In many real-world applications (e.g., email networks, social networks, and phone call networks), the relationships between entities can be modeled as a temporal graph, in which each edge is associated with a timestamp representing the interaction time. As a fundamental task in temporal graph analysis, triangle counting has received much attention, and several triangle models have been developed, including δ-temporal triangle, sliding-window triangle, and (δ 1,3 , δ 1,2 , δ 2,3 )-temporal triangle. In particular, the δ-temporal triangle, requiring the gap of timestamps of any two edges within it to be bounded by a threshold δ, has been demonstrated effective in many real applications, such as cohesiveness analysis, transitivity, clustering coefficient, and graph classification. In this paper, we study fast algorithms for counting δ-temporal triangles in a given query time window. We first propose an online algorithm, which enumerates all edges in the graph and for each edge, calculates how many δ-temporal triangles end with the edge. We further develop an efficient index-based solution, which maps δ-temporal triangles into points of the 2-dimensional space and further compactly organizes these points using hierarchical structures. Besides, we study the problem of binary δ-temporal triangle counting by considering the existence of δ-temporal triangle among three vertices. Experiments on large temporal graphs show that our online algorithm is up to 70× faster than the state-of-the-art algorithm, and our index-based algorithm is up to 10 8 × faster than the online algorithm.

As a building block of many graph-based areas, the s-t path enumeration problem aims to find all paths between s and t by satisfying a given constraint, e.g., hop numbers. In many real-world scenarios, graphs are multi-attribute, where vertices and edges are associated with numerical attributes, such as expense or distance in road networks. However, existing methods have not fully leveraged all attributes in s-t path analysis. Hence, in this paper, we study the problem of skyline path enumeration, which aims to identify paths that balance multiple attributes, ensuring that no skyline result is dominated by another, thus meeting diverse user needs. To efficiently tackle this problem, we design a task-oriented core attribute index, called CAI, to rule out all redundant vertices and edges not located in any skyline path. Additionally, we introduce a hop-dependency label propagation strategy to construct the CAI index in parallel, improving the indexing process. Based on this index, we further design a CAI-based querying strategy that reduces fruitless explorations between candidate vertices not in the same skyline path, significantly optimizing query processing time. Experimental evaluations on fifteen real-world graphs show that CAI outperforms existing methods by up to four orders of magnitude in speed while demonstrating enhanced scalability and well-bound memory costs.

Heterogeneous information networks (HINs) are prevalent in various domains, including bibliographic information networks, social media, and knowledge graphs. As a fundamental topic in HIN mining, community mining has found various real applications, such as recommendation, biological data analysis, and event organization. Most existing works often rely on meta-paths, relational constraints, spectral partitioning, label propagation, and network representation to define the communities. However, almost all these works do not explicitly consider the structural similarity between vertices, which plays a vital role in modeling communities and also ignore the specific roles of vertices. In this paper, we propose a novel community model, called structurally similar community (SSC) , which models the HIN communities by explicitly considering the structural similarity between vertices. In particular, SSC can not only support various structural similarity measures, but also identify different roles of the vertices in the community, such as cores, non-cores, hubs, and outliers. Based on the SSC, we develop fast online and index-based algorithms that support both efficient searching and detecting SSCs in large HINs, where the former one searches an SSC containing a specific query vertex while the latter one detects all the SSCs from the HIN. Extensive experiments on real-world datasets demonstrate the effectiveness of SSC model in revealing meaningful communities and the high efficiency of our proposed algorithms.

With the popularity of GPS-equipped smart devices, spatial crowdsourcing (SC) techniques have attracted growing attention in both academia and industry. In existing trajectory-aware task assignment approaches, tasks assigned to a worker may be far apart from each other, resulting in a higher detour cost as the worker needs to deviate from the original trajectory more often than necessary. Motivated by the above observations, we investigate a trajectory-aware task coalition assignment (TCA) problem and prove it to be NP-hard. The goal is to maximize the number of assigned tasks by assigning task coalitions to workers based on their preferred trajectories. To tackle the TCA problem, we develop a batch-based three-stage framework consisting of task grouping, planning, and assignment. Extensive experiments on real and synthetic datasets demonstrate the effectiveness and efficiency of the proposed algorithms.

Sparse matrices are often used to model the interactions among different objects and they are prevalent in many areas including e-commerce, social network, and biology. As one of the fundamental matrix operations, the sparse matrix chain multiplication (SMCM) aims to efficiently multiply a chain of sparse matrices, which has found various real-world applications in areas like network analysis, data mining, and machine learning. The efficiency of SMCM largely hinges on the order of multiplying the matrices, which further relies on the accurate estimation of the sparsity values of intermediate matrices. Existing matrix sparsity estimators often struggle with large sparse matrices, because they suffer from the accuracy issue in both theory and practice. To enable efficient SMCM, in this paper we introduce a novel row-wise sparsity estimator (RS-estimator), a straightforward yet effective estimator that leverages matrix structural properties to achieve efficient, accurate, and theoretically guaranteed sparsity estimation. Based on the RS-estimator, we propose a novel ordering algorithm for determining a good order of efficient SMCM. We further develop an efficient parallel SMCM algorithm by effectively utilizing multiple CPU threads. We have conducted experiments by multiplying various chains of large sparse matrices extracted from five real-world large graph datasets, and the results demonstrate the effectiveness and efficiency of our proposed methods. In particular, our SMCM algorithm is up to three orders of magnitude faster than the state-of-the-art algorithms.

Directed graphs are prevalent in social networks, web networks, and communication networks. A well-known concept of the directed graph is the D-core, or ( k, l )-core, which is the maximal subgraph in which each vertex has an in-degree not less than k and an out-degree not less than l. Computing the non-empty D-cores for all possible values of k and l , a.k.a. D-core decomposition, has found versatile applications spanning social network analysis, community search, and graph visualization. However, existing algorithms of D-core decomposition suffer from efficiency and scalability issues on large graphs, because serial peeling-based algorithms are limited by single-core utilization, while skyline coreness-based methods exhibit notably high time complexity. To tackle these issues, in this paper, we propose efficient parallel algorithms for D-core decomposition by leveraging the computational prowess of multicore CPUs. Specifically, we first propose a novel algorithm that computes the D-cores for each possible k value, by exploiting an implicit level-by-level vertex removal strategy, which not only diminishes dependencies between vertices but also maintains a time complexity akin to that of sequential algorithms. We further develop an advanced algorithm by introducing a novel concept of D-shell, which allows us to curtail redundant computations by reducing the necessary k values when computing corresponding D-cores, and deriving D-cores with larger k values from the D-cores currently computed based on D-shell. Extensive experiments on ten real-world large graphs show that our algorithms are highly efficient and scalable, and the advanced algorithm is up to two orders of magnitude faster than the state-of-the-art parallel decomposition algorithm with 32 threads.

With the popularity of GPS-equipped smart devices, spatial crowdsourcing (SC) techniques have attracted growing attention in both academia and industry. A fundamental problem in SC is assigning location-based tasks to workers under spatial-temporal constraints. In many real-life applications, workers choose tasks on the basis of their preferred trajectories. However, by existing trajectory-aware task assignment approaches, tasks assigned to a worker may be far apart from each other, resulting in a higher detour cost as the worker needs to deviate from the original trajectory more often than necessary. Motivated by the above observations, we investigate a trajectory-aware task coalition assignment (TCA) problem and prove it to be NP-hard. The goal is to maximize the number of assigned tasks by assigning task coalitions to workers based on their preferred trajectories. For tackling the TCA problem, we develop a batch-based three-stage framework consisting of task grouping, planning, and assignment. First, we design greedy and spanning grouping approaches to generate task coalitions. Second, to gain candidate task coalitions for each worker efficiently, we design task-based and trajectory-based pruning strategies to reduce the search space. Furthermore, a 2-approximate algorithm, termed MST-Euler, is proposed to obtain a route among each worker and task coalition with a minimal detour cost. Third, the MST-Euler Greedy (MEG) algorithm is presented to compute an assignment that results in the maximal number of tasks assigned and a parallel strategy is introduced to boost its efficiency. Extensive experiments on real and synthetic datasets demonstrate the effectiveness and efficiency of the proposed algorithms.

As a fundamental problem in graph data mining, Densest Subgraph Discovery (DSD) aims to find the subgraph with the highest density from a graph. It has been studied for several decades and found a large number of real-world applications, such as network community detection, regulatory motif discovery in DNA, graph index construction, and fake follower detection. Although there are many existing DSD algorithms, they are often not scalable or efficient to process large-scale graphs, since most of them are serial algorithms and can only leverage the computing resource of a single CPU core. To tackle these issues, in this paper we propose efficient parallel algorithms for solving the DSD problems on both undirected and directed graphs at scale. Our main idea is to use the k-cores (a kind of dense subgraph) to approximate the densest subgraph in the undirected graphs, and then propose efficient parallel algorithms for computing the cores by optimizing the iterative process and also reducing the number of iterations. We further extend this idea for directed graphs by introducing a novel concept, named w-induced subgraph, to avoid unnecessary enumerations of x or y when searching [x,y]-cores (a kind of directed dense subgraph to approximate the densest). To verify the scalability and efficiency of the proposed algorithms, we have conducted extensive experiments on 12 large real-world graphs, and four of them are billion-scale. The experimental results show that our proposed algorithms outperform the state-of-the-art algorithms on both undirected and directed graphs, in terms of scalability and efficiency.

As an important cohesive subgraph model in bipartite graphs, the (α, β)-core (a.k.a. bi-core) has found a wide spectrum of real-world applications, such as product recommendation, fraudster detection, and community search. In these applications, the bipartite graphs are often large and dynamic, where vertices and edges are inserted and deleted frequently, so it is costly to recompute (α, β)-cores from scratch when the graph has changed. Recently, a few works have attempted to study how to maintain (α, β)-cores in the dynamic bipartite graph, but their performance is still far from perfect, due to the huge size of graphs and their frequent changes. To alleviate this issue, in this paper we present efficient (α, β)-core maintenance algorithms over bipartite graphs. We first introduce a novel concept, called bi-core numbers, for the vertices of bipartite graphs. Based on this concept, we theoretically analyze the effect of inserting and deleting edges on the changes of vertices' bi-core numbers, which can be further used to narrow down the scope of the updates, thereby reducing the computational redundancy. We then propose efficient (α, β)-core maintenance algorithms for handling the edge insertion and edge deletion respectively, by exploiting the above theoretical analysis results. Finally, extensive experimental evaluations are performed on both real and synthetic datasets, and the results show that our proposed algorithms are up to two orders of magnitude faster than the state-of-the-art approaches.

In this paper, for the first time, we introduce the concepts of window-CCs and window-SCCs on undirected and directed temporal graphs, respectively. We then study the queries of window-CC and window-SCC by developing several efficient index-based query solutions. The space costs of the best indices are linear to the sizes of the temporal graphs. The extensive experimental evaluation on 12 real-world datasets demonstrates the high efficiency and effectiveness of the proposed solutions. In the future, we will develop distributed index construction algorithms, which would be useful for very large temporal graphs containing billions of edges. In the future, we will implement our algorithms by using a distributed computing platform (e.g., Pregel), which would be very useful when the temporal graph is too large to be kept by a single machine.

Recently, the topic of influential community search has gained much attention. Given a graph, it aims to find communities of vertices with high importance values from it. Existing works mainly focus on conventional homogeneous networks, where vertices are of the same type. Thus, they cannot be applied to heterogeneous information networks (HINs) like bibliographic networks and knowledge graphs, where vertices are of multiple types and their importance values are of heterogeneity (i.e., for vertices of different types, their importance meanings are also different). In this paper, we study the problem of influential community search over large HINs. We introduce a novel community model, called heterogeneous influential community (HIC), or a set of closely connected vertices that are of the same type and high importance values, using the meta-path-based core model. An HIC not only captures the importance of vertices in a community, but also considers the influence on meta-paths connecting them. To search the HICs, we mainly consider meta-paths with two and three vertex types. Then, we develop basic algorithms by iteratively peeling vertices with low importance values, and further propose advanced algorithms by identifying the key vertices and designing pruning strategies that allow us to quickly eliminate vertices with low importance values. Extensive experiments on four real large HINs show that our solutions are effective for searching HICs, and the advanced algorithms significantly outperform baselines.

Influential community search aims to find cohesive subgraphs (communities) with considerable influence. It is a fundamental graph management operator that can play a crucial role in biological network analysis, activity organization, and other real-life applications. Existing research on influential community search is mainly focused on deterministic graphs with the assumption that influences between entities are certain. This assumption is invalid in many cases because it ignores the uncertainty which is an inherent property of influence. Against this backdrop, in this paper, we introduce an uncertain influential community model, namely$(k, \eta)$-influential community, based on which the influential community search problem over uncertain graphs is formulated. Furthermore, we propose an online approach by integrating a peeling-pruning strategy that can progressively refine the given uncertain graph to find the$(k, \eta)$-influential communities. To further improve the search performance, two novel indexes, ICU-Index and FICU-Index, are developed to organize the$(k, \eta)$-influential communities at different probabilistic intervals. The indexes decompose the probabilistic interval into multiple subintervals and based on this, the$(k, \eta)$-influential communities are divided into different groups in turn. Compared with ICU-Index, FICU-Index requires considerably less space with the introduction of two optimization strategies. These indexes help obtain results of an influential community search problem more efficiently. Extensive experiments on large real and synthetic datasets demonstrate the efficiency and effectiveness of our proposed algorithms.

As a typical most-to-most connected quasi-biclique model, k-biplex is a superset of bicliques, which allows nodes on each side of a fully connected subgraph to lose at most$k$connections. In this paper, we investigate the maximum biplex search problem for the first time. The goal here is to find a k-biplex with the maximum number of edges and we have proved that the problem is NP-hard. It is widely used in fraudulent reviewer group detection, gene expression analysis, social recommendation, and other real-life applications. To solve this problem, a maximum k-biplex search algorithm (MBS) is first presented by integrating two pruning strategies, including degree-based and 2-hop-based pruning. In addition, we define a new dense subgraph over bipartite graphs,$\langle x, y\rangle$-core, and develop a core-based maximum k-biplex search algorithm (MBS-Core) which can significantly reduce the search space with the introduction of a core-based graph reduction technique. In particular, it only needs to search these cores instead of the entire graph to obtain the maximum k-biplex. Moreover, a parallel algorithm and a heuristic algorithm are developed to achieve better query performance on larger-scale bipartite graphs. Extensive experiments have been conducted on real-life and synthetic datasets to verify the efficiency and effectiveness of the proposed algorithms. Our results show that MBS-Core is up to 3 orders of magnitude faster than the existing approaches.

As one of the most fundamental problems in graph data mining, the densest subgraph discovery (DSD) problem has found a broad spectrum of real applications, such as social network community detection, graph index construction, regulatory motif discovery in DNA, fake follower detection, and so on. Theoretically, DSD closely relates to other fundamental graph problems, such as network flow and bipartite matching. Triggered by these applications and connections, DSD has garnered much attention from the database, data mining, theory, and network communities. In this tutorial, we first highlight the importance of DSD in various applications and the unique challenges that need to be addressed. Subsequently, we classify existing DSD solutions into several groups, which cover around 50 research papers published in many well-known venues (e.g., SIGMOD, PVLDB, TODS, WWW), and conduct a thorough review of these solutions in each group. Afterwards, we analyze and compare the models and solutions in these works. Finally, we point out a list of promising future research directions. We believe that this tutorial not only helps researchers have a better understanding of existing densest subgraph models and solutions, but also provides them insights for future study.