Bipartite graphs serve as a natural model for representing relationships between two different types of entities. When analyzing bipartite graphs, butterfly counting is a fundamental research problem that aims to count the number of butterflies (i.e., 2x2 bicliques) in a given bipartite graph. While this problem has been extensively studied in the literature, existing algorithms usually necessitate access to a large portion of the entire graph, presenting challenges in real scenarios where graphs are extremely large and I/O costs are expensive. In this paper, we study the butterfly counting problem under the query model, where the following query operations are permitted: degree query, neighbor query, and vertex-pair query. We propose TLS, a practical two-level sampling algorithm that can estimate the butterfly count accurately while accessing only a limited graph structure, achieving significantly lower query costs under the standard query model. TLS also incorporates several key techniques to control the variance, including "small-degree-first sampling" and "wedge sampling via small subsets". To ensure theoretical guarantees, we further introduce two novel techniques: "heavy-light partition" and "guess-and-prove", integrated into TLS. With these techniques, we prove that the algorithm can achieve a (1+eps) accuracy for any given approximation parameter 0 < eps < 1 on general bipartite graphs with a promised time and query complexity. In particular, the promised time is sublinear when the input graph is dense enough. Extensive experiments on 15 datasets demonstrate that TLS delivers robust estimates with up to three orders of magnitude lower query costs and runtime compared to existing solutions.

Bipartite graphs are commonly used to model relationships between two distinct entities in real-world applications, such as user-product interactions, user-movie ratings and collaborations between authors and publications. A butterfly (a 2×2 bi-clique) is a critical substructure in bipartite graphs, playing a significant role in tasks like community detection, fraud detection, and link prediction. As more real-world data is presented in a streaming format, efficiently counting butterflies in streaming bipartite graphs has become increasingly important. However, most existing algorithms typically assume that duplicate edges are absent, which is hard to hold in real-world graph streams, as a result, they tend to sample edges that appear multiple times, leading to inaccurate results. The only algorithm designed to handle duplicate edges is FABLE, but it suffers from significant limitations, including high variance, substantial time complexity, and memory inefficiency due to its reliance on a priority queue. To overcome these limitations, we introduce DEABCpro (Duplicate-Edge-Aware Butterfly Counting), an innovative method that uses bucket-based priority sampling to accurately estimate the number of butterflies, accounting for duplicate edges. Compared to existing methods, DEABCpro significantly reduces memory usage by storing only the essential sampled edge data while maintaining high accuracy. We provide rigorous proofs of the unbiasedness and variance bounds for DEABCpro, ensuring they achieve high accuracy. We compare DEABCpro with state-of-the-art algorithms on real-world streaming bipartite graphs. The results show that our DEABCpro outperforms existing methods in memory efficiency and accuracy, while also achieving significantly higher throughput.

Account risk detection, which aims to identify accounts at forced liquidation risk within financial account-asset bipartite graphs, is crucial for ensuring financial market stability and economic resilience. Although traditional node-classification-based anomaly detection techniques can be applied to this task, these approaches often exhibit two key limitations: (1) insufficient consideration of asset fluctuations, resulting in unsatisfactory accuracy; and (2) scalability challenges, making them unsuitable for large-scale financial graphs. To address these issues, we propose RiskGuard, a novel framework for account risk detection that integrates auxiliary asset prediction and gradient-based sampling. First, we introduce an auxiliary asset prediction paradigm to capture the critical influence of asset fluctuations on account risk. Rather than solely predicting account risk, our unified model employs aTemporal-AttentionNet(TANet) to jointly predict asset fluctuations and account risk. This auxiliary task enables the model to learn fluctuation-aware asset representations, significantly enhancing prediction accuracy. To overcome scalability challenges, we design GLUE, an online graph sampler leveraging gradient entropy. GLUE dynamically adjusts sampling weights based on model gradients and graph structure, prioritizing high-entropy nodes in the neighborhood for improved efficiency. Extensive experiments on five financial datasets demonstrate that RiskGuard outperforms existing techniques in accuracy while achieving high efficiency in processing large financial graphs.

Hierarchical directed acyclic graph (DAG) is an essential model for representing terminologies and their hierarchical relationships, such as Disease Ontology and ImageNet categories. Due to the vast number of terminologies and complex structures in a large DAG, it becomes challenging for humans to effectively analyze and explore the hierarchical relationships they encode. Therefore, summarizing hierarchical DAGs is essential for enhancing the interpretability and visualization of the underlying hierarchy. Beyond visual data exploration, hierarchical DAG summarization also supports a range of applications, such as biomedical ontology analytics, snippet generation for information search, and summarized recommendation. In this article, we address a new problem of finding k representative vertices to summarize a hierarchical DAG. To capture diverse summarization and identify important vertices, we design a summary score function that reflects vertices diversity coverage and structure correlation. The studied problem is theoretically proven to be NP-hard. To tackle it efficiently, we propose a greedy algorithm with an approximation guarantee that iteratively adds vertices with significant summary contributions to the answers. To further enhance the answer quality, we introduce a subtree extraction-based method that is proven to achieve higher-quality answers. Additionally, we develop a scalable algorithm, \(\mathsf {k}\) - \(\mathsf {PCGS}\) , which employs candidate pruning and DAG compression for large-scale hierarchical DAGs. For the query-dependent problem, we propose an index-based method and several optimization techniques to improve efficiency. Extensive experiments on large real-world datasets demonstrate the effectiveness and efficiency of our proposed algorithms.

k-truss is a widely studied cohesive sub graph model that has gained significant attention over the past decades. Truss decomposition, a fundamental task in graph analysis, aims to compute the largest k for which an edge belongs to a k-truss. However, directly performing truss decomposition on sensitive graphs risks exposing the private information of user connections in real-world applications. Edge local differential privacy (edge LDP) is extensively used to protect the privacy of edges in graph analysis. This paper, for the first time, addresses the problem of truss decomposition under edge LDP. A naive approach allows each vertex to perturb its neighbor list locally and generate a noisy graph for truss decomposition. However, it often produces excessive truss number estimations, since the noisy graph is generally much denser and fails to preserve the input graph structure. To obtain more accurate estimates, we propose the Local algorithm that leverages the local information during the truss decomposition process. Furthermore, to avoid adding substantial noise to truss numbers to satisfy edge LDP, we introduce the Global algorithm that optimizes the noise scale of support numbers, enhancing the accuracy of truss decom-position results. We further propose the Global * algorithm that eliminates the need for vertices to download noisy edges by utilizing noisy degrees to adjust support numbers during truss decomposition, achieving high accuracy with significantly lower communication costs. Extensive experiments on 9 real-world datasets demonstrate the effectiveness and efficiency of our proposed algorithms.

Edge connectivity is a crucial concept in graph theory, which often serves as a fundamental metric for analyzing and improving the robustness, efficiency, and reliability of various types of networks. A graph is k-connected if it is still connected after removing arbitrarily fewer than$k$edges. Sparse certificate is the lossless compression of a graph in terms of k-connectivity with a bounded size, which enables processing connectivity-driven queries in a more efficient way. Existing studies mainly focus on computing a sparse certificate in static graphs. Motivated by the prevalence of highly dynamic graphs, we aim to propose algorithms to maintain a sparse certificate when an edge is in-serted or deleted. We propose efficient algorithms to significantly improve the theoretical running time for both edge insertion and edge deletion compared with the baseline. We also propose a novel strategy for the search process in edge deletion. The strategy improves the efficiency compared with other potential methods with a theoretical guarantee. Extensive performance studies have been conducted on fourteen real-world datasets. The results demonstrate the significant advantages of our algorithms.

The$k$-truss is a popular cohesive subgraph model for graph analysis, which requires each edge in the subgraph to be contained in at least$k-2$triangles, each consists three pairwisely connected edges. In this paper, we study the$k$-truss breaking problem (TBP) that aims to find the smallest set of edges whose removal makes the graph free of$k$-truss. The problem has been formulated in the literature with applications in community deception, critical connection identification, etc. However, existing solutions cannot scale to large graphs. We observe that chosen edges in a high-quality solution usually have high triangle support, while most share triangles with a significant number of easy-breaking edges (i.e., low-support edges). Motivated by these, we propose the Easy-Breaking Heuristic (EBH) that prioritizes the candidate edges based on their impact on easy-breaking edges. We also design several optimizations to further enhance the performance of EBH. Additionally, we extend our framework to efficiently handle the$k$-truss minimization problem (TMP), which aims to identify a set of at most$b$edges whose removal minimizes the size of the remaining k-truss. Extensive experiments demonstrate that our proposed algorithm outperforms state-of-the-art approaches by up to three orders of magnitude in efficiency when solving TBP, while maintaining comparable effectiveness. Additionally, our proposed algorithm achieves up to four orders of magnitude improvement in efficiency for TMP, along with generally better effectiveness.

Counting the number of triangles in a graph is a fundamental task and has been extensively studied recently. In real-world applications, continuously releasing the triangle count of a graph poses a significant privacy risk for users. To protect sensitive edge information from a central server, we study the problem of estimating the number of triangles under edge local differential privacy (edge LDP). Existing approaches adopt a multi-round computing scheme, allowing the vertices to perform local triangle counting using the noisy graph constructed in the previous round. However, these algorithms not only restrict the noisy graph that can be downloaded to each vertex, but also have coarse upper bounds for the scale of noise added to the estimates. In this paper, we propose a vertex-centric triangle counting algorithm under edge LDP, which improves data utility by leveraging a larger part of the noisy adjacency matrix. Our approach fully exploits the local graph structure to obtain refined estimates of per-vertex triangle counts. We also devise tight bounds for global sensitivities to not only comply with privacy requirements but also control the scale of added noise. Furthermore, we perform a rigorous analysis of the L2 loss of our unbiased estimators and design optimizations for allocating the privacy budget to minimize L2 loss based on the input graph. Extensive experiments on 12 datasets validate the effectiveness and efficiency of our proposed algorithms.

High-dimensional vector similarity search (HVSS) is critical for many data processing and AI applications. However, traditional HVSS methods often require extensive data access for distance calculations, leading to inefficiencies. Triangle-inequality-based lower bound pruning is a widely used technique to reduce the number of data access in low-dimensional spaces but becomes less effective in high-dimensional settings. This is attributed to the ''distance concentration'' phenomenon, where the lower bounds derived from the triangle inequality become too small to be useful. To address this, we propose TRIM, which enhances the effectiveness of traditional triangle-inequality-based pruning in high-dimensional vector similarity search using two key ways: (1) optimizing landmark vectors used to form the triangles, and (2) relaxing the lower bounds derived from the triangle inequality, with the relaxation degree adjustable according to user's needs. TRIM is a versatile operation that can be seamlessly integrated into both memory-based (e.g., HNSW, IVFPQ) and disk-based (e.g., DiskANN) HVSS methods, reducing distance calculations and disk access. Extensive experiments show that TRIM enhances memory-based methods, improving graph-based search by up to 90% and quantization-based search by up to 200%, while achieving a pruning ratio of up to 99%. It also reduces I/O costs by up to 58% and improves efficiency by 102% for disk-based methods, while preserving high query accuracy. Our source code is available at https://github.com/petrizhang/TRIM.

Missing data imputation, which aims to impute the missing values in the raw datasets, is crucial for modern data-driven models like large language models (LLMs). Despite its importance, existing solutions either 1) only support numerical and categorical data or 2) show an unsatisfactory performance due to their design prioritizing text data and overlooking intrinsic characteristics of tabular data. In this paper, we propose UnIMP, a Un ified IMP utation framework that leverages LLM and high-order message passing to enhance the imputation of mixed-type data, including numerical, categorical, and text data. Specifically, we first introduce a cell-oriented hypergraph to model the table. We then propose BiHMP, an efficient Bidirectional High-order Message-Passing network to aggregate global-local and high-order information while capturing the inter-column heterogeneity and intra-column homogeneity. To align the capacity of the LLM with the information aggregated by BiHMP, we introduce Xfusion, which, together with BiHMP, acts as adapters for the LLM. We follow a pre-training and fine-tuning pipeline to train UnIMP, integrating two optimizations: chunking technique, which divides tables into smaller chunks to enhance efficiency; and progressive masking technique, which gradually adapts the model to learn more complex data patterns. Both theoretical proofs and empirical experiments on 10 real-world datasets highlight the superiority of UnIMP over existing techniques.

Cohesive subgraph computation on bipartite graphs has drawn significant research interest recently. As a popular cohesive subgraph model,$k$-bitruss is defined as the maximal subgraph where each edge is contained in at least$k$butterflies (i.e., a (2, 2)-biclique). The bitruss decomposition problem is widely studied, which aims to compute all$k$-bitrusses for$k \geq 0$. The state-of-the-art CPU-based solutions require extensive costs to construct an index structure for grouping butterflies, leading to scalability challenges on large bipartite graphs. In this paper, we explore bitruss decomposition with GPU by leveraging the parallel computing capabilities of GPU architectures. As the index-based approach requires extensive space and the memory resources of GPUs are limited, we proposeGBiD, which is a peeling-based algorithm on GPUs that utilizes a block-centric computation scheme to enable space-efficient bitruss decomposition without any indexing structure. In addition, cost-aware common neighbor exploration and neighbor list accessing optimizations are proposed to enhanceGBiDby reducing the cost of enumerating butterflies and accessing the graph structure during the peeling process. Extensive experiments conducted on 10 real-world datasets demonstrate that our proposed techniques significantly surpass existing CPU-based solutions in terms of both space and time efficiency.

Bipartite graphs are commonly used to model relationships between two distinct types of entities, such as customer-product relationships in e-commerce platforms and protein-protein interactions in bioinformatics. Enumerating all maximal bicliques from a bipartite graph is a fundamental graph mining problem that has been widely used in many real-world applications including community search and spam detection. Existing algorithms for maximal biclique enumeration can struggle to scale to large graphs with a vast number of maximal bicliques. In this paper, we propose a novel and highly-efficient algorithm for maximal biclique enumeration in bipartite graphs using prefix trees. Specifically, a prefix tree is a data structure that stores lists of elements as paths in the tree, and we observe that a maximal biclique can be represented uniquely by the vertices in one of its vertex layers and stored compactly in prefix trees. The process of our algorithm is divided into two steps. First, we find the lower layer vertices of all maximal bicliques and organize them in a prefix tree (i.e., the result tree). During this step, we transform the original time-consuming operations of checking maximality and filtering candidates for vertex sets into determining uniqueness and performing extraction from a prefix tree at each level of the recursion. Second, we use the result tree to obtain the upper layer vertices of the maximal bicliques by computing the common neighbors of vertices in the tree. In this step, we further optimize the computation for intersections of vertex sets by compressing the neighbors of each vertex and memoization. In addition, we also propose a pre-processing method based on the order of traversal on the prefix tree to reduce memory usage. We conduct extensive experiments on 10 real-world datasets, and the results demonstrate that the proposed algorithm outperforms existing solutions by up to one order of magnitude.

Butterfly counting on bipartite graphs has gained increasing attention in past decades. Inevitably, butterfly counts can reveal the presence of certain edges, posing a privacy risk in real applications. Edge local differential privacy (edge LDP), which requires each vertex to perturb its neighbors locally, has been applied to protect edge privacy in graphs. This paper, for the first time, investigates butterfly counting on bipartite graphs with edge LDP. Although a straightforward approach that allows each vertex to perturb its incident edges locally to construct a noisy graph and perform butterfly counting preserves edge LDP, it often results in severe over-counting and significant bias since the resulting noisy graph is generally much denser than the input graph. To obtain unbiased butterfly counts, we propose a multiple-round interaction algorithm to allow the vertices to download the noisy graph and compute local motif counts. Moreover, to avoid adding substantial noise to satisfy edge LDP, we further propose the Download-free Butterfly. Estimation (DBE) algorithm, which captures motif transformation probabilities and relies on motif counts from the noisy graph to yield unbiased butterfly estimates. DBE significantly enhances accuracy via reduced communication between vertices and the data curator. Extensive experiments on 14 datasets validate the effectiveness and efficiency of our proposed techniques.

In many real-world scenarios, relationships between two different entities can be naturally represented as bipartite graphs, such as author-paper, user-item, and people-location. Cohesive subgraph search, which aims to find densely connected subgraphs, is a popular research topic on bipartite graphs. While various cohesive subgraph models are proposed on bipartite graphs, none of them consider the temporal dimension, which expresses dynamic changes occurring in cohesive subgraphs over time. In this paper, we propose the first cohesive subgraph model$(\alpha,\ \beta,\ \mathcal{T})$-core on temporal bipartite graphs. Given degree constraints$\alpha$and$\beta$, as well as a time window$\mathcal{T}=[t_{s},t_{e}],(\alpha,\beta,\ \mathcal{T})$-core guarantees that each vertex in the upper or lower layer has at least$\alpha$or$\beta$neighbors, respectively, within the snapshot over the time window$\mathcal{T}$. An intuitive solution to compute the$(\alpha,\ \beta,\ \mathcal{T})$-core is to iteratively remove the vertices that do not satisfy the degree constraints in the snapshot, which suffers from inefficiency and is impractical on large temporal bipartite graphs. Therefore, we turn to index-based methods to enhance query performance. To support efficient arbitrary$(\alpha,\ \beta,\ \mathcal{T})$-core queries, we propose a vertex-partitioning historical index called VH-Index and a time-partitioning historical index called TH-Index. Note that these two indexes need to store$(\alpha,\ \beta,\ \mathcal{T})$-core for each possible combination of$\alpha, \beta$, and$a\mathcal{T}$and incur large construction costs. Therefore, we further propose a temporal intersection index called TH*-Index to strike a balance between the efficiency of query processing and the space cost of the index. We develop both sequential and parallel algorithms for efficiently constructing the temporal-intersection index. Extensive experiments are conducted on 10 real-world temporal bipartite graphs to validate the effectiveness of the$(\alpha,\ \beta,\ \mathcal{T})$-core model and the efficiency of our proposed algorithms.

Reverse$k$nearest neighbor search$(\mathbf{R}k\mathbf{NNS})$plays an important role in various data processing and analysis tasks, seeking to pinpoint data considering the query data$q$among their$k$nearest neighbors. As large models gain popularity, processing high-dimensional vectors has become more and more widespread. However, existing$\mathbf{R}k\mathbf{NNS}$solutions face inefficiency when handling large-scale high-dimensional vectors due to their sensitivity to data dimensions and sizes during index construction or the verification of numerous candidate results in the query phase. Motivated by these challenges and the inherent intricacies of high-dimensional data processing, in this paper, we study an approximate version of the$\mathbf{R}k\mathbf{NNS}$problem$(\mathbf{R}k\mathbf{ANNS})$for high-dimensional vectors, aiming to offer efficient and practical solutions. To this end, we propose a new proximity-graph-based index called HAMG, which enables finding the query results within$k$hops from$q$. We also present a user-friendly query algorithm on HAMG that can adaptively adjust the search scope based on the desired query recall of users. To further enhance the query process, two pruning strategies are proposed to reduce the number of candidates requiring verification. Extensive experiments validate that HAMG scales well for data dimensions and sizes, and our query algorithm improves query efficiency by up to two orders of magnitude while maintaining comparable query accuracy against existing approaches.

As the field of data science continues to evolve, bipartite graphs have emerged as a fundamental structure in numerous applications, drawing significant interest from both academic and industrial communities. Bipartite graphs are a specific type of graph consisting of two distinct sets of vertices, where connections only occur between vertices of different sets. Examples include e-commerce networks and biological networks. Analytics of bipartite graphs has become an important research topic in the era of big data. This tutorial aims to shed light on analysis methods for bipartite graphs, categorizing them into three areas: classical models, learning-based models, and application-driven models. We start by outlining the importance of bipartite graph analytics, and the unique challenges that need to be addressed. Then, we conduct a thorough review of existing works on bipartite graph analytics. We also compare and analyze the models and solutions in these works. Finally, we point out new research directions.

Periodic group behaviors often exist in temporal interaction networks, such as monthly group meetings, quarterly animal migrations, and yearly birthday parties. In real life, these events are usually quasi-periodic, meaning that the time intervals between two adjacent events are nearly constant but not exactly constant. Most existing studies mainly focus on identifying exact periodic group behaviors, which may result in an incomplete detection of periodic patterns in temporal networks. To fill this gap, we focus on a quasi-periodic community mining problem, which aims to find the most representative cohesive sub graphs, including the quasi-periodic$k$-core and quasi-periodic k-clique. The number of quasi-periodic communities is much larger than that of periodic communities, since the number of quasi-periodic sub-sequences is larger than that of periodic sub-sequences in a given time sequence. To efficiently compute the quasi-periodic communities, we propose a novel two-stage framework. In the first stage, the framework checks whether the time sequence of each vertex contains quasi-periodic sub-sequences. To this end, we develop a new structure, the DAG oracle, which comprises a set of concise DAGs that enables rapid extraction of all quasi-periodic sub-sequences. Based on the DAG oracle, we can easily compute all quasi-periodic sub-sequences for every vertex. In the second stage, the framework computes local quasi-periodic subgraphs that contain the vertex, which allows for the application of existing community mining algorithms. Given the large number of these subgraphs, we propose several carefully -designed pruning rules to further reduce redundant computations. Extensive experiments on 5 real-life datasets demonstrate the efficiency and effectiveness of our proposed solutions.

Network resilience is a critical ability of a network to maintain its functionality against disturbances. A network is resilient/robust when a large portion of the nodes are to be better engaged in the network, i.e., they are less likely to leave given the changes on the network. Existing studies validate that the engagement of a node can be well captured by its coreness on network topology. Therefore, it is promising to maximize the number of nodes with increasing coreness values. In this paper, we propose and study thefollower maximization problem: maximizing the resilience gain (the number of coreness-increased vertices) via anchoring a set of vertices within a given budget. We prove that the problem is NP-hard and W[2]-hard, and it is NP-hard to approximate within an O(n^1-ε ) factor. We first propose an advanced greedy approach, followed by a time-dependent framework designed to quickly find high-quality results. The framework is initialized by the advanced greedy algorithm and incorporates novel techniques for optimizing the search space. The effectiveness and efficiency of our solution are verified with extensive experiments on 8 real-life datasets. Our source codes are available at https://github.com/Tsyxxxka/Follower-Maximization.

Bipartite graphs, formed by two vertex layers, arise as a natural fit for modeling the relationships between two groups of entities. In bipartite graphs, common neighborhood computation between two vertices on the same vertex layer is a basic operator, which is easily solvable in general settings. However, it inevitably involves releasing the neighborhood information of vertices, posing a significant privacy risk for users in real-world applications. To protect edge privacy in bipartite graphs, in this paper, we study the problem of estimating the number of common neighbors of two vertices on the same layer under edge local differential privacy (edge LDP). The problem is challenging in the context of edge LDP since each vertex on the opposite layer of the query vertices can potentially be a common neighbor. To obtain efficient and accurate estimates, we propose a multiple-round framework that significantly reduces the candidate pool of common neighbors and enables the query vertices to construct unbiased estimators locally. Furthermore, we improve data utility by incorporating the estimators built from the neighbors of both query vertices and devise privacy budget allocation optimizations. These improve the estimator's robustness and consistency, particularly against query vertices with imbalanced degrees. Extensive experiments on 15 datasets validate the effectiveness and efficiency of our proposed techniques.

We study the problem of missing data imputation, which is a fundamental task in the area of data quality that aims to impute the missing data to achieve the completeness of datasets. Though the recent distribution-modeling-based techniques (e.g., distribution generation and distribution matching) can achieve state-of-the-art performance in terms of imputation accuracy, we notice that (1) they deploy a sophisticated deep learning model that tends to be overfitting for missing data imputation; (2) they directly rely on a global data distribution while overlooking the local information. Driven by the inherent variability in both missing data and missing mechanisms, in this paper, we explore the uncertain nature of this task and aim to address the limitations of existing works by proposing an uNcertainty-driven netwOrk for Missing data Imputation, termed NOMI. NOMI has three key components, i.e., the retrieval module, the neural network gaussian process imputator (NNGPI) and the uncertainty-based calibration module. NOMI~ runs these components sequentially and in an iterative manner to achieve a better imputation performance. Specifically, in the retrieval module, NOMI~ retrieves local neighbors of the incomplete data samples based on the pre-defined similarity metric. Subsequently, we design NNGPI~ that merges the advantages of both the Gaussian Process and the universal approximation capacity of neural networks. NNGPI~ models the uncertainty by learning the posterior distribution over the data to impute missing values while alleviating the overfitting issue. Moreover, we further propose an uncertainty-based calibration module that utilizes the uncertainty of the imputator on its prediction to help the retrieval module obtain more reliable local information, thereby further enhancing the imputation performance. We also demonstrate that our NOMI~ can be reformulated as an instance of the well-known Expectation Maximization (EM) algorithm, highlighting the strong theoretical foundation of our proposed methods. Extensive experiments are conducted over 12 real-world datasets. The results demonstrate the excellent performance of NOMI in terms of both accuracy and efficiency.

Community search has aroused widespread interest in the past decades. Among existing solutions, the learning-based models exhibit outstanding performance in terms of accuracy by leveraging labels to 1) train the model for community score learning, and 2) select the optimal threshold for community identification. However, labeled data are not always available in real-world scenarios. To address this notable limitation of learning-based models, we propose a pre-trained graph Trans former based community search framework that uses Zero label (i.e., unsupervised), termed TransZero. TransZero has two key phases, i.e., the offline pre-training phase and the online search phase. Specifically, in the offline pre-training phase, we design an efficient and effective community search graph transformer ( CSGphormer ) to learn node representation. To pre-train CSGphormer without the usage of labels, we introduce two self-supervised losses, i.e., personalization loss and link loss, motivated by the inherent uniqueness of node and graph topology, respectively. In the online search phase, with the representation learned by the pre-trained CSGphormer , we compute the community score without using labels by measuring the similarity of representations between the query nodes and the nodes in the graph. To free the framework from the usage of a label-based threshold, we define a new function named expected score gain to guide the community identification process. Furthermore, we propose two efficient and effective algorithms for the community identification process that run without the usage of labels. Extensive experiments over 10 public datasets illustrate the superior performance of TransZero regarding both accuracy and efficiency.

Hypergraphs, which use hyperedges to capture groupwise interactions among different entities, have gained increasing attention recently for their versatility in effectively modeling real-world networks. In this paper, we study the problem of computing hyper-triangles (formed by three fully-connected hyperedges), which is a basic structural unit in hypergraphs. Although existing approaches can be adopted to compute hyper-triangles by exhaustively examining hyperedge combinations, they overlook the structural characteristics distinguishing different hyper-triangle patterns. Consequently, these approaches lack specificity in computing particular hyper-triangle patterns and exhibit low efficiency. In this paper, we unveil a new formation pathway for hyper-triangles, transitioning from hyperedges to hyperwedges before assembling into hyper-triangles, and classify hyper-triangle patterns based on hyperwedges. Leveraging this insight, we introduce a two-step framework to reduce the redundant checking of hyperedge combinations. Under this framework, we propose efficient algorithms for computing a specific pattern of hyper-triangles. Approximate algorithms are also devised to support estimated counting scenarios. Furthermore, we introduce a fine-grained hypergraph clustering coefficient measurement that can reflect diverse properties of hypergraphs based on different hyper-triangle patterns. Extensive experimental evaluations conducted on 11 real-world datasets validate the effectiveness and efficiency of our proposed techniques.

The rapid rise and prevalence of social platforms have created great demands on effective schemes to limit the influence of negative information, e.g., blocking key vertices for influence minimization. However, there is currently no system providing practical schemes to solve the negative influence minimization problem with a blocking budget effectively and efficiently in the literature. In this demo, we present IMinimize, the first interactive system that provides audiences with vertex-blocking schemes over different budgets and demonstrates via visualization for comparison vividly and directly, aiming to help minimize the negative influence spreading in networks. Our IMinimize system applies an advanced greedy algorithm to select blocked vertices with both high efficiency and effectiveness. Furthermore, we extend IMinimize to the application of epidemic controlling and prevention and show the usability of IMinimize through two case studies of real-life applications.

Information cascade in online social networks can be rather negative, e.g., the spread of rumors may trigger panic. To limit the influence of misinformation in an effective and efficient manner, the influence minimization (IMIN) problem is studied in the literature: given a graph G and a seed set S, blocking at most b vertices such that the influence spread of the seed set is minimized. In this paper, we are the first to prove the IMIN problem is NP-hard and hard to approximate. Due to the hardness of the problem, existing works resort to greedy solutions and use Monte-Carlo Simulations to solve the problem. However, they are cost-prohibitive on large graphs since they have to enumerate all the candidate blockers and compute the decrease of expected spread when blocking each of them. To improve the efficiency, we propose the AdvancedGreedy algorithm (AG) based on a new graph sampling technique that applies the dominator tree structure, which can compute the decrease of the expected spread of all candidate blockers at once. Besides, we further propose the GreedyReplace algorithm (GR) by considering the relationships among candidate blockers. Extensive experiments on 8 real-life graphs demonstrate that our AG and GR algorithms are significantly faster than the state-of-the-art by up to 6 orders of magnitude, and GR can achieve better effectiveness with its time cost close to AG.

Biclique percolation community (BPC) search is a fundamental problem in bipartite graph analysis and have many applications. Existing online approach has to enumerate all the maximal bicliques and compute the results based on these bicliques. Considering the large number of maximal bicliques in real graphs and the high frequency of BPC search requests issued in real applications, existing approach is cost prohibitive to obtain the result. Motivated by this, we devise an index-based (BPC-Index) approach to address the problem. Based on the index, we can obtain the result in near-optimal time with well-bounded index space. We further devise an efficient index construction algorithm. Moreover, we also extend our indexing method to address the personalized BPC search problem, which is one of the most common variants of BPC search. We conduct extensive experiments on 10 real bipartite graphs, and the experimental results demonstrate the effectiveness of the BPC model, and the efficiency of our BPC search algorithms and index construction algorithms. Remarkably, our approach can achieve up to 8 orders of magnitude speedup compared to the existing online approach.

Quantifying node importance on engagement dynamics is critical to support network stability. We can motivate or retain the users in a social platform according to their importance s.t. the network is more sustainable. Existing studies validate that the coreness of a node is the "best practice" on network topology to estimate the engagement of the node. In this paper, the importance of a node is the effect on the engagement of other nodes when its engagement is strengthened or weakened. Specifically, the importance of a node is quantified via two novel concepts: the anchor power to measure the engagement effect of node strengthening (i.e., the overall coreness gain) and the collapse power to measure the engagement effect of node weakening (i.e., the overall coreness loss). We find the computation of the two concepts can be naturally integrated into a shell component-based framework, and propose a unified static algorithm to compute both the anchored and collapsed followers. For evolving networks, efficient maintenance techniques are designed to update the follower sets of each node, which is faster than redoing the static algorithm by around 3 orders of magnitude. Extensive experiments on real-life data demonstrate the effectiveness of our model and the efficiency of our algorithms.

In this paper, we study the k-clique densest subgraph problem, which detects the subgraph that maximizes the ratio between the number of k-cliques and the number of vertices in it. The problem has been extensively studied in the literature and has many applications in a wide range of fields such as biology and finance. Existing solutions rely heavily on repeatedly computing all the k-cliques, which are not scalable to handle large k values on large-scale graphs. In this paper, by adapting the idea of "pivoting", we propose the SCT*-Index to compactly organize the k-cliques. Based on the SCT*-Index, our SCTL algorithm can directly obtain the k-cliques from the index and efficiently achieve near-optimal approximation. To further improve SCTL, we propose SCTL* that includes novel graph reductions and batch-processing optimizations to reduce the search space and decrease the number of visited k-cliques, respectively. As evaluated in our experiments, SCTL* significantly outperform existing approaches by up to two orders of magnitude. In addition, we propose a sampling-based approximate algorithm that can provide reasonable approximations for any k value on billion-scale graphs. Extensive experiments on 12 real-world graphs validate both the efficiency and effectiveness of the proposed techniques.

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.

In this article, we propose the$(\alpha,\beta,\eta)$-core model, which is the first cohesive subgraph model on uncertain bipartite graphs. To capture the uncertainty of relationships/edges,$\eta$-degree is adopted to measure the vertex engagement level, which is the largest integer$k$such that the probability of a vertex having at least$k$neighbors is not less than$\eta$. Given degree constraints$\alpha$and$\beta$, and a probability threshold$\eta$, the$(\alpha,\beta,\eta)$-core requires that each vertex on the upper or lower level have$\eta$-degree no less than$\alpha$or$\beta$, respectively. An$(\alpha,\beta,\eta)$-core can be obtained by iteratively removing the vertices with$\eta$-degrees below the degree constraints. Apart from the online computation algorithm, we propose a probability-aware index to strike a balance between time and space costs. To efficiently build such an index, we design a top-down index construction algorithm to allow computation sharing. Then, we show how to parallelize our query algorithms and index construction algorithms. In addition, we study community search on uncertain bipartite graphs by adopting the$(\alpha,\beta,\eta)$-core model. Extensive experiments are conducted on 13 datasets to validate the efficiency and effectiveness of our proposed techniques.

Bipartite graphs are widely used to model relationships between two types of entities. Community search retrieves densely connected subgraphs containing a query vertex, which has been extensively studied on unipartite graphs. However, it remains largely unexplored on bipartite graphs. Moreover, all existing cohesive subgraph models on bipartite graphs only measure the structure cohesiveness while overlooking the edge weight. In this paper, we study the significant (alpha, beta)-community search problem on weighted bipartite graphs. Given a query vertex q, we aim to find the significant (alpha, beta)-community R of q which adopts (alpha, beta)-core to characterize the engagement level of vertices, and maximizes the minimum edge weight (significance) within R. To support fast retrieval of R, we first obtain the maximal connected subgraph of (alpha, beta)-core containing q (the (alpha, beta)-community), and the search space is limited to this subgraph with a much smaller size than the original graph. A novel index structure is presented to support retrieving the (alpha, beta)-community in optimal time. Efficient index maintenance techniques are also proposed to handle dynamic graphs. To further obtain R, we develop peeling and expansion algorithms. The experimental results on real graphs validate the effectiveness and efficiency of our proposed techniques.

Abstract Bipartite graphs are of great importance in many real-world applications. Butterfly, which is a complete $$2 \times 2$$ 2 × 2 biclique, plays a key role in bipartite graphs. In this paper, we investigate the problem of efficient counting the number of butterflies. The most advanced techniques are based on enumerating wedges which is the dominant cost of counting butterflies. Nevertheless, the existing algorithms cannot efficiently handle large-scale bipartite graphs. This becomes a bottleneck in large-scale applications. In this paper, instead of the existing layer-priority-based techniques, we propose a vertex-priority-based paradigm $${\mathsf {BFC}}$$ BFC - $${\mathsf {VP}}$$ VP to enumerate much fewer wedges; this leads to a significant improvement of the time complexity of the state-of-the-art algorithms. In addition, we present cache-aware strategies to further improve the time efficiency while theoretically retaining the time complexity of $${\mathsf {BFC}}$$ BFC - $${\mathsf {VP}}$$ VP . We also show that our proposed techniques can work efficiently in external and parallel contexts. Moreover, we study the butterfly counting problem on batch-dynamic graphs. Specifically, given a bipartite graph G and a batch-update of edges B , we aim to maintain the number of butterflies in G . To tackle this problem, fast vertex-priority-based algorithms are proposed with optimizations for reducing the computation of existing wedges in G . Our extensive empirical studies demonstrate that the proposed techniques significantly outperform the baseline solutions on real datasets.

Bipartite networks, which model relationships between two different types of entities, are prevalent in many real-world applications. On bipartite networks, the cascading node departure undermines the networks' ability to provide sustainable services, which makes reinforcing bipartite networks a vital problem. Although network reinforcement is extensively studied on unipartite networks, it remains largely unexplored on bipartite graphs. On bipartite networks, ($\alpha, \beta$) -core is a stable structure that ensures different minimum engagement levels of the vertices from different layers, and we aim to reinforce bipartite networks by maximizing the ($\alpha, \beta$) -core. Specifically, given a bipartite network$G$, degree constraints$\alpha$and$\beta$, budgets$b_{1}$and$b_{2}$, we aim to find$b_{1}$upper layer vertices and$b_{2}$lower layer vertices as anchors and bring them into the ($\alpha, \beta$) -core s.t. the number of non-anchor vertices entering in the ($\alpha, \beta$) -core is maximized. We prove the problem is NP-hard and propose a heuristic algorithm FILVER to solve the problem. FILVER runs$b_{1}+b_{2}$iterations and choose the best anchor in each iteration. Under a filter-verification framework, it reduces the pool of candidate anchors (in the filter stage) and computes the resulting ($\alpha, \beta$) - core for each anchor vertex more efficiently (in the verification stage). In addition, filter-stage optimizations are proposed to further reduce “dominated” anchors and allow computation-sharing across iterations. To optimize the verification stage, we explore the cumulative effect of placing multiple anchors, which effectively reduces the number of running iterations. Extensive experiments on 18 real-world datasets and a billion-scale synthetic dataset validate the effectiveness and efficiency of our proposed techniques.

Bipartite graphs are naturally used to model relationships between two different types of entities. On bipartite graphs, maximum biclique search is a fundamental problem that aims to find the complete bipartite subgraph (biclique) with the maximum number of edges and is widely adopted for many applications such as anomaly detection in E-commerce and social network analysis. However, maximum biclique search only identifies the biclique whose size is globally maximum, whereas fast microscopic (personalized) analysis is needed in many real-world scenarios. For instance, when a suspected user is identified in an E-commerce network (e.g., a user-product network), it is important to quickly find the anomalous group containing the user and send the group of users for further human expert investigation. To fill this research gap, for the first time, we study the efficient personalized maximum biclique search problem, which aims to find the maximum biclique containing a specific query vertex in real-time. Apart from online computation algorithms, we explore index-based approaches and propose the PMBC-Index. With the PMBC-Index, the query algorithm is up to five orders of magnitude faster than the baseline algorithms. Furthermore, effective pruning strategies and parallelization techniques are devised to support efficient index construction. Extensive experiments on 10 real-world graphs validate both the effectiveness and the efficiency of our proposed techniques.

Bipartite graph is a widely used model to describe relationships between two different types of entities. Exploring graph hierarchy with cohesive subgraphs has been extensively studied on unipartite graphs, while only a few works focus on bipartite graphs. In this paper, we propose the bipartite hierarchy, which is the first model to discover the hierarchical structure of bipartite graphs based on the concept of$(\alpha_{2}\beta){-}$core and graph connectivity. Notably,$(\alpha, \beta)-\text{core}$is a vertex- centric model that conforms to the special structure of bipartite graphs (i.e., formed by two different vertex layers). Accordingly, the bipartite hierarchy has two parts (i.e., the upper and lower hierarchies) to record the hierarchical relationships among upper and lower vertices, respectively. We theoretically prove that the bipartite hierarchy is space-efficient (i.e., its space cost is linear to the graph size) and clearly illustrate its structure via visualization. In addition, efficient algorithms for building the bipartite hierarchy are proposed by utilizing the nested property of$(\alpha, \beta)-\text{core}$. Since bipartite graphs can be dynamically changed in real-world scenarios, we also study the bipartite hierarchy maintenance algorithms against the edge insertion/deletion cases. These algorithms can effectively identify the affected regions to limit computation scope and avoid re-building the bipartite hierarchy from scratch. Extensive experiments on 10 real-world graphs not only demonstrate the effectiveness of the proposed bipartite hierarchy but also validate the efficiency of our hierarchy construction and maintenance algorithms.

Bipartite graphs are extensively used to model relationships between two different types of entities. In many real-world bipartite graphs, relationships are naturally uncertain due to various reasons such as data noise, measurement error and imprecision of data, leading to uncertain bipartite graphs. In this paper, we propose the ($\alpha,\beta,\eta$)-core model, which is the first cohesive subgraph model on uncertain bipartite graphs. To capture the uncertainty of relationships/edges,$\eta$-degree is adopted to measure the vertex engagement level, which is the largest integer$k$such that the probability of a vertex having at least$k$neighbors is not less than$\eta$. Given degree constraints$\alpha$and$\beta$, and a probability threshold$\eta$, the ($\alpha, \beta, \eta$)-core requires that each vertex on the upper or lower level have$\eta$-degree no less than$\alpha$or$\beta$, respectively. An ($\alpha, \beta, \eta$)-core can be derived by iteratively removing a vertex with$\eta$-degree below the degree constraint and updating the$\eta$-degrees of its neighbors. This incurs prohibitively high cost due to the$\eta$-degree computation and updating, and is not scalable to large bipartite graphs. This motivates us to develop index-based approaches. We propose a basic full index that stores ($\alpha, \beta, \eta$)-core for all possible$\alpha, \beta$, and$\eta$combinations, thus supporting optimal retrieval of the vertices in any ($\alpha, \beta, \eta$)-core. Due to its long construction time and high space complexity, we further propose a probability-aware index to achieve a balance between time and space costs. To efficiently build the probability-aware index, we design a bottom-up index construction algorithm and a top-down index construction algorithm. Extensive experiments are conducted on real-world datasets with generated edge probabilities under different distributions, which show that (1) ($\alpha,\beta,\eta$)-core is an effective model; (2) index construction and query processing are significantly sped up by the proposed techniques.

Driven by real-life applications in geo-social networks, we study the problem of computing radius-bounded$k$-cores (RB-$k$-cores) that aims to find communities satisfying both social and spatial constraints. In particular, the model$k$-core (i.e., the subgraph where each vertex has at least$k$neighbors) is used to ensure the social cohesiveness, and a radius-bounded circle is used to restrict the locations of users in an RB-$k$-core. We explore several algorithmic paradigms to compute RB-$k$-cores, including a triple-vertex-based paradigm, a binary-vertex-based paradigm, and a paradigm utilizing the concept of rotating circles. The rotating-circle-based paradigm is further enhanced by several pruning techniques to achieve better efficiency. In addition, to find representative RB-$k$-cores, we study the diversified radius-bounded$k$-core search problem, which finds$t$RB-$k$-cores to cover the most number of vertices. We first propose a baseline algorithm that identifies the distinctive RB-$k$-cores after finding all the RB-$k$-cores. Beyond this, we design algorithms that can efficiently maintain the top-$t$candidate RB-$k$-cores and also achieve a guaranteed approximation ratio. Experimental studies on both real and synthetic datasets demonstrate that our proposed techniques can efficiently compute (diversified) RB-$k$-cores. Moreover, our techniques can be used to compute the minimum-circle-bounded$k$-core and significantly outperform the existing techniques.

Bipartite graphs arise naturally when modeling two different types of entities such as user-item, author-paper, and director-board. In recent years, driven by numerous real-world applications in these networks, mining cohesive structures in bipartite graphs becomes a popular research topic. In this paper, we propose the first cohesive-structure-based bipartite graph analytics system, CohBGA. The key innovative features of our system are as follows. Firstly, we involve several cohesive-structure-based models and statistics in our system to analyze bipartite graphs at different levels of granularity. Secondly, CohBGA has a user-friendly and interactive visual interface with various functional tools to meet users' diverse query requirements. Thirdly, we implement state-of-the-art algorithms in CohBGA to support efficient query processing. Furthermore, as a generic framework is designed in CohBGA, CohBGA is going to be an open-source bipartite graph analytics platform that allows researchers to evaluate the effectiveness of more cohesive-structure-based models and algorithms for bipartite graphs.

In many real-world applications, bipartite graphs are naturally used to model relationships between two types of entities. Community discovery over bipartite graphs is a fundamental problem and has attracted much attention recently. However, all existing studies overlook the weight (e.g., influence or importance) of vertices in forming the community, thus missing useful properties of the community. In this paper, we propose a novel cohesive subgraph model named Pareto-optimal (α β), which is the first to consider both structure cohesiveness and weight of vertices on bipartite graphs. The proposed Pareto-optimal (α β) model follows the concept of (α, β)-core by imposing degree constraints for each type of vertices, and integrates the Pareto-optimality in modelling the weight information from two different types of vertices. An online query algorithm is developed to retrieve Pareto-optimal (α β) with the time complexity of O(p. m) where p is the number of resulting communities, and m is the number of edges in the bipartite graph G. To support efficient query processing over large graphs, we also develop index-based approaches. A complete index i is proposed, and the query algorithm based on i achieves linear query processing time regarding the result size (i.e., the algorithm is optimal). Nevertheless, the index i incurs prohibitively expensive space complexity. To strike a balance between query efficiency and space complexity, a space-efficient compact index 𝕀 is proposed. Computation-sharing strategies are devised to improve the efficiency of the index construction process for the index 𝕀. Extensive experiments on 9 real-world graphs validate both the effectiveness and the efficiency of our query processing algorithms and indexing techniques.

Bipartite graphs are widely used to model relation-ships between two types of entities. Community search retrieves densely connected subgraphs containing a query vertex, which has been extensively studied on unipartite graphs. However, community search on bipartite graphs remains largely unexplored. Moreover, all existing cohesive subgraph models on bipartite graphs can only be applied to measure the structure cohesiveness between two sets of vertices while overlooking the edge weight in forming the community. In this paper, we study the significant (α, β)-community search problem on weighted bipartite graphs. Given a query vertex q, we aim to find the significant (α, β)-community ℛ of q which adopts (α, β)-core to characterize the engagement level of vertices, and maximizes the minimum edge weight (significance) within ℛ.To support fast retrieval of ℛ, we first retrieve the maximal connected subgraph of (α, β)-core containing the query vertex (the (α, β)-community), and the search space is limited to this subgraph with a much smaller size than the original graph. A novel index structure is presented which can be built in O(δ·m) time and takes O(δ·m) space where m is the number of edges in G, δ is bounded by √m and is much smaller in practice. Utilizing the index, the (α, β)-community can be retrieved in optimal time. To further obtain ℛ, we develop peeling and expansion algorithms to conduct searches by shrinking from the (α, β)-community and expanding from the query vertex, respectively. The experimental results on real graphs not only demonstrate the effectiveness of the significant (α, β)-community model but also validate the efficiency of our query processing and indexing techniques.

With the advent of a wide spectrum of recent applications, querying heterogeneous information networks (HINs) has received a great deal of attention from both academic and industrial societies. HINs involve objects (vertices) and links (edges) that are classified into multiple types; examples include bibliography networks, knowledge networks, and user-item networks in E-business. An important component of these HINs is the cohesive subgraph, or a subgraph containing vertices that are densely connected internally. Searching cohesive subgraphs over HINs has found many real applications, such as community search, product recommendation, fraud detection, and so on. Consequently, how to design effective cohesive subgraph models and how to efficiently search cohesive subgraphs on large HINs become important research topics in the era of big data. In this tutorial, we first highlight the importance of cohesive subgraph search over HINs in various applications and the unique challenges that need to be addressed. Subsequently, we conduct a thorough review of existing works of cohesive subgraph search over HINs. Then, we analyze and compare the models and solutions in these works. Finally, we point out new research directions. We believe that this tutorial not only helps researchers to have a better understanding of existing cohesive subgraph search models and solutions, but also provides them insights for future study.

Bipartite graphs are naturally used to model relationships between two different types of entities, such as people-location, author-paper, and customer-product. When modeling real-world applications like disease outbreaks, edges are often enriched with temporal information, leading to temporal bipartite graphs. While reachability has been extensively studied on (temporal) unipartite graphs, it remains largely unexplored on temporal bipartite graphs. To fill this research gap, in this paper, we study the reachability problem on temporal bipartite graphs. Specifically, a vertex u reaches a vertex w in a temporal bipartite graph G if u and w axe connected through a series of consecutive wedges with time constraints. Towards efficiently answering if a vertex can reach the other vertex, we propose an index-based method by adapting the idea of 2-hop labeling. Effective optimization strategies and parallelization techniques are devised to accelerate the index construction process. To better support real-life scenarios, we further show how the index is leveraged to efficiently answer other types of queries, e.g., single-source reachability query and earliest-arrival path query. Extensive experiments on 16 real-world graphs demonstrate the effectiveness and efficiency of our proposed techniques.

Cohesive subgraph mining in bipartite graphs becomes a popular research topic recently. An important structure k-bitruss is the maximal cohesive subgraph where each edge is contained in at least k butterflies (i.e., (2,2)-bicliques). In this paper, we study the bitruss decomposition problem which aims to find all the k-bitrusses for k ≥ 0. The existing bottom-up techniques need to iteratively peel the edges with the lowest butterfly support. In this peeling process, these techniques are time-consuming to enumerate all the supporting butterflies for each edge. To relax this issue, we first propose a novel online index - the BE-Index which compresses butterflies into k-blooms (i.e., (2,k)-bicliques). Based on the BE-Index, the new bitruss decomposition algorithm BiT-BU is proposed, along with two batch-based optimizations, to accomplish the butterfly enumeration of the peeling process in an efficient way. Furthermore, the BiT-PC algorithm is devised which is more efficient against handling the edges with high butterfly supports. We theoretically show that our new algorithms significantly reduce the time complexities of the existing algorithms. Also, we conduct extensive experiments on real datasets and the results demonstrate that our new techniques can speed up the state-of-the-art techniques by up to two orders of magnitude.

Bipartite networks are of great importance in many real-world applications. In bipartite networks, butterfly (i.e., a complete 2 x 2 biclique) is the smallest non-trivial cohesive structure and plays a key role. In this paper, we study the problem of efficient counting the number of butterflies in bipartite networks. The most advanced techniques are based on enumerating wedges which is the dominant cost of counting butterflies. Nevertheless, the existing algorithms cannot efficiently handle large-scale bipartite networks. This becomes a bottleneck in large-scale applications. In this paper, instead of the existing layer-priority-based techniques, we propose a vertex-priority-based paradigm BFC-VP to enumerate much fewer wedges; this leads to a significant improvement of the time complexity of the state-of-the-art algorithms. In addition, we present cache-aware strategies to further improve the time efficiency while theoretically retaining the time complexity of BFC-VP. Moreover, we also show that our proposed techniques can work efficiently in external and parallel contexts. Our extensive empirical studies demonstrate that the proposed techniques can speed up the state-of-the-art techniques by up to two orders of magnitude for the real datasets.

Driven by real-life applications in geo-social networks, in this paper, we investigate the problem of computing the radius-bounded k-cores (RB-k-cores) that aims to find cohesive subgraphs satisfying both social and spatial constraints on large geo-social networks. In particular, we use k-core to ensure the social cohesiveness and we use a radius-bounded circle to restrict the locations of users in a RB-k-core. We explore several algorithmic paradigms to compute RB-k-cores, including a triple vertex-based paradigm, a binary-vertex-based paradigm, and a paradigm utilizing the concept of rotating circles. The rotating circle-based paradigm is further enhanced with several pruning techniques to achieve better efficiency. The experimental studies conducted on both real and synthetic datasets demonstrate that our proposed rotating-circle-based algorithms can compute all RB-k-cores very efficiently. Moreover, it can also be used to compute the minimum-circle-bounded k-core and significantly outperforms the existing techniques for computing the minimum circle-bounded k-core.

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Metric probabilistic range queries (MPRQ) have received substantial attention due to their utility in multimedia and text retrieval, decision making, etc. Existing MPRQ studies generally aim to improve query efficiency and resource usage. In contrast, we define and offer solutions to why-not questions on MPRQ. Given an original metric probabilistic range query and a why-not set W of uncertain objects that are absent from the query result, a why-not question on MPRQ explains why the uncertain objects in W do not appear in the query result, and provides refinements of the original query and/or W with the minimal penalty, so that the uncertain objects in W appear in the result of the refined query. Specifically, we propose a framework that consists of three efficient solutions, one that modifies the original query, one that modifies the why-not set, and one that modifies both the original query and the why-not set. Extensive experiments using both real and synthetic data sets offer insights into the properties of the proposed algorithms, and show that they are effective and efficient.