Cohesive subgraph mining has been extensively studied and finds numerous graph mining applications such as link farm identification, community detection, and product recommendation. Among various cohesive subgraph structures, the$k$-truss is particularly notable for its strong structural cohesiveness based on triangles. However, the classical$k$-truss problem aims to find the$k$-truss with the maximum number of vertices, which is often extremely large and complex in practice. To fully leverage the benefits of the$k$-truss, we consider a novel problem called theminimum$k$-truss problem, which seeks to identify a$k$-truss with the minimum number of vertices, where$k\geq 2$is a positive integer. We first formally prove the NP-hardness of the problem. We then design a baseline algorithmMTEnumthat is based on the vertex enumeration and a heuristic method for computing an upper bound. Despite these efforts,MTEnumstill faces practical efficiency issues which may be due to the fact that the$k$-truss lacks the hereditary property. To address this issue, we develop a novel decomposition-based frameworkDSA, which elegantly transforms the problem into a sequence of problems that are based on a new cohesive subgraph model callededge-based$s$-plex ($s$-eplex). With the hereditary property of$s$-eplex, we design a branch-and-bound algorithm with several customized techniques for the newly formulated problem. Extensive experiments demonstrate the effectiveness of our studied problem and the efficiency of our proposed algorithmDSA. In particular,DSAruns up to five orders of magnitude faster than the baselineMTEnum.

Maximal clique enumeration (MCE) is crucial for tasks like community detection and biological network analysis. Existing algorithms typically adopt the branch-and-bound frame-work with the vertex-oriented Bron-Kerbosch (BK) branching strategy, which forms the sub-branches by expanding the partial clique with a vertex. In this paper, we present a novel approach, HBBMC, a hybrid framework combining vertex-oriented BK branching and edge-oriented BK branching, where the latter adopts a branch-and-bound framework which forms the sub-branches by expanding the partial clique with a edge. This hybrid strategy enables more effective pruning and helps achieve a worst-case time complexity better than the best-known one under a condition which holds for the majority of real-world graphs. To further enhance efficiency, we introduce an early termination technique, which leverages the topological information of the graphs and constructs the maximal cliques directly without branching. Our early termination technique is applicable to all branch-and-bound frameworks. Extensive experiments demonstrate the superior performance of our techniques.

Cohesive subgraph mining is a fundamental problem in graph theory with numerous real-world applications, such as social network analysis and protein-protein interaction modeling. Among various cohesive subgraphs, the γ-quasi-clique is widely studied for its flexibility in requiring each vertex to connect to at least a γ proportion of other vertices in the subgraph. However, solving the maximum γ-quasi-clique problem is NP-hard and further complicated by the lack of the hereditary property, which makes designing efficient pruning strategies challenging. Existing algorithms, such as DDA and FastQC, either struggle with scalability or exhibit significant performance declines for small values of γ. In this paper, we propose a novel algorithm, IterQC, which reformulates the maximum γ-quasi-clique problem as a series of k-plex problems that possess the hereditary property. IterQC introduces a non-trivial iterative framework and incorporates two key optimization techniques: (1) the pseudo lower bound (pseudo LB) technique, which leverages information across iterations to improve the efficiency of branch-and-bound searches, and (2) the preprocessing technique that reduces problem size and unnecessary iterations. Extensive experiments demonstrate that IterQC achieves up to four orders of magnitude speedup and solves significantly more graph instances compared to state-of-the-art algorithms DDA and FastQC.

The s -bundle, as a cohesive subgraph model which relaxes the clique, remains connected whenever fewer than n-s vertices are removed, where n is the number of vertices inside. Finding the largest s -bundle is a fundamental problem and has diverse applications in various fields such as social network analysis, graph visualization, and bioinformatics. Existing studies for solving the problem follow the same branch-and-bound framework and improve the efficiency by developing pruning techniques. As a result, all share the same worst-case time complexity of O* (2 n ), where O* suppresses the polynomial factors. In this paper, we propose a new branch-and-bound algorithm, called SymBD, which achieves improved theoretical guarantees and practical performance. It adopts the existing Symmetric-BK branching strategy whose performance highly depends on the ordering of vertices. We explore various vertex orderings for improving the performance. In particular, we propose two novel vertex orderings based on the local vertex connectivity. With the proposed vertex orderings, SymBD improves the worst-case time complexity to O* (λ n s ) where λ s is strictly less than 2. To further boost the practical efficiency, we introduce a heuristic algorithm for computing a large initial solution and a divide-and-conquer strategy. Extensive experiments on 664 graphs demonstrate that our algorithm is up to five orders of magnitude faster than existing solutions.

Community search has widespread applications in areas such as advertising, friend recommendation, and protein complex identification. In this paper, we revisit the Size-bounded Community Search (SCS) problem, which aims to identify a connected subgraph containing a query vertex q and between l and h vertices, while maximizing the minimum degree of the subgraph. Existing state-of-the-art exact solutions for SCS face challenges in practical efficiency due to ineffective strategies for searching candidate solutions and suboptimal method for finding optimal solution. To address these challenges, we propose a novel branch-and-bound algorithm that efficiently locating a subset of candidate solutions with favorable structural properties, forming the basis for three progressively refined frameworks to determine the optimal solution. Furthermore, we enhance practical performance through a new heuristic, two reduction rules, and a query decomposition technique. Extensive experiments on 12 large real-world graphs demonstrate that our most efficient framework significantly outperforms state-of-the-art methods, achieving an average speedup of two orders of magnitude while consistently identifying communities with higher cohesion.

Given two input graphs, finding the largest subgraph that occurs in both, i.e., finding the maximum common subgraph, is a fundamental operator for evaluating the similarity between two graphs in graph data analysis. Existing works for solving the problem are of either theoretical or practical interest, but not both. Specifically, the algorithms with a theoretical guarantee on the running time are known to be not practically efficient; algorithms following the recently proposed backtracking framework called McSplit, run fast in practice but do not have any theoretical guarantees. In this paper, we propose a new backtracking algorithm called RRSplit, which at once achieves better practical efficiency and provides a non-trivial theoretical guarantee on the worst-case running time. To achieve the former, we develop a series of reductions and upper bounds for reducing redundant computations, i.e., the time for exploring some unpromising branches of exploration that hold no maximum common subgraph. To achieve the latter, we formally prove that RRSplit incurs a worst-case time complexity which matches the best-known complexity for the problem. Finally, we conduct extensive experiments on four benchmark graph collections, and the results demonstrate that our algorithm outperforms the practical state-of-the-art by several orders of magnitude.

Finding cohesive subgraphs from a directed graph is a fundamental approach to analyze directed graph data. We consider a new model called directed$(k,\ell )$-plex for a cohesive directed subgraph, which is generalized from the concept of$k$-plex that is only applicable to undirected graphs. Directed$(k,\ell )$-plex (or DPlex) has the connection requirements on both inbound and outbound directions of each vertex inside, i.e., each vertex disconnects at most$k$vertices and is meanwhile not pointed to by at most$\ell$vertices. In this paper, we study the maximum DPlex search problem which finds a DPlex with the most vertices. We formally prove the NP-hardness of the problem. We then design a heuristic algorithm calledDPHeuris, which finds a DPlex with the size close to the maximum one and runs practically fast in polynomial time. Furthermore, we propose a branch-and-bound algorithm calledDPBBto find the exact maximum DPlex and develop effective graph reduction strategies for boosting the empirical performance. We also consider the problem of querying personalized maximum DPlex, and design a new method calledDPBBQfor the problem. Finally, we conduct extensive experiments on real directed graphs. The experimental results show that (1) our heuristic method can quickly find a near-optimal solution and (2) our branch-and-bound method runs up to six orders of magnitude faster than other baselines.

The$k$-vertex connected ($k$-VC) subgraph, which remains connected with fewer than$k$vertices being removed, is an essential structure in graph mining. It has found many applications, such as survivable network design and web search optimization. However, existing studies focus on mining maximal$k$-VCs, which are excessively large yet less cohesive in real applications. In this paper, we study theminimum$k$-VC search (MinVC)problem, seeking to find a$k$-VC with the minimum number of vertices. We formally prove that this problem is NP-hard and then propose two algorithms to obtain the exact solution. The basic method, calledEnum, follows a branch-and-bound framework with some pruning rules, which directly enumerates all possible vertex sets. Nonetheless, it suffers from the efficiency issues due to the non-hereditary property of the$k$-VC model. To address this challenge, we propose an advanced method, calledVCtoB, which divides theMinVCproblem into several new sub-problems, called thefixed-size$k$-VCproblems. Each of them can be solved efficiently by exploiting the hereditary property of the$s$-bundle model. Finally, our empirical experiments on 139 real-world networks demonstrate thatVCtoBachieves performance improvement of up to six orders of magnitude over the baseline.

Finding cohesive subgraphs from a directed graph is a fundamental approach to analyze directed graph data. We consider a new model called directed$(k,\ell)$-plex for a cohesive directed subgraph, which is generalized from the concept of$k$-plex that is only applicable to undirected graphs. Directed$(k,\ell)$-plex has the connection requirements on both inbound and outbound directions of each vertex inside, i.e., each vertex disconnects at most$K$vertices and is meanwhile not pointed to by at most$\ell$vertices. In this paper, we study the maximum directed$(k, \ell)$-plex search problem which finds a directed$(k, \ell)$-plex with the most vertices. We formally prove the NP-hardness of the problem. We then design a heuristic algorithm called DPHeuris, which finds a directed$(k, \ell)$-plex with the size close to the maximum one and runs practically fast in polynomial time. Furthermore, we propose a branch-and-bound algorithm called DPBB to find the exact maximum directed$(k, \ell)$-plex and develop effective graph reduction strategies for boosting the empirical performance. Finally, we conduct extensive experiments on real directed graphs. The experimental results show that (1) our heuristic method can quickly find a near-optimal solution and (2) our branch-and-bound method runs up to six orders of magnitude faster than other baselines.

k-clique listing is a vital graph mining operator with diverse applications in various networks. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial k-clique with a vertex. These algorithms have the time complexity of O(k · m · (δ/2)k-2 ), where m is the number of edges in the graph and δ is the degeneracy of the graph. In this paper, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial k-clique with two vertices that connect each other (which correspond to an edge ). We explore various edge orderings for EBBkC such that it achieves a time complexity of O( m · δ + k · m · (τ/2)k-2 ), where τ is an integer related to the maximum truss number of the graph and we have τ < δ. The time complexity of EBBkC is better than that of VBBkC algorithms for k>3 since both O(m · δ) and O(k · m · (τ/2)k-2 ) are bounded by O(k · m · (δ/2)k-2 ). Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC based algorithms with the early termination technique consistently and largely outperform the state-of-the-art (VBBkC based) algorithms.

k -plexes relax cliques by allowing each vertex to disconnect to at most k vertices. Finding a maximum k -plex in a graph is a fundamental operator in graph mining and has been receiving significant attention from various domains. The state-of-the-art algorithms all adopt the branch-reduction-and-bound (BRB) framework where a key step, called reduction-and-bound (RB), is used for narrowing down the search space. A common practice of RB in existing works is SeqRB, which sequentially conducts the reduction process followed by the bounding process once at a branch. However, these algorithms suffer from the efficiency issues. In this paper, we propose a new alternated reduction-and-bound method AltRB for conducting RB. AltRB first partitions a branch into two parts and then alternatively and iteratively conducts the reduction process and the bounding process at each part of a branch. With newly-designed reduction rules and bounding methods, AltRB is superior to SeqRB in effectively narrowing down the search space in both theory and practice. Further, to boost the performance of BRB algorithms, we develop efficient and effective pre-processing methods which reduce the size of the input graph and heuristically compute a large k -plex as the lower bound. We conduct extensive experiments on 664 real and synthetic graphs. The experimental results show that our proposed algorithm kPEX with AltRB and novel preprocessing techniques runs up to two orders of magnitude faster and solves more instances than state-of-the-art algorithms.

Enumerating maximal k-biplexes (MBPs) of a bipartite graph has been used for applications such as fraud detection. Nevertheless, there usually exists an exponential number of MBPs, which brings up two issues when enumerating MBPs, namely the effectiveness issue (many MBPs are of low values) and the efficiency issue (enumerating all MBPs is not affordable on large graphs). Existing proposals of tackling this problem impose constraints on the number of vertices of each MBP to be enumerated, yet they are still not sufficient (e.g., they require to specify the constraints, which is often not user-friendly, and cannot control the number of MBPs to be enumerated directly). Therefore, in this paper, we study the problem of finding K MBPs with the most edges called MaxBPs, where K is a positive integral user parameter. The new proposal well avoids the drawbacks of existing proposals (i.e., the number of MBPs to be enumerated is directly controlled and the MBPs to be enumerated tend to have high values since they have more edges than the majority of MBPs). We formally prove the NP-hardness of the problem. We then design two branch-and-bound algorithms, among which, the better one called FastBB improves the worst-case time complexity to O*(γkn), where O* suppresses the polynomials, γk is a real number that relies on k and is strictly smaller than 2, and n is the number of vertices in the graph. For example, for k=1, γk is equal to 1.754. We further introduce three techniques for boosting the performance of the branch-and-bound algorithms, among which, the best one called PBIE can further improve the time complexity to O*(γkd3) for large sparse graphs, where d is the maximum degree of the graph (note that d<

Mining cohesive subgraphs from a graph is a fundamental problem in graph data analysis. One notable cohesive structure is γ-quasi-clique (QC), where each vertex connects at least a fraction γ of the other vertices inside. Enumerating maximal γ-quasi-cliques (MQCs) of a graph has been widely studied and used for many applications such as community detection and significant biomolecule structure discovery. One common practice of finding all MQCs is to (1) find a set of QCs containing all MQCs and then (2) filter out non-maximal QCs. While quite a few algorithms have been developed (which are branch-and-bound algorithms) for finding a set of QCs that contains all MQCs, all focus on sharpening the pruning techniques and devote little effort to improving the branching part. As a result, they provide no guarantee on pruning branches and all have the worst-case time complexity of O*(2n), where O* suppresses the polynomials and n is the number of vertices in the graph. In this paper, we focus on the problem of finding a set of QCs containing all MQCs but deviate from further sharpening the pruning techniques as existing methods do. We pay attention to both the pruning and branching parts and develop new pruning techniques and branching methods that would suit each other better towards pruning more branches both theoretically and practically. Specifically, we develop a new branch-and-bound algorithm called FastQC based on newly developed pruning techniques and branching methods, which improves the worst-case time complexity to O*(αkn), where αk is a positive real number strictly smaller than 2. Furthermore, we develop a divide-and-conquer strategy for boosting the performance of FastQC. Finally, we conduct extensive experiments on both real and synthetic datasets, and the results show that our algorithms are up to two orders of magnitude faster than the state-of-the-art on real datasets.

Cohesive subgraph discovery is an important problem in bipartite graph mining. In this paper, we focus on one kind of cohesive structure, called k-biplex, where each vertex of one side is disconnected from at most k vertices of the other side. We consider the large maximal k-biplex enumeration problem which is to list all those maximal k-biplexes with the number of vertices at each side at least a non-negative integer . This formulation, we observe, has various applications and targets to find non-redundant results by excluding non-maximal ones. Existing approaches suffer from massive redundant computations and can only run on small and moderate datasets. Towards improving scalability, we propose an efficient tree-based algorithm with two advanced strategies and powerful pruning techniques. Experimental results on real and synthetic datasets show the superiority of our algorithm over existing approaches.

Cohesive subgraph discovery is an important problem in bipartite graph mining. In this paper, we focus on one kind of cohesive structure, called$k$-biplex, where each vertex of one side is disconnected from at most$k$vertices of the other side. We consider the large maximal$k$-biplex enumeration problem which is to list all those maximal$k$-biplexes with the number of vertices at each side at least a non-negative integer$\theta$. This formulation aims to find non-redundant results by excluding non-maximal ones and has various applications. Existing approaches suffer from massive redundant computations and can only run on small and moderate datasets. Towards improving scalability, we propose an efficient tree-based algorithm with two advanced strategies and powerful pruning techniques. Experimental results show the superiority of our algorithm over existing approaches.

Mining maximal subgraphs with cohesive structures from a bipartite graph has been widely studied. One important cohesive structure on bipartite graphs is k-biplex, where each vertex on one side disconnects at most k vertices on the other side. In this paper, we study the maximal k-biplex enumeration problem which enumerates all maximal k-biplexes. Existing methods suffer from efficiency and/or scalability issues and have the time of waiting for the next output exponential w.r.t. the size of the input bipartite graph (i.e., an exponential delay). In this paper, we adopt a reverse search framework called bTraversal, which corresponds to a depth-first search (DFS) procedure on an implicit solution graph on top of all maximal k-biplexes. We then develop a series of techniques for improving and implementing this framework including (1) carefully selecting an initial solution to start DFS, (2) pruning the vast majority of links from the solution graph of bTraversal, and (3) implementing abstract procedures of the framework. The resulting algorithm is called iTraversal, which has its underlying solution graph significantly sparser than (around 0.1% of) that of bTraversal. Besides, iTraversal provides a guarantee of polynomial delay. Our experimental results on real and synthetic graphs, where the largest one contains one billion edges, show that our algorithm is up to four orders of magnitude faster than existing algorithms.

The advent of online applications such as social networks has led to an unprecedented scale of data and complex relationships among data. Hypergraphs are introduced to represent complex relationships that may involve more than two entities. A hypergraph is a generalized form of a graph, where edges are generalized to hyperedges. Each hyperedge may consist of any number of vertices. The flexibility of hyperedges also brings challenges in distributed hypergraph processing. In particular, a hypergraph is more difficult to be partitioned and distributed among$k$workers with balanced partitions. In this paper, we propose to convert a hypergraph into an intersection graph before partitioning by leveraging the inherent shared relationships among hypergraphs. We explore the intersection graph construction method and the corresponding partition strategy which can achieve the goal of evenly distributing vertices and hyperedges across workers, while yielding a significant communication reduction. We also design a distributed processing framework named$Hyraph$that can directly run hypergraph analysis algorithms on our intersection graphs. Experimental results on real datasets confirm the effectiveness of our techniques and the efficiency of the$Hyraph$framework.

Densest subgraph discovery (DSD) is a fundamental problem in graph mining. It has been studied for decades, and is widely used in various areas, including network science, biological analysis, and graph databases. Given a graph G , DSD aims to find a subgraph D of G with the highest density (e.g., the number of edges over the number of vertices in D ). Because DSD is difficult to solve, we propose a new solution paradigm in this paper. Our main observation is that the densest subgraph can be accurately found through a k -core (a kind of dense subgraph of G ), with theoretical guarantees. Based on this intuition, we develop efficient exact and approximation solutions for DSD. Moreover, our solutions are able to find the densest subgraphs for a wide range of graph density definitions, including clique-based- and general pattern-based density. We have performed extensive experimental evaluation on both real and synthetic datasets. Our results show that our algorithms are up to four orders of magnitude faster than existing approaches.