Cohesive subgraphs discovery in networks

Searching for a community based on query nodes in a graph is a fundamental problem and has been extensively investigated. Most of the existing approaches focus on finding a community in a social network, and very few studies consider location-based social networks where users can check in locations. In this paper we propose the GeoSocial Community Search problem (GCS) which aims to find a social community and a cluster of spatial locations that are densely connected in a location-based social network simultaneously. The GCS can be useful for marketing and user/location recommendation. To the best of our knowledge, this is the first work to find a social community and a cluster of spatial locations that are densely connected from location-based social networks. We prove that the problem is NP-hard, and is not in APX, unless P = NP. To solve this problem, we propose three algorithms: core-based basic algorithm, top-down greedy removing algorithm, and an expansion algorithm. Finally, we report extensive experimental studies that offer insights into the efficiency and effectiveness of the proposed solutions.

There has been significant interest in the study of the problem of community search in large networks. Given one or more query nodes, this problem aims to discover densely connected subgroups containing these nodes. Various algorithms have been proposed to solve this challenging problem using different measures or a variety of cohesive subgraphs. In this paper, given an undirected graph and a set of query nodes, we study the community search using novel several cohesive subgraph models. More precisely, we propose to exploit several cohesive structures in a unified framework to find densely communities for query nodes in large complex networks. First, we review some existing cohesive structures. Next, to make these structures more effective models of communities, we focus on interesting configurations that are larger and more cohesive by fulfilling some constraints. The new structures obtained allow to ensure a larger density on the discovered communities and overcome some weaknesses of existing models. Finally, empirical results show the effectiveness of our framework to find communities for query nodes in a variety of real graphs.

Recently, the community search problem has attracted significant attention, due to its wide spectrum of real-world applications such as event organization, friend recommendation, advertisement in e-commence, and so on. Given a query vertex, the community search problem finds dense subgraph that contains the query vertex. In social networks, users have multiple check-in locations, influence score, and profile information (keywords). Most previous studies that solve the CS problem over social networks usually neglect such information in a community. In this paper, we propose a novel problem, named community search over spatial-social networks (TCS-SSN), which retrieves community with high social influence, small traveling time, and covering certain keywords. In order to tackle the TCS-SSN problem over the spatial-social networks, we design effective pruning techniques to reduce the problem search space. We also propose an effective indexing mechanism, namely social-spatial index, to facilitate the community query, and develop an efficient query answering algorithm via index traversal. We verify the efficiency and effectiveness of our pruning techniques, indexing mechanism, and query processing algorithm through extensive experiments on real-world and synthetic data sets under various parameter settings.

Community search is a query-dependent graph task to find communities containing a given set of query vertices, which is useful for personalized search and recommendation. Recently, community search over multilayer networks has gained attention thanks to its strong ability to capture cross-layer relationships among diverse entities from multiple domains. This brings significant advantages against the classical studies of community search over only single-layer graphs. However, most existing multilayer community models suffer from two major limitations: 1) failure to identify informative communities with the most layers when a multilayer graph is associated with a large number of layers; 2) missing to distinguish the degree of connections in internal layers and cross-layers. To tackle the above limitations, this paper proposes a novel multilayer subgraph model called <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(k, d)$</tex>-core. A <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(k,d)$</tex>-core based community requires that every two layers have enough <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> internal layer connections and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d$</tex> cross-layer connections for each vertex in this community. We formulate the problem of multilayer community search (MCS-problem), which finds a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(k,d)$</tex>-core connected subgraph <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H$</tex> containing query vertices to achieve the largest number of cross-layers. For cross-layer connectivity, we consider two-fold definitions of full-layer and path-layer connectivities. First, we consider a strong definition of full-layer connectivity, which constrains that every two layers are connected in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H$</tex>. We show that the MCS-problem under full-layer connectivity is NP-hard. We propose two methods of exact exploration and heuristic search for finding M CS answers. Second, to improve the efficiency of community search, we further study a relaxation of path-layer connectivity, allowing two layers to be connected via a path of immediate layers. Then, we develop a fast search algorithm to identify path-layer-based communities and then refine them to full-layer answers. Furthermore, we develop a novel <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(k,d){-}$</tex> core index that effectively captures essential <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(k,d)$</tex>-core structure, including the neighborhood information, the layer connectivities, and the internal/cross-layer corenesses. Extensive experiments on nine real-world multilayer graphs demonstrate the effectiveness and efficiency of our M CS model and algorithms.

Community search (CS) on graphs returns the largest densely connected vertex subset containing a query vertex, namely k-community, where every vertex's degree in the induced subgraph is not less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> . It has significant influence in many real-life applications including event organization and friend recommendation. Many complex networks such as social networks and protein-protein interaction (PPI) networks are often modeled as uncertain graphs. In this paper, we identify and study the problem of reliable community search on uncertain graphs (UCS for short). Given an uncertain graph, a query vertex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$q$</tex> , a positive integer <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> and a probability threshold θ, the reliable community, viz., (k, θ) -community, of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$q$</tex> is the largest vertex subset, so that the probability of every vertex to be in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$q$</tex> 's k-community is not less than θ. We prove that it is a NP-hard problem. We propose two novel pruning strategies to reduce the candidate set to a much smaller size. We develop an efficient index, namely CD-index, with which the pruning process can be done in optimal time. We also present efficient sampling algorithms on top of stratified sampling and lazy sampling to accelerate the search under accuracy guarantees. Extensive experiments using four real-world datasets demonstrate the superior performance of proposed algorithms to the state-of-the-art approaches.

Communities serve as basic structures for understanding the organization of many real-world networks, such as social, biological, collaboration, and communication networks. Recently, community search over large graphs has attracted significantly increasing attention, from simple and static graphs to evolving, attributed, location-based graphs. Different from the well-studied problem of community detection that finds all communities in an entire network, community search is to find the cohesive communities w.r.t. the query nodes. In this tutorial, we survey the state-of-the-art of community search on various kinds of networks across different application areas such as densely-connected community search, attