Graph neural networks (GNNs) are deep learning models designed for graph-structured data that have achieved strong results across domains--social networks, knowledge graphs, bioinformatics, transportation, World Wide Web, and finance--on tasks such as node and graph classification, link prediction, entity resolution, question answering, recommendation, and fraud detection. Explaining the decisions of high-performing, yet ''black-box'' GNNs remains both challenging and essential. The initial five years have produced tremendous progress with many GNN explainers (e.g., GNNExplainer, PGExplainer, SubgraphX, PGMExplainer, GraphLime, GCFExplainer, CF2, GNN-LRP) that identify the influential nodes, edges, subgraphs, and features aiming to explain the output of GNNs.

Sparse Matrix-Matrix Multiplication (SpMM) is a fundamental operation in graph computing and analytics. However, the irregularity of real-world graphs poses significant challenges to achieving efficient SpMM for graph data on GPUs. Recently, the introduction of new efficient computing cores within GPUs offers new opportunities for acceleration. In this paper, we present HC-SpMM, a pioneering algorithm that leverages Hybrid GPU Cores (Tensor cores and CUDA cores) to accelerate SpMM for graphs. To adapt to the computing characteristics ofdifferent GPU cores, we investigate the impact of sparse graph features on the performance of different cores, develop a data partitioning technique for the graph adjacency matrix, and devise a novel strategy for intelligently selecting the most efficient cores for processing each submatrix. Additionally, we optimize it by considering memory access and thread utilization. To support complex graph computing workloads, we integrate HC-SpMM into the GNN training pipeline. Furthermore, we propose a kernel fusion strategy to enhance data reuse, as well as a cost-effective graph layout reorganization method to mitigate the irregularity of real-world graphs, better fitting the computational models of hybrid GPU cores. Extensive experiments on 14 real-world datasets demonstrate that HC-SpMM achieves an average speedup of 1.33× and 1.23× over state-of-the-art SpMM kernels and GNN frameworks.

Subspace clustering seeks to identify subspaces that segment a set of n data points into k (k« n) groups, which has emerged as a powerful tool for analyzing data from various domains, especially images and videos. Recently, several studies have demonstrated the great potential of subspace clustering models for partitioning vertices in attributed graphs, referred to as SCAG. However, these works either demand significant computational overhead for constructing the nxn self-expressive matrix, or fail to incorporate graph topology and attribute data into the subspace clustering framework effectively, and thus, compromise result quality.

Approximate Nearest Neighbor Search (ANNS) in high-dimensional spaces finds extensive applications in databases, information retrieval, recommender systems, etc. While graph-based methods have emerged as the leading solution for ANNS due to their superior query performance, they still face several challenges, such as struggling with local optima and redundant computations. These issues arise because existing methods (i) fail to fully exploit the topological information underlying the proximity graph G, and (ii) suffer from severe distribution mismatches between the base data and queries in practice. To this end, this paper proposes GATE, high-tier proximity Graph with Adaptive Topology and Query Awareness, as a lightweight and adaptive module atop the graph-based indexes to accelerate ANNS. Specifically, GATE formulates the critical problem to identify an optimal entry point in the proximity graph for a given query, facilitating faster online search. By leveraging the inherent clusterability of high-dimensional data, GATE first extracts a small set of hub nodes V as candidate entry points. Then, resorting to a contrastive learning-based two-tower model, GATE encodes both the structural semantics underlying G and the query-relevant features into the latent representations of these hub nodes V. A navigation graph index on V is further constructed to minimize the model inference overhead. Extensive experiments demonstrate that GATE achieves a 1.2-2.0X speed-up in query performance compared to state-of-the-art graph-based indexes.

Maximum Inner Product Search (MIPS) is a fundamental challenge in machine learning and information retrieval, particularly in high-dimensional data applications. Existing approaches to MIPS either rely solely on Inner Product (IP) similarity, which faces issues with local optima and redundant computations, or reduce the MIPS problem to the Nearest Neighbor Search under the Euclidean metric via space projection, leading to topology destruction and information loss. Despite the divergence of the two paradigms, we argue that there is no inherent binary opposition between IP and Euclidean metrics. By stitching IP and Euclidean in the design of indexing and search algorithms, we can significantly enhance MIPS performance. Specifically, this paper explores the theoretical and empirical connections between these two metrics from the MIPS perspective. Our investigation, grounded in graph-based search, reveals that different indexing and search strategies offer distinct advantages for MIPS, depending on the underlying data topology. Building on these insights, we introduce a novel graph-based index called Metric-Amphibious Graph (MAG) and a corresponding search algorithm, Adaptive Navigation with Metric Switch (ANMS). To facilitate parameter tuning for optimal performance, we identify three statistical indicators that capture essential data topology properties and correlate strongly with parameter tuning. Extensive experiments on 12 real-world datasets demonstrate that MAG outperforms existing state-of-the-art methods, achieving up to 4x search speedup while maintaining adaptability and scalability.

Approximate nearest neighbor search (ANNS) in high-dimensional vector spaces has a wide range of real-world applications. Numerous methods have been proposed to handle ANNS efficiently, while graph-based indexes have gained prominence due to their high accuracy and efficiency. However, the indexing overhead of graph-based indexes remains substantial. With exponential growth in data volume and increasing demands for dynamic index adjustments, this overhead continues to escalate, posing a critical challenge. In this paper, we introduce Tagore, a fas T library a ccelerated by G PUs f or graph ind e xing, which has powerful capabilities of constructing refinement-based graph indexes such as NSG and Vamana. We first introduce GNN-Descent, a GPU-specific algorithm for efficient k-Nearest Neighbor (k-NN) graph initialization. GNN-Descent speeds up the similarity comparison by a two-phase descent procedure and enables highly parallelized neighbor updates. Next, aiming to support various k-NN graph pruning strategies, we formulate a universal pruning procedure termed CFS and devise two generalized GPU kernels for parallel processing complex dependencies in neighbor relationships. For large-scale datasets exceeding GPU memory capacity, we propose an asynchronous GPU-CPU-disk indexing framework with a cluster-aware caching mechanism to minimize the I/O pressure on the disk. Extensive experiments on 7 real-world datasets exhibit that Tagore achieves 1.32x to 112.79x speedup while maintaining the index quality.

Community search on attributed graphs (CSAG) is a fundamental topic in graph data mining. Given an attributed graph G and a query node q , CSAG seeks a structural- and attribute-cohesive subgraph from G that contains q . Exact methods based on graph traversal are time-consuming, especially for large graphs. Approximate methods improve efficiency by pruning the search space with heuristics but still take hundreds of milliseconds to tens of seconds to respond, hindering their use in time-sensitive applications. Moreover, pruning strategies are typically tailored to specific algorithms and their cohesiveness metrics, making them difficult to generalize. To address this, we study a general approach to accelerate various CSAG methods. We first present a proximity graph-based, cohesiveness-aware hierarchical index that accommodates different cohesiveness metrics. Then, we present two optimizations to enhance the index's navigability and reliability. Finally, we design a compressed storage structure for space-efficient indexing. Experiments on real-world datasets show that integrating our index with existing mainstream CSAG methods results in an average 30.7× speedup while maintaining a comparable or even better attribute cohesiveness.

In high-dimensional vector spaces, Approximate Nearest Neighbor Search (ANNS) is a key component in database and artificial intelligence infrastructures. Graph-based ANNS methods, particularly HNSW, have emerged as leading solutions, offering an impressive trade-off between search efficiency and accuracy. Many vector databases utilize graph indexes as their core algorithms, benefiting from various optimizations to enhance search performance. However, the high indexing time associated with graph algorithms poses a significant challenge, especially given the increasing volume of data, query processing complexity, and dynamic index maintenance demand. This has rendered indexing time a critical performance metric for users. In this paper, we comprehensively analyze the underlying causes of the low graph indexing efficiency on modern CPUs, identifying that distance computation dominates indexing time, primarily due to high memory access latency and suboptimal arithmetic operation efficiency. We demonstrate that distance comparisons during index construction can be effectively performed using compact vector codes at an appropriate compression error. Drawing from insights gained through integrating existing compact coding methods in the graph indexing process, we propose a novel compact coding strategy, named Flash, designed explicitly for graph indexing and optimized for modern CPU architectures. By minimizing random memory accesses and maximizing the utilization of SIMD (Single Instruction, Multiple Data) instructions, Flash significantly enhances cache hit rates and arithmetic operations. Extensive experiments conducted on eight real-world datasets, ranging from ten million to one billion vectors, exhibit that Flash achieves a speedup of 10.4× to 22.9× in index construction efficiency, while maintaining or improving search performance.

Maximum Inner Product Search (MIPS) for high-dimensional vectors is pivotal across databases, information retrieval, and artificial intelligence. Existing methods either reduce MIPS to Nearest Neighbor Search (NNS) while suffering from harmful vector space transformations, or attempt to tackle MIPS directly but struggle to mitigate redundant computations due to the absence of the triangle inequality. This paper presents a novel theoretical framework that equates MIPS with NNS without requiring space transformation, thereby allowing us to leverage advanced graph-based indices for NNS and efficient edge pruning strategies, significantly reducing unnecessary computations. Despite a strong baseline set by our theoretical analysis, we identify and address two persistent challenges to further refine our method: the introduction of the P roximity Graph with S pherical P athway (PSP), designed to mitigate the issue of MIPS solutions clustering around large-norm vectors, and the implementation of A daptive E arly T ermination (AET), which efficiently curtails the excessive exploration once an accuracy bottleneck is reached. Extensive experiments reveal that our method is superior to existing state-of-the-art techniques in search efficiency, scalability, and practical applicability. Compared with state-of-the-art graph-based methods, it achieves an average 35% speed-up in query processing and a 3× reduction in index size. Notably, our approach has been validated and deployed in the search engines of Shopee, a well-known online shopping platform. Our code and an industrial-scale dataset for offline evaluation will also be released to address the absence of e-commerce data in public benchmarks.

Given an attributed graph$G$and a query node$q$, Community Search over Attributed Graphs (CS-AG) aims to find a structure- and attribute-cohesive subgraph from$G$that contains$q$. Although CS-AG has been widely studied, they still face three challenges. (1) Exact methods based on graph traversal are time-consuming, especially for large graphs. Some tailored indices can improve efficiency, but introduce nonnegligible storage and maintenance overhead. (2) Approximate methods with a loose approximation ratio only provide a coarse-grained evaluation of a community's quality, rather than a reliable evaluation with an accuracy guarantee in runtime. (3) Attribute cohesiveness metrics often ignores the important correlation with the query node$q$. We formally define our CS-AG problem atop a$q- \mathbf{centric}$attribute cohesiveness metric considering both textual and numerical attributes, for$k-\mathbf{core}$model on homogeneous graphs. We show the problem is NP-hard. To solve it, we first propose an exact baseline with three pruning strategies. Then, we propose an index-free sampling-estimation-based method to quickly return an approximate community with an accuracy guarantee, in the form of a confidence interval. Once a good result satisfying a user-desired error bound is reached, we terminate it early. We extend it to heterogeneous graphs,$k-\mathbf{truss}$model, and size-bounded CS. Comprehensive experimental studies on ten real-world datasets show its superiority, e.g., at least$1.54\times (41.1\times$on average) faster in response time and a reliable relative error (within a user-specific error bound) of attribute cohesiveness is achieved.

Subgraph matching has garnered increasing attention for its diverse real-world applications. Given the dynamic nature of real-world graphs, addressing evolving scenarios with-out incurring prohibitive overheads has been a focus of research. However, existing approaches for dynamic subgraph matching often proceed serially, retrieving incremental matches for each updated edge individually. This approach falls short when handling batch data updates, leading to a decrease in system throughput. Leveraging the parallel processing power of GPUs, which can execute a massive number of cores simultaneously, has been widely recognized for performance acceleration in various domains. Surprisingly, systematic exploration of subgraph matching in the context of batch-dynamic graphs, particularly on a GPU platform, remains untouched. In this paper, we bridge this gap by introducing an efficient framework, GAMMA (GPU-Accelerated Batch-Dynamic Subgraph Matching). Our approach features a DFS-based warp-centric batch-dynamic subgraph matching algorithm. To ensure load balance in the DFS-based search, we propose warp-level work stealing via shared memory. Additionally, we introduce coalesced search to reduce redundant computations. Comprehensive experiments demonstrate the superior performance of GAMMA. Compared to state-of-the-art algorithms, GAMMA showcases a performance improvement up to hundreds of times.

Counting ($p$, q)-bicliques in bipartite graphs poses a foundational challenge with broad applications, from densest sub-graph discovery in algorithmic research to personalized content recommendation in practical scenarios. Despite its significance, current leading ($p$, q)-biclique counting algorithms fall short, particularly when faced with larger graph sizes and clique scales. Fortunately, the problem's inherent structure, allowing for the independent counting of each biclique starting from every vertex, combined with a substantial set intersections, makes it highly amenable to parallelization. Recent successes in GPU-accelerated algorithms across various domains motivate our exploration into harnessing the parallelism power of GPUs to efficiently address the ($p$, q)-biclique counting challenge. We introduce GBC (GPU-based Biclique Counting), a novel approach designed to enable efficient and scalable ($p$, q)-biclique counting on GPUs. To address major bottleneck arising from redundant comparisons in set intersections (occupying an average of 90% of the runtime), we introduce a novel data structure that hashes adjacency lists into truncated bitmaps to enable efficient set intersection on GPUs via bit-wise AND operations. Our in-novative hybrid DFS-BFS exploration strategy further enhances thread utilization and effectively manages memory constraints. A composite load balancing strategy, integrating pre-runtime and runtime workload allocation, ensures equitable distribution among threads. Additionally, we employ vertex reordering and graph partitioning strategies for improved compactness and scalability. Experimental evaluations on eight real-life and two synthetic datasets demonstrate that GBC outperforms state-of-the-art algorithms by a substantial margin. In particular, GBC achieves an average speedup of$497.8\times$, with the largest instance achieving a remarkable$1217.7\times$speedup when$p=q=8$.

We investigate the problem of multimodal search of target modality, where the task involves enhancing a query in a specific target modality by integrating information from auxiliary modalities. The goal is to retrieve relevant objects whose contents in the target modality match the specified multimodal query. The paper first introduces two baseline approaches that integrate techniques from the Database, Information Retrieval, and Computer Vision communities. These baselines either merge the results of separate vector searches for each modality or perform a single-channel vector search by fusing all modalities. However, both baselines have limitations in terms of efficiency and accuracy as they fail to adequately consider the varying importance of fusing information across modalities. To overcome these limitations, the paper proposes a novel framework, Multimodal Search of Target Modality, called MUST. Our framework employs a hybrid fusion mechanism, combining different modalities at multiple stages. Notably, we leverage vector weight learning to determine the importance of each modality, thereby enhancing the accuracy of joint similarity measurement. Additionally, the proposed framework utilizes a fused proximity graph index, enabling efficient joint search for multimodal queries. MUST offers several other advantageous properties, including a plug-gable design to integrate any advanced embedding techniques, user flexibility to customize weight preferences, and modularized index construction. Extensive experiments on real-world datasets demonstrate the superiority of MUST over the baselines in terms of both search accuracy and efficiency. Our framework achieves over 10× faster search times while attaining an average of 93% higher accuracy. Furthermore, MUST exhibits scalability to datasets containing more than 10 million data elements.

Given a graph G and a query node q, community search (CS) aims to find a structurally cohesive subgraph from G that contains q. CS is widely used in many real-world applications, such as online recommendation and expert finding. Recently, the rise of learning-based CS methods has garnered extensive research interests, showcasing the promising potential of neural solutions. However, there remains room for optimization: (1) They initialize node features via classical methods, e.g., one-hot, random, and position encoding, which may fall short in capturing valuable community cohesiveness-related features. (2) The reliance on GCN or GCN-like models poses challenges in scaling to large graphs. (3) Existing methods do not adapt well to dynamic graphs, often requiring retraining from scratch. To handle this, we present CSFormer, a scalable CS based on Graph Transformer. First, we present a novel l-hop neighborhood community vector based on n-order h-index to represent each node's community features, generating a sequence of feature vectors by varying the neighborhood scope l. Then, we build a Transformer backbone to learn a good graph embedding that carries rich community features, based on which we perform a prediction-filtering-based online CS to efficiently return a community of q. We extend CSFormer to dynamic graphs and various community models. Extensive experiments on seven real-world graphs show our solution's superiority on effectiveness, e.g., we attain an average improvement of 20.6% in F1-score compared to the latest competitors.

Generating explanations for graph neural networks (GNNs) has been studied to understand their behaviors in analytical tasks such as graph classification. Existing approaches aim to understand the overall results of GNNs rather than providing explanations for specific class labels of interest, and may return explanation structures that are hard to access, nor directly queryable. We propose GVEX, a novel paradigm that generates Graph Views for GNN EXplanation. (1) We design a two-tier explanation structure called explanation views. An explanation view consists of a set of graph patterns and a set of induced explanation subgraphs. Given a database G of multiple graphs and a specific class label l assigned by a GNN-based classifier M, it concisely describes the fraction of G that best explains why l is assigned by M. (2) We propose quality measures and formulate an optimization problem to compute optimal explanation views for GNN explanation. We show that the problem is Σ2P-hard. (3) We present two algorithms. The first one follows an explain-and-summarize strategy that first generates high-quality explanation subgraphs which best explain GNNs in terms of feature influence maximization, and then performs a summarization step to generate patterns. We show that this strategy provides an approximation ratio of 1/2. Our second algorithm performs a single-pass to an input node stream in batches to incrementally maintain explanation views, having an anytime quality guarantee of 1/4-approximation. Using real-world benchmark data, we experimentally demonstrate the effectiveness, efficiency, and scalability of GVEX. Through case studies, we showcase the practical applications of GVEX.

High-dimensional vector similarity search (HVSS) is gaining prominence as a powerful tool for various data science and AI applications. As vector data scales up, in-memory indexes pose a significant challenge due to the substantial increase in main memory requirements. A potential solution involves leveraging disk-based implementation, which stores and searches vector data on high-performance devices like NVMe SSDs. However, implementing HVSS for data segments proves to be intricate in vector databases where a single machine comprises multiple segments for system scalability. In this context, each segment operates with limited memory and disk space, necessitating a delicate balance between accuracy, efficiency, and space cost. Existing disk-based methods fall short as they do not holistically address all these requirements simultaneously. In this paper, we present Starling, an I/O-efficient disk-resident graph index framework that optimizes data layout and search strategy within the segment. It has two primary components: (1) a data layout incorporating an in-memory navigation graph and a reordered disk-based graph with enhanced locality, reducing the search path length and minimizing disk bandwidth wastage; and (2) a block search strategy designed to minimize costly disk I/O operations during vector query execution. Through extensive experiments, we validate the effectiveness, efficiency, and scalability of Starling. On a data segment with 2GB memory and 10GB disk capacity, Starling can accommodate up to 33 million vectors in 128 dimensions, offering HVSS with over 0.9 average precision and top-10 recall rate, and latency under 1 millisecond. The results showcase Starling's superior performance, exhibiting 43.9x higher throughput with 98% lower query latency compared to state-of-the-art methods while maintaining the same level of accuracy.

Similarity search, the task of identifying objects most similar to a given query object under a specific metric, has gathered significant attention due to its practical applications. However, the absence of coordinate information to accelerate similarity search and the high computational cost of measuring object similarity hinder the efficiency of existing CPU-based methods. Additionally, these methods struggle to meet the demand for high throughput data management. To address these challenges, we propose GTS, a GPU-based tree index designed for the parallel processing of similarity search in general metric spaces, where only the distance metric for measuring object similarity is known. The GTS index utilizes a pivot-based tree structure to efficiently prune objects and employs list tables to facilitate GPU computing. To efficiently manage concurrent similarity queries with limited GPU memory, we have developed a two-stage search method that combines batch processing and sequential strategies to optimize memory usage. The paper also introduces an effective update strategy for the proposed GPU-based index, encompassing streaming data updates and batch data updates. Additionally, we present a cost model to evaluate search performance. Extensive experiments on five real-life datasets demonstrate that GTS achieves efficiency gains of up to two orders of magnitude over existing CPU baselines and up to 20x efficiency improvements compared to state-of-the-art GPU-based methods.

Recent years have witnessed a growing trend toward employing deep reinforcement learning (Deep-RL) to derive heuristics for combinatorial optimization (CO) problems on graphs. Maximum Coverage Problem (MCP) and its probabilistic variant on social networks, Influence Maximization (IM), have been particularly prominent in this line of research. In this paper, we present a comprehensive benchmark study that thoroughly investigates the effectiveness and efficiency of five recent Deep-RL methods for MCP and IM. These methods were published in top data science venues, namely S2V-DQN, Geometric-QN, GCOMB, RL4IM, and LeNSE. Our findings reveal that, across various scenarios, the Lazy Greedy algorithmconsistently outperforms allDeep-RL methods for MCP. In the case of IM, theoretically sound algorithms like IMM and OPIM demonstrate superior performance compared to Deep-RL methods in most scenarios. Notably, we observe anabnormal phenomenonin IM problem where Deep-RL methods slightly outperform IMM and OPIM when the influence spread nearly does not increase as the budget increases. Furthermore, our experimental results highlight common issues when applying Deep-RL methods to MCP and IM in practical settings. Finally, we discuss potential avenues for improving Deep-RL methods. Our benchmark study sheds light on potential challenges in current deep reinforcement learning research for solving combinatorial optimization problems.

Retrieval-augmented Large Language Models (LLMs) have reshaped traditional query-answering systems, offering unparalleled user experiences. However, existing retrieval techniques often struggle to handle multi-modal query contexts. In this paper, we present an interactive M ulti-modal Q uery A nswering (MQA) system, empowered by our newly developed multi-modal retrieval framework and navigation graph index, integrated with cutting-edge LLMs. It comprises five core components: Data Preprocessing, Vector Representation, Index Construction, Query Execution, and Answer Generation, all orchestrated by a dedicated coordinator to ensure smooth data flow from input to answer generation. One notable aspect of MQA is its utilization of contrastive learning to assess the significance of different modalities, facilitating precise measurement of multimodal information similarity. Furthermore, the system achieves efficient retrieval through our advanced navigation graph index, refined using computational pruning techniques. Another highlight of our system is its pluggable processing framework, allowing seamless integration of embedding models, graph indexes, and LLMs. This flexibility provides users diverse options for gaining insights from their multi-modal knowledge base. A preliminary video introduction of MQA is available at https://youtu.be/xvUuo2ZIqWk.

Given a graph G and a query node q, community search (CS) seeks a cohesive subgraph from G that contains q. CS has gained much research interests recently. In the database research community, researchers aim to find the most cohesive subgraph satisfying a specific community model (e.g., k-core or k-truss) via graph traversal. These works obtain good precision, however suffering from the low efficiency issue. In the AI research community, a new thought of using the deep learning model to support CS without relying on graph traversal emerges. Supervised end-to-end models using GCN are presented, which perform efficiently, but leave a large room for precision improvement. None of them can achieve a good balance between the efficiency and effectiveness. This motivates our solution: First, we present an offline community-injected graph embedding method to preserve the community’s cohesiveness features into the learned node representations. Second, we resort to a proximity graph (PG) built from node representations, to quickly return the community online. Moreover, we develop a self-augmented method based on KL divergence to further optimize node representations. Extensive experiments on seven real-world graphs show our solution’s superiority on effectiveness (at least 39.3% improvement) and efficiency (one to two orders of magnitude faster).

We investigate the novel problem of voting-based opinion maximization in a social network: Find a given number of seed nodes for a target campaigner, in the presence of other competing campaigns, so as to maximize a voting-based score for the target campaigner at a given time horizon.The bulk of the influence maximization literature assumes that social network users can switch between only two discrete states, inactive and active, and the choice to switch is frozen upon one-time activation. In reality, even when having a preferred opinion, a user may not completely despise the other opinions, and the preference level may vary over time due to social influence. To this end, we employ models rooted in opinion formation and diffusion, and use several voting-based scores to determine a user’s vote for each of the multiple campaigners at a given time horizon.Our problem is NP-hard and non-submodular for various scores. We design greedy seed selection algorithms with quality guarantees for our scoring functions via sandwich approximation. To improve the efficiency, we develop random walk and sketch-based opinion computation, with quality guarantees. Empirical results validate our effectiveness, efficiency, and scalability.

Computing the densest subgraph is a primitive graph operation with critical applications in detecting communities, events, and anomalies in biological, social, Web, and financial networks. In this paper, we study the novel problem of Most Probable Densest Subgraph (MPDS) discovery in uncertain graphs: Find the node set that is the most likely to induce a densest subgraph in an uncertain graph. We further extend our problem by considering various notions of density, e.g., clique and pattern densities, studying the top-k MPDSs, and finding the node set with the largest containment probability within densest subgraphs. We show that it is #P-hard to compute the probability of a node set inducing a densest subgraph. We then devise sampling-based efficient algorithms, with end-to-end accuracy guarantees, to compute the MPDS. Our thorough experimental results and real-world case studies on brain and social networks validate the effectiveness, efficiency, and usefulness of our solution.

Bipartite graphs characterize relationships between two different sets of entities, like actor-movie, user-item, and author-paper. The butterfly, a 4-vertices 4-edges (2,2)-biclique, is the simplest cohesive motif in a bipartite graph and is the fundamental component of higher-order substructures. Counting and enumerating the butterflies offer significant benefits across various applications, including fraud detection, graph embedding, and community search. While the corresponding motif, the triangle, in the unipartite graphs has been widely studied in both static and temporal settings, the extension of butterfly to temporal bipartite graphs remains unexplored. In this paper, we investigate the temporal butterfly counting and enumeration problem: count and enumerate the butterflies whose edges establish following a certain order within a given duration. Towards efficient computation, we devise a non-trivial baseline rooted in the state-of-the-art butterfly counting algorithm on static graphs, further, explore the intrinsic property of the temporal butterfly, and develop a new optimization framework with a compact data structure and effective priority strategy. The time complexity is proved to be significantly reduced without compromising on space efficiency. In addition, we generalize our algorithms to practical streaming settings and multi-core computing architectures. Our extensive experiments on 11 large-scale real-world datasets demonstrate the efficiency and scalability of our solutions.

The social network host has knowledge of the network structure and user characteristics and can earn a profit by providing merchants with viral marketing campaigns. We investigate the problem of host profit maximization by leveraging performance incentives and user flexibility. To incentivize the host's performance, we propose setting a desired influence threshold that would allow the host to receive full payment, with the possibility of a small bonus for exceeding the threshold. Unlike existing works that assume a user's choice is frozen once they are activated, we introduce the Dynamic State Switching model to capture "comparative shopping" behavior from an economic perspective, in which users have the flexibilities to change their minds about which product to adopt based on the accumulated influence and propaganda strength of each product. In addition, the incentivized cost of a user serving as an influence source is treated as a negative part of the host's profit. The host profit maximization problem is NP-hard, submodular, and non-monotone. To address this challenge, we propose an efficient greedy algorithm and devise a scalable version with an approximation guarantee to select the seed sets. As a side contribution, we develop two seed allocation algorithms to balance the distribution of adoptions among merchants with small profit sacrifice. Through extensive experiments on four real-world social networks, we demonstrate that our methods are effective and scalable.

Expert finding is crucial for a wealth of applications in both academia and industry. Given a user query and trove of academic papers, expert finding aims at retrieving the most relevant experts for the query, from the academic papers. Existing studies focus on embedding-based solutions that consider academic papers’ textual semantic similarities to a query via document representation and extract the top- n experts from the most similar papers. Beyond implicit textual semantics, however, papers’ explicit relationships (e.g., co-authorship) in a heterogeneous graph (e.g., DBLP) are critical for expert finding, because they help improve the representation quality. Despite their importance, the explicit relationships of papers generally have been ignored in the literature. In this article, we study expert finding on heterogeneous graphs by considering both the explicit relationships and implicit textual semantics of papers in one model. Specifically, we define the cohesive ( k , 𝒫)-core community of papers w.r.t. a meta-path 𝒫 (i.e., relationship) and propose a ( k , 𝒫)-core based document embedding model to enhance the representation quality. Based on this, we design a proximity graph-based index (PG-Index) of papers and present a threshold algorithm (TA)-based method to efficiently extract top- n experts from papers returned by PG-Index. We further optimize our approach in two ways: (1) we boost effectiveness by considering the ( k , 𝒫)-core community of experts and the diversity of experts’ research interests, to achieve high-quality expert representation from paper representation; and (2) we streamline expert finding, going from “extract top- n experts from top- m ( m> n ) semantically similar papers” to “directly return top- n experts”. The process of returning a large number of top- m papers as intermediate data is avoided, thereby improving the efficiency. Extensive experiments using real-world datasets demonstrate our approach’s superiority.

Finding relevant experts in specified areas is often crucial for a wide range of applications in both academia and industry. Given a user input query and a large amount of academic knowledge (e.g., academic papers), expert finding aims to find and rank the experts who are most relevant to the given query, from the academic knowledge. Existing studies mainly focus on the embedding-based solutions that (1) consider academic papers' textual semantic similarities to a given query through document representation models and (2) extract the$\mathbf{top-}n$experts with the greatest similarities. Beyond the implicit textual semantics of papers, however, the papers' explicit relationships (e.g., co-authorship, citation, and same-topic relationship) in a heterogeneous academic graph (e.g., DBLP) are critical for document representation, insofar as they help improve the expert finding quality. Despite their importance, the explicit relationships of papers generally have been ignored in the literature. In this paper, we study the academic expert finding on heterogeneous graphs by considering the explicit relationships besides the implicit textual semantics of papers in one representation model. Specifically, we first define the$(k,\mathcal{P})$-core to denote a cohesive community of papers that are closely connected via a meta-path$\mathcal{P}(\mathcal{P}$indicates the different relationships of papers). We then propose an offline$(k,\mathcal{P})$-core based document embedding model to capture papers' various explicit relationships for representation. Moreover, by using papers' embeddings, we present an online threshold algorithm (TA)-based method to efficiently return top-n experts via a carefully designed proximity graph-based index (PG-Index). We extend our approach to support multiple relationships simultaneously for representation. Extensive experiments over real-world datasets demonstrate the effectiveness and efficiency of our approach.

Graph summarization is beneficial in a wide range of applications, such as visualization, interactive and exploratory analysis, approximate query processing, reducing the on-disk storage footprint, and graph processing in modern hardware. However, the bulk of the literature on graph summarization surprisingly overlooks the possibility of having edges of different types. In this article, we study the novel problem of producing summaries of multi-relation networks, i.e., graphs where multiple edges of different types may exist between any pair of nodes. Multi-relation graphs are an expressive model of real-world activities, in which a relation can be a topic in social networks, an interaction type in genetic networks, or a snapshot in temporal graphs. The first approach that we consider for multi-relation graph summarization is a two-step method based on summarizing each relation in isolation, and then aggregating the resulting summaries in some clever way to produce a final unique summary. In doing this, as a side contribution, we provide the first polynomial-time approximation algorithm based on the k -Median clustering for the classic problem of lossless single-relation graph summarization. Then, we demonstrate the shortcomings of these two-step methods, and propose holistic approaches, both approximate and heuristic algorithms, to compute a summary directly for multi-relation graphs. In particular, we prove that the approximation bound of k -Median clustering for the single relation solution can be maintained in a multi-relation graph with proper aggregation operation over adjacency matrices corresponding to its multiple relations. Experimental results and case studies (on co-authorship networks and brain networks) validate the effectiveness and efficiency of the proposed algorithms.

Network reliability measures the probability that a target node is reachable from a source node in an uncertain graph, i.e., a graph where every edge is associated with a probability of existence. In this paper, we investigate the novel and fundamental problem of adding a small number of edges in the uncertain network for maximizing the reliability between a given pair of nodes. We study the$\mathbf {NP}$-hardness and the approximation hardness of our problem, and design effective, scalable solutions. Furthermore, we consider extended versions of our problem (e.g., multiple source and target nodes can be provided as input) to support and demonstrate a wider family of queries and applications, including sensor network reliability maximization and social influence maximization. Experimental results validate the effectiveness and efficiency of the proposed algorithms.

Network reliability measures the probability that a target node is reachable from a source node in an uncertain graph, i.e., a graph where every edge is associated with a probability of existence. In this paper, we investigate the novel and fundamental problem of adding a small number of edges in the uncertain network for maximizing the reliability between a given pair of nodes. We study the NP-hardness and the approximation hardness of our problem, and design effective, scalable solutions. Furthermore, we consider extended versions of our problem (e.g., multiple source and target nodes can be provided as input) to support and demonstrate a wider family of queries and applications, including sensor network reliability maximization and social influence maximization.

Uncertain, or probabilistic, graphs have been increasingly used to represent noisy linked data in many emerging applications, and have recently attracted the attention of the database research community. A fundamental problem on uncertain graphs is the s-t reliability, which measures the probability that a target node t is reachable from a source node s in a probabilistic (or uncertain) graph, i.e., a graph where every edge is assigned a probability of existence. Due to the inherent complexity of the s-t reliability estimation problem (#P-hard), various sampling and indexing based efficient algorithms were proposed in the literature. However, since they have not been thoroughly compared with each other, it is not clear whether the later algorithm outperforms the earlier ones. More importantly, the comparison framework, datasets, and metrics were often not consistent (e.g., different convergence criteria were employed to find the optimal number of samples) across these works. We address this serious concern by re-implementing six state-of-the-art s-t reliability estimation methods in a common system and code base, using several medium and large-scale, real-world graph datasets, identical evaluation metrics, and query workloads. Through our systematic and in-depth analysis of experimental results, we report surprising findings, such as many follow-up algorithms can actually be several orders of magnitude inefficient, less accurate, and more memory intensive compared to the ones that were proposed earlier. We conclude by discussing our recommendations on the road ahead.

-1mmWe study the novel problem of jointly finding the top- k seed nodes and the top- r relevant tags for targeted influence maximization in a social network. The bulk of the research on influence maximization assumes that the influence diffusion probabilities across edges are fixed, and the top- k seed users are identified to maximize the cascade in the entire graph. However, in real-world applications, edge probabilities typically depend on the information being cascaded, e.g., in social influence networks, the probability that a tweet of some user will be re-tweeted by her followers depends on whether the tweet contains specific hashtags. In addition, a campaigner often has a specific group of target customers in mind. In this work, we model such practical constraints, and investigate the novel problem of jointly finding the top-k seed nodes and the top- r relevant tags that maximize the influence inside a target set of users. Due to the hardness of the influence maximization problem, we develop heuristic solutions --- with smart indexing, iterative algorithms, and good initial conditions, which target high-quality, efficiency, and scalability. -1mm

This paper demonstrates PERC --- our system for crowdsourced entity resolution with human errors. Entity Resolution (ER) is a critical step in data cleaning and analytics. Although many machine-based methods existed for ER task, crowdsourcing is becoming increasingly important since humans can provide more insightful information for complex tasks, e.g., clustering of images and natural language processing. However, human workers still make mistakes due to lack of domain expertise or seriousness, ambiguity, or even malicious intent. To this end, we present a system, called PERC (probabilistic entity resolution with crowd errors), which adopts an uncertain graph model to address the entity resolution problem with noisy crowd answers. Using our framework, the problem of ER becomes equivalent to finding the maximum-likelihood clustering. In particular, we propose a novel metric called "reliability" to measure the quality of a clustering, which takes into account both the connected-ness inside and across all clusters. PERC then automatically selects the next question to ask the crowd that maximally increases the "reliability" of the current clustering. This demonstration highlights (1) a reliability-based next crowd-sourcing framework for crowdsourced ER, which does not require any user-defined threshold, and no apriori information about the error rate of the crowd workers, (2) it improves the ER quality by 15% and reduces the crowdsourcing cost by 50% compared to state-of-the-art methods, and (3) its GUI can interact with users to help them compare different crowdsourced ER algorithms, their intermediate ER results as they progress, and their selected next crowdsourcing questions in a user-friendly manner. Our demonstration video is at: https://www.youtube.com/watch?v=rQ7nu3b8zXY.

Crowdsourcing is becoming increasingly important in entity resolution tasks due to their inherent complexity such as clustering of images and natural language processing. Humans can provide more insightful information for these difficult problems compared to machine-based automatic techniques. Nevertheless, human workers can make mistakes due to lack of domain expertise or seriousness, ambiguity, or even due to malicious intents. The bulk of literature usually deals with human errors via majority voting or by assigning a universal error rate over crowd workers. However, such approaches are incomplete, and often inconsistent, because the expertise of crowd workers are diverse with possible biases, thereby making it largely inappropriate to assume a universal error rate for all workers over all crowdsourcing tasks. We mitigate the above challenges by considering an uncertain graph model, where the edge probability between two records A and B denotes the ratio of crowd workers who voted YES on the question if A and B are same entity. To reflect independence across different crowdsourcing tasks, we apply the notion of possible worlds, and develop parameter-free algorithms for both next crowdsourcing and entity resolution tasks. In particular, for next crowdsourcing, we identify the record pair that maximally increases the reliability of the current clustering. Since reliability takes into account the connected-ness inside and across all clusters, this metric is more effective in deciding next questions, in comparison with state-of-the-art works, which consider local features, such as individual edges, paths, or nodes to select next crowdsourcing questions. Based on detailed empirical analysis over real-world datasets, we find that our proposed solution, PERC (probabilistic entity resolution with imperfect crowd) improves the quality by 15% and reduces the overall cost by 50% for the crowdsourcing-based entity resolution.