Structural graph clustering (SCAN) is one of the most popular graph clustering paradigms, and has attracted plenty of attention recently. Existing solutions assume that the input graphs is homogeneous, i.e., the vertices are of the same type. However, in many real applications, such as bibliographic networks and knowledge graphs, the input graphs is heterogeneous information networks which consist of multi-typed and interconnected objects, which makes SCAN cannot be applied to cluster. Therefore, in this paper, we study the SCAN problem over heterogeneous information networks. Based on the concept of meta-path, we propose two new structural graph clustering models first. Following these two new models, we design new algorithms to support the efficient clustering of a heterogeneous information network. We conduct extensive experiments on six real heterogeneous information networks, and the results demonstrate the effectiveness of our new models and the efficiency of our proposed clustering algorithms.

Bipartite graphs are widely used to model relationships among diverse entities in domains such as gene co-expression networks, collaboration networks, and customer-product interactions. A fundamental problem in analyzing bipartite graphs is the maximum balanced biclique (MBBC) search, which identifies the maximum fully connected subgraph with an equal number of vertices on both sides in the given bipartite graph. Despite its utility, the MBBC model suffers from practical limitations: its strict all-to-all connectivity and exact size-equality requirements make it impractical for noisy, incomplete real-world bipartite data. To overcome these limitations, we propose the maximum balanced k-biplex (MBKBP) model, which relaxes the stringent requirements of MBBC. In MBKBP, each vertex is allowed to miss up to k neighbors on the opposite side of the bipartite graph, and a user-defined parameter$\delta$ensures approximate balance between the two vertex sets. This flexibility enhances robustness to noise, accommodates incomplete data, and broadens the model's applicability. To compute the MBKBP in a given bipartite graph, a baseline approach involves enumerating all maximal balanced k-biplexes and identifying the largest one. However, as confirmed by our experiments, this approach is computationally inefficient. To address this challenge, we introduce the concept of$(z_{L},\ z_{R})$search space and propose a new framework to compute the MBKBP. By generating a series of smaller$(z_{L)}z_{R})$search spaces, our framework significantly reduces the number of maximal k-biplexes that need to be explored. Additionally, we leverage the$\delta$-balance property to refine the search spaces further and develop three categories of pruning rules to minimize computational overhead. Extensive experiments on real-world bipartite graphs demonstrate that our algorithm achieves up to three orders of magnitude speedup compared to baseline approache, showcasing its efficiency and practicality for bipartite graph analysis.

Graphs have been widely used to represent the relationships of entities in real-world applications [1], [2]. k-truss model is a typical cohesive subgraph model and has received considerable attention due to its unique cohesive properties on degree and bounded diameter [3], [4].

As a popular data structure, bipartite graph is widely used to model the complex relationships between two types of entities widely in many real world application domains[1]. Detecting cohesive subgraphs, such as biclique, is a fundamental problem in graph analysis[2], [3]. Given a bipartite graph$G$, a subgraph$B=(X,\ Y)$is a biclique if$B$is a complete subgraph.

Top- k Nearest Neighbors ( k NN) problem on road network has numerous applications on location-based services. As direct search using the Dijkstra's algorithm results in a large search space, a plethora of complex-index-based approaches have been proposed to speedup the query processing. However, even with the current state-of-the-art approach, long query processing delays persist, along with significant space overhead and prohibitively long indexing time. In this paper, we depart from the complex index designs prevalent in existing literature and propose a simple index named KNN-Index. With KNN-Index, we can answer a k NN query optimally and progressively with small and size-bounded index. To improve the index construction performance, we propose a bidirectional construction algorithm which can effectively share the common computation during the construction. Theoretical analysis and experimental results on real road networks demonstrate the superiority of KNN-Index over the state-of-the-art approach in query processing performance, index size, and index construction efficiency.

Computing the reachability between two vertices in a graph is a fundamental problem in graph data analysis. Most of the existing works assume that the edges in the graph have no labels, but in many real application scenarios, edges naturally come with edge-labels, and label constraints may be placed on the edges appearing on a valid path between two query vertices. Therefore, we study the label-constrained reachability (LCR) queries in this paper, where we are given a source vertex s , a target vertex t , a label set Δ, and the goal is to check whether there exists any path from s to t such that all the labels of edges on the path belong to Δ. A plethora of methods have been proposed in the literature to support the LCR queries. All these methods take the assumption that the graph is resident in the main memory of a machine. Nevertheless, the graphs in many real application scenarios are generally big and may not reside in memory. In these cases, existing methods suffer from serious scalability problem, i.e., result in huge I/O costs. Motivated by this, in this paper, we study the I/O efficient LCR query problem and aim to efficiently answer the LCR queries when the graph cannot fit in the main memory. To achieve this goal, we propose a reduction-based indexing approach. We introduce two elegant graph reduction operators which aims to reduce the size of the graph loaded in memory while preserving the LCR information among the remaining vertices. With these two operators, we devise an index named LCR-Index and propose algorithms to adaptively construct the index based on the available memory. Equipped with LCR-Index, we can answer a LCR query by only scanning the LCR-Index sequentially. Experiments demonstrate our query processing algorithm can handle graphs with billions of edges.

