techniqueAT-Dijkstraand abottom-upcompression method to further reduce the label size, save construction time and speedup query answering. Experiments on real world road network show that our approach outperforms the state-of-art fastest path index approaches and can speed up the query answering by hundreds of times.

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.

Online social networking companies devote much effort to maximize the engagement of users. Most existing methods assume that the budget is already known and focus on heuristic solutions to maximize the number of users that keep engaged. However, many real-life applications aim to keep the engagement of the majority of users in the social network. In such scenarios, the budget is flexible, and the goal is to minimize the budget used to keep these users active. In this paper, we define and study the anchored k-core budget minimization problem. Given an undirected graph G, a degree constraint k and a quota q for engaged users, this problem aims to minimize the budget that assures at least q users not in k-core keeping engaged.We propose CLOCK, an efficient greedy algorithm to solve the anchored k-core budget minimization problem. The main idea is to delete the anchored vertex with the lowest score in every iteration while maintaining the required number of engaged users. By considering the influence among the anchored vertices, a scoring function is well designed to identify the anchored vertices with high quality. Then we propose two merging strategies to get better anchored vertices. In addition, we dynamically maintain all anchored vertices by a local update strategy which avoids many redundant calculations. Finally, we conduct extensive experiments on real world datasets, demonstrating the effectiveness and efficiency of CLOCK.

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.

Node embedding learns a low-dimensional representation for each node in the graph. Recent progress on node embedding shows that proximity matrix factorization methods gain superb performance and scale to large graphs with millions of nodes. Existing approaches first define a proximity matrix and then learn the embeddings that fit the proximity by matrix factorization. Most existing matrix factorization methods adopt the same proximity for different tasks, while it is observed that different tasks and datasets may require different proximity, limiting their representation power.

Empirical entropy is a classic concept in data mining and the foundation of many other important concepts like mutual information. However, computing the exact empirical entropy/mutual information on large datasets can be expensive. Some recent research work explores sampling techniques on the empirical entropy/mutual information to speed up the top-k and filtering queries. However, their solution still aims to return the exact answers to the queries, resulting in high computational costs.

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.

The intelligent transportation systems, e.g., DiDi and Uber, have served as essential travel tools for customers, which foster plenty of studies for the location-based queries on road network. In particular, given a set O of objects and a query point q on a road network, the k Nearest Neighbor (kNN) query returns the k nearest objects in O with the shortest road network distance to q. In literature, most existing solutions for kNN queries tend to reduce the query time, indexing storage, or throughput of the kNN queries while overlooking the correctness of the queries caused by query-query and update-query conflicts. In our work, we propose a grid-based framework on conflict-aware kNN queries on moving objects which aims to optimize system throughput while guaranteeing query correctness. In particular, we first propose efficient index structures and new query algorithms that significantly improve the throughput. We further present novel scheduling algorithms that aim to avoid conflicts and improve the system throughput. Moreover, we devise approximate solutions that provide a controllable trade-off between the conflict of kNN queries and system throughput. Finally, we propose a cost-based dispatching strategy to assign the kNN results to the corresponding queries. Extensive experiments on real-world data demonstrate the effectiveness and efficiency of our proposed solutions over alternatives.

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.

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.

We are pleased to present a special issue of Data Science and Engineering (DSE), which contains a collection of five papers from the WISE 2019 conference, and two regular submission papers.The International Conference on Web Information Systems Engineering (WISE) has established itself as a community aiming at high-quality research and offering the ground for advancing efforts in topics related to Web information systems.WISE 2019 focuses on research, development, and applications in relation to Web information systems, including a wide range of topics, such as blockchain, deep learning, machine learning, recommender systems, data mining, Web-based applications, graph learning, knowledge graphs, graph mining, text mining, and crowdsourcing.WISE 2019 was held in Hong Kong, China, during January 19-21, 2020.WISE 2019 attracted a total of 211 research paper submissions.The conference program committee selected 50 full research papers (with an acceptance rate of 23.7%), five demonstration papers, and three tutorial papers to be presented at the conference and published in the proceedings [1,2].The conference program also included keynote presentations by

Given a social network G , a cost associated with each node, and a budget B , the budgeted influence maximization (BIM) problem aims to find a set S of nodes, denoted as the seed set, that maximizes the expected number of influenced users under the constraint that the total cost of the users in S is no larger than B. The current state-of-the-art practical solution for BIM problem provides a (1-1/ e /2 --- ε)-approximate (≈ 0.316 --- ε) result and is still inefficient on large networks. We first show that we can improve the approximation guarantee to 1 --- 1/ e β --- ε where 1 --- 1/ e β = (1 --- β) (1 --- 1/ e ), achieving a better approximation guarantee (≈ 0.355 --- ε). Next, we apply the reverse sampling based technique, a popular technique for classic influence maximization, to our studied BIM problem. However, it is non-trivial to design efficient solutions for large scale networks even the reverse sampling based technique is applied. On one hand, it is unclear how to derive tight bounds for the nodes selected by the greedy algorithm under the budgeted scenario, where each time it selects the seed node with the highest benefit-cost ratio. With tighter bounds, the algorithm can terminate as soon as the approximation ratio is satisfied, thus saving the running cost. On the other hand, the number of nodes selected under BIM problem may be quite large since it may greedily select many nodes with large benefit-cost ratio but with low costs. The time complexity of existing influence maximization algorithms heavily depends on the size of the seed set. To tackle such challenging issues, we first present new bound estimation techniques for the BIM problem. Next, we present new node selection strategies to alleviate the dependency to the size of the seed set. Extensive experiments show that our proposed solution is far more efficient than alternatives.

Route scheduling on time-dependent road network is slow due to its problem complexity of Ω(T(|V|log |V|+|E|)), where T is the size of the result's time-dependent function, |V| is the number of vertices and |E| is the number of edges. To make things worse, T grows larger as the route becomes longer or the query time interval becomes bigger, especially for a fastest path profile query whose time interval is 24 hours. In this paper, we aim to answer the fastest path profile query on time-dependent road network faster by extending the 2-hop labeling approach, which is fast in answering shortest distance query on the static graph. However, building an index on a time-dependent graph is both time and space consuming, so currently only online-search approach exist. Apparently, its query answering power is limited by the online searching. To solve this problem, we first propose the time-dependent hop on large road network by partitioning it into smaller sub-graphs. The index is built within and between the partitions, and is retrieved from disk during query answering with the help of sampling. Moreover, we propose an online approximation technique AT-Dijkstra and a bottom-up compression method to further reduce the label size, save construction time and speedup query answering. Experiments on real world road network show that our approach outperforms the state-of-art fastest path index approaches and can speed up the query answering by hundreds of times.

Given a set O of objects and a query point q on a road network, the k Nearest Neighbor (kNN) query returns the k nearest objects in O with the shortest road network distance to q. These kNN queries find many applications in location-based services, e.g., ride-hailing services, where each taxi is regarded as an object. In such applications, objects are constantly moving such that even for the same query point, the correct answer of a kNN query may vary with time. Ideally, the returned answer should be adequately correct with respect to the moving object set. However, in literature, all existing solutions for kNN queries mainly focus on reducing the query time, indexing storage, or throughput of the kNN queries with little focus on their correctness. Motivated by this, we propose a framework on correctness-aware kNN queries which aim to optimize system throughput while guaranteeing query correctness on moving objects. We formally define the serializable-kNN query that ensures the correctness of the query answer when considering moving objects and dependencies of different queries. We propose several techniques to optimize the throughput of serializable-kNN queries: firstly, we propose efficient index structures and new query algorithms that significantly improve the throughput; we further present novel scheduling algorithms that aim to avoid conflicts and improve the system throughput. Moreover, we devise approximate solutions that provide a controllable trade-off between the correctness of kNN queries and system throughput. Extensive experiments on real-world data demonstrate the effectiveness and efficiency of our proposed solutions over alternatives.

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 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.

Massive amounts of images textually annotated by different users are provided by social image websites, e.g., Flickr. Social images are always associated with various information, such as visual features, tags, and users. In this paper, we utilize hypergraph instead of ordinary graph to model social images, since relations among various information are more sophisticated than pairwise. Based on the hypergraph, we propose HIRT, a scalable image retrieval and tagging system, which uses Personalized PageRank to measure vertex similarity, and employs top-k search to support image retrieval and tagging. To achieve good scalability and efficiency, we develop parallel and approximate top-k search algorithms with quality guarantees. Experiments on a large Flickr dataset confirm the effectiveness and efficiency of our proposed system HIRT compared with existing state-of-the-art hypergraph based image retrieval system. In addition, our parallel and approximate top-k search methods are verified to be more efficient than the state-of-the-art methods and meanwhile achieve higher result quality.

Given a directed 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. The average of the personalized PageRank score of t with respect to each source node υ∈ V is exactly the PageRank score π( t ) of node t , which denotes the overall importance of node t in the graph. A heavy hitter of node t is a node whose contribution to π( t ) is above a φ fraction, where φ is a value between 0 and 1. Finding heavy hitters has important applications in link spam detection, classification of web pages, and friend recommendations.

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).

Twitter provides us a convenient channel to get access to the immediate information about major events. However, it is challenging to acquire a clean and complete set of event-related data due to the characteristics of tweets, eg short and noisy. In this paper, we propose a semi-supervised method to obtain high quality event-related tweets from Twitter stream, in terms of precision and recall. Specifically, candidate event-related tweets are selected based on a set of keywords. We propose to generate and update these keywords dynamically along the event development. To be included in this keyword set, words are evaluated based on single word properties, property based on co-occurred words, and changes of word importance over time. Our solution is capable of capturing keywords of emerging aspects or aspects with increasing importance along event evolvement. By leveraging keyword importance information and a few labeled tweets, we propose a semi-supervised expectation maximization process to identify event-related tweets. This process significantly reduces human effort in acquiring high quality tweets. Experiments on three real world datasets show that our solution outperforms state-of-the-art approaches by up to 10% in F1 measure.

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.

Influence maximization is a combinatorial optimization problem that finds important applications in viral marketing, feed recommendation, etc. Recent research has led to a number of scalable approximation algorithms for influence maximization, such as TIM + and IMM , and more recently, SSA and D-SSA . The goal of this paper is to conduct a rigorous theoretical and experimental analysis of SSA and D-SSA and compare them against the preceding algorithms. In doing so, we uncover inaccuracies in previously reported technical results on the accuracy and efficiency of SSA and D-SSA , which we set right. We also attempt to reproduce the original experiments on SSA and D-SSA , based on which we provide interesting empirical insights. Our evaluation confirms some results reported from the original experiments, but it also reveals anomalies in some other results and sheds light on the behavior of SSA and D-SSA in some important settings not considered previously. We also report on the performance of SSA-Fix , our modification to SSA in order to restore the approximation guarantee that was claimed for but not enjoyed by SSA . Overall, our study suggests that there exist opportunities for further scaling up influence maximization with approximation guarantees.

In a constrained shortest path (CSP) query, each edge in the road network is associated with both a length and a cost. Given an origin s , a destination t , and a cost constraint θ , the goal is to find the shortest path from s to t whose total cost does not exceed θ . Because exact CSP is NP-hard, previous work mostly focuses on approximate solutions. Even so, existing methods are still prohibitively expensive for large road networks. Two main reasons are (i) that they fail to utilize the special properties of road networks and (ii) that most of them process queries without indices; the few existing indices consume large amounts of memory and yet have limited effectiveness in reducing query costs. Motivated by this, we propose COLA , the first practical solution for approximate CSP processing on large road networks. COLA exploits the facts that a road network can be effectively partitioned, and that there exists a relatively small set of landmark vertices that commonly appear in CSP results. Accordingly, COLA indexes the vertices lying on partition boundaries, and applies an on-the-fly algorthm called α - Dijk for path computation within a partition, which effectively prunes paths based on landmarks. Extensive experiments demonstrate that on continent-sized road networks, COLA answers an approximate CSP query in sub-second time, whereas existing methods take hours. Interestingly, even without an index, the α-Dijk algorithm in COLA still outperforms previous solutions by more than an order of magnitude.

Personalized PageRank (PPR) computation is a fundamental operation in web search, social networks, and graph analysis. Given a graph G , a source s, and a target t, the PPR query Π( s, t ) returns the probability that a random walk on G starting from s terminates at t. Unlike global PageRank which can be effectively pre-computed and materialized, the PPR result depends on both the source and the target, rendering results materialization infeasible for large graphs. Existing indexing techniques have rather limited effectiveness; in fact, the current state-of-the-art solution, BiPPR, answers individual PPR queries without pre-computation or indexing, and yet it outperforms all previous index-based solutions. Motivated by this, we propose HubPPR, an effective indexing scheme for PPR computation with controllable tradeoffs for accuracy, query time, and memory consumption. The main idea is to pre-compute and index auxiliary information for selected hub nodes that are often involved in PPR processing. Going one step further, we extend HubPPR to answer top- k PPR queries, which returns the k nodes with the highest PPR values with respect to a source s , among a given set T of target nodes. Extensive experiments demonstrate that compared to the current best solution BiPPR, HubPPR achieves up to 10x and 220x speedup for PPR and top- k PPR processing, respectively, with moderate memory consumption. Notably, with a single commodity server, HubPPR answers a top- k PPR query in seconds on graphs with billions of edges, with high accuracy and strong result quality guarantees.

A public transportation network can often be modeled as a timetable graph where (i) each node represents a station; and (ii) each directed edge (u,v) is associated with a timetable that records the departure (resp. arrival) time of each vehicle at station u (resp. v). Several techniques have been proposed for various types of route planning on timetable graphs, e.g., retrieving the route from a node to another with the shortest travel time. These techniques, however, either provide insufficient query efficiency or incur significant space overheads.

Data deduplication stands as a building block for data integration and data cleaning. The state-of-the-art techniques focus on how to exploit crowdsourcing to improve the accuracy of deduplication. However, they either incur significant overheads on the crowd or offer inferior accuracy.

Reachability queries are a fundamental type of queries on graphs that find important applications in numerous domains. Although a plethora of techniques have been proposed for reachability queries, most of them require that the input graph is static, i.e., they are inapplicable to the {\em dynamic} graphs (e.g., social networks and the Semantic Web) commonly encountered in practice. There exist a few techniques that can handle dynamic graphs, but none of them can scale to sizable graphs without significant loss of efficiency. To address this deficiency, this paper presents a novel study on reachability indices for large dynamic graphs. We first introduce a general indexing framework that summarizes a family of reachability indices with the best performance among the existing techniques for static graphs. Then, we propose general and efficient algorithms for handling vertex insertions and deletions under the proposed framework. In addition, we show that our update algorithms can be used to improve the existing reachability techniques on static graphs, and we also propose a new approach for constructing a reachability index from scratch under our framework. We experimentally evaluate our solution on a large set of benchmark datasets, and we demonstrate that our solution not only supports efficient updates on dynamic graphs, but also provides even better query performance than the state-of-the-art techniques for static graphs.

This paper investigates two types of graph queries: single source distance (SSD) queries and single source shortest path (SSSP) queries. Given a node v in a graph G, an SSD query from v asks for the distance from $v$ to any other node in G, while an SSSP query retrieves the shortest path from v to any other node. These two types of queries find important applications in graph analysis, especially in the computation of graph measures. Most of the existing solutions for SSD and SSSP queries, however, require that the input graph fits in the main memory, which renders them inapplicable for the massive disk-resident graphs commonly used in web and social applications. There are several techniques that are designed to be I/O efficient, but they all focus on undirected and/or unweighted graphs, and they only offer sub-optimal query efficiency. To address the deficiency of existing work, this paper presents Highways-on-Disk (HoD), a disk-based index that supports both SSD and SSSP queries on directed and weighted graphs. The key idea of HoD is to augment the input graph with a set of auxiliary edges, and exploit them during query processing to reduce I/O and computation costs. We experimentally evaluate HoD on both directed and undirected real-world graphs with up to billions of nodes and edges, and we demonstrate that HoD significantly outperforms alternative solutions in terms of query efficiency.