Shortest path query on road networks is a fundamental problem to support many location-based services and wide variant applications. Contraction Hierarchies(CH) is widely adopted to accelerate the shortest path query by leveraging shortcuts among vertices. However, the state-of-the-art CH construction method named$\mathsf{VCHCons}$suffers from inefficiencies due to their strong reliance on pre-determined vertex order. This leads to the generation of a large number of invalid shortcuts and the limit of parallel processing capability. Motivated by it, in this paper, an innovative CH construction algorithm called$\mathsf{ECHCons}$is devised following an edge-centric paradigm, which addresses the issue of invalid shortcut production by introducing a novel edge-ordering strategy. Furthermore, it optimizes shortcut calculation within a dynamically constructed optimal subgraph, which is significantly smaller than the original network, thus shrinking the traversal space during index construction. To further enhance efficiency and overcome the limitations in parallelism inherent to$\mathsf{VCHCons}$, our approach leverages batch contraction of edges and introduces a well-defined lower bound technique to unlock more efficient parallel computation resources. Our approach provides both theoretical guarantee and practical advancement in CH construction. Extensive and comprehensive experiments are conducted on real road networks. The experimental results demonstrate the effectiveness and efficiency of our proposed approach.

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.

Shortest path counting computes the number of shortest paths between two vertices on a graph, which can be used in the applications such as social network search and POI (Point of Interest) recommendation. The state-of-the-art approach leverages index to speed up the query processing. However, this approach incurs not only significant space overheads but also prohibitive indexing time, which makes it inapplicable to handle such queries on large graphs. Motivated by this, in this paper, we aim to propose a new solution to scale up the shortest path counting. To achieve this goal, we first propose a novel size-tunable indexing framework, which allows users to tune the index space consumption based on their requirements for query processing efficiency and available memory. Based on the size-tunable indexing framework, we devise a new parallel paradigm to accelerate index construction. We conduct experiments on 15 real graphs and the experimental results demonstrate that our new approach significantly outperforms the state-of-the-art approach regarding the index space cost and index construction cost, and is able to handle billion-scale graphs that the state-of-the-art approach cannot process with less than 5 milliseconds query processing time on all test cases.

Abstract In the real world, road networks with weight and label on edges can be applied in several application domains. The shortest path query with label restrictions has been receiving increasing attention recently. To efficiently answer such kind of queries, a novel index, namely Contraction Hierarchies with Label Restrictions (CHLR), is proposed in the literature. However, existing studies mainly focus on the static road networks and do not support the CHLR maintenance when the road networks are dynamically changed. Motivated by this, in this paper, we investigate the CHLR maintenance problem in dynamic road networks. We first devise a baseline approach to update CHLR by recomputing the potential affected shortcuts. However, many shortcuts recomputed in baseline do not change in fact, which leads to unnecessary overhead of the baseline. To overcome the drawbacks of baseline, we further propose a novel CHLR maintenance algorithm which can only travel little shortcuts through an update propagate chain with accuracy guarantee. Moreover, an optimization strategy is presented to further improve the efficiency of index maintenance. Considering the frequency of edge changes, we also propose a batch index maintenance algorithm to handle batch edge changes which can process a large number of edge changes at once. Furthermore, a parallel method is proposed to further accelerate calculations. Extensive and comprehensive experiments are conducted on real road networks. The experimental results demonstrate the efficiency and effectiveness of our proposed algorithms.

$k$-truss model is a typical cohesive subgraph model and has been received considerable attention recently. However, the$k$-truss model only considers the direct common neighbors of an edge, which restricts its ability to reveal fine-grained structure information of the graph. Motivated by this, in this paper, we propose a new model named$(k, \tau)$-truss that considers the higher-order neighborhood ($\tau$hop) information of an edge. Based on the$(k, \tau)$-truss model, we study the higher-order truss decomposition problem which computes the$(k, \tau)$-trusses for all possible$k$values regarding a given$\tau$. Higher-order truss decomposition can be used in the applications such as community detection and search, hierarchical structure analysis, and graph visualization. To address this problem, we first propose a bottom-up decomposition paradigm in the increasing order of$k$values to compute the corresponding$(k, \tau)$-truss. Based on the bottom-up decomposition paradigm, we further devise three optimization strategies to reduce the unnecessary computation. We evaluate our proposed algorithms on real datasets and synthetic datasets, the experimental results demonstrate the efficiency, effectiveness and scalability of our proposed algorithms.

Clique is one of the most fundamental models for cohesive subgraph mining in network analysis. Existing clique model mainly focuses on unsigned networks. However, in real world, many applications are modeled as signed networks with positive and negative edges. As the signed networks hold their own properties different from the unsigned networks, the existing clique model is inapplicable for the signed networks. Motivated by this, we propose the balanced clique model that considers the most fundamental and dominant theory, structural balance theory, for signed networks. Following the balanced clique model, we study themaximalbalancedcliqueenumeration problem (${\mathsf {MBCE}}$) which computes all the maximal balanced cliques in a given signed network. Moreover, in some applications, users prefer a unique and representative balanced clique with maximum size rather than all balanced cliques. Thus, we also study themaximumbalancedcliquesearch problem (${\mathsf {MBCS}}$) which computes the balanced clique with maximum size. We show that${\mathsf {MBCE}}$problem and${\mathsf {MBCS}}$problem are both NP-Hard. For the${\mathsf {MBCE}}$problem, a straightforward solution is to treat the signed network as two unsigned networks and leverage the off-the-shelf techniques for unsigned networks. However, such a solution is inefficient for large signed networks. To address this problem, in this paper, we first propose a new maximal balanced clique enumeration algorithm by exploiting the unique properties of signed networks. Based on the new proposed algorithm, we devise two optimization strategies to further improve the efficiency of the enumeration. For the${\mathsf {MBCS}}$problem, we first propose a baseline solution. To overcome the huge search space problem of the baseline solution, we propose a new search framework based on search space partition. To further improve the efficiency of the new framework, we propose multiple optimization strategies regarding to redundant search branches and invalid candidates. We conduct extensive experiments on large real datasets. The experimental results demonstrate the efficiency, effectiveness and scalability of our proposed algorithms for${\mathsf {MBCE}}$problem and${\mathsf {MBCS}}$problem.

Bipartite graphs are widely used to capture the relationships between two types of entities. In bipartite graph analysis, finding the maximum balanced biclique (MBB) is an important problem with numerous applications. A biclique is balanced if its two disjoint vertex sets are of equal size. However, in real-world scenarios, each vertex is associated with a weight to denote its properties, such as influence, i.e., weighted bipartite graph. For weighted bipartite graphs, the previous studies for MBB are no longer applicable due to the ignorance of weight. To fill the gap, in this paper, we propose a reasonable definition of “balance” by restricting the weight difference between two sides of a biclique within$k$. Given a weighted bipartite graph$G$and a constraint$k$, we aim to find the maximum$k$-balanced biclique (Max$k$BB) with the maximum weight. To address the problem, we first propose an approach based on biclique enumeration on single side of$G$following the Branch-and-Bound framework. To improve the performance, we further devise three optimization strategies to prune invalid search branches. Moreover, we utilize graph reduction strategy to reduce the redundant search space. Extensive experiments are conducted on 12 real bipartite datasets to demonstrate the efficiency, effectiveness and scalability of our proposed algorithms. The experimental results show that our algorithms can address MBB detection problem efficiently, and the case study demonstrates the effectiveness of our model compared with MBB model.

Structural clustering (SCAN) is one of the most popular graph clustering paradigms. However, SCAN assumes that the input graph is undirected and can not cluster the directed graphs. To address this problem, in this paper, we propose a new structural clustering model based on SCAN to cluster directed graphs. Following the new model, we propose an index-based approach to support the efficient clustering of a given graph. Moreover, we also devise efficient index maintenance algorithms to handle the case that the input graph is dynamically updated. We conduct extensive experiments on nine real directed graphs, one of which contains more than 2 billion edges. The results demonstrate the effectiveness and efficiency of our proposed methods.

Clique is one of the most fundamental models for cohesive subgraph mining in network analysis. Existing clique model mainly focuses on unsigned networks. In real world, however, many applications are modeled as signed networks with positive and negative edges. As the signed networks hold their own properties different from the unsigned networks, the existing clique model is inapplicable for the signed networks. Motivated by this, we propose the balanced clique model that considers the most fundamental and dominant theory, structural balance theory, for signed networks, and study the maximal balanced clique enumeration problem which computes all the maximal balanced cliques in a given signed network. We show that the maximal balanced clique enumeration problem is NP-Hard. A straightforward solution for the maximal balanced clique enumeration problem is to treat the signed network as two unsigned networks and leverage the off-the-shelf techniques for unsigned networks. However, such a solution is inefficient for large signed networks. To address this problem, in this paper, we first propose a new maximal balanced clique enumeration algorithm by exploiting the unique properties of signed networks. Based on the new proposed algorithm, we devise two optimization strategies to further improve the efficiency of the enumeration. We conduct extensive experiments on large real and synthetic datasets. The experimental results demonstrate the efficiency, effectiveness and scalability of our proposed algorithms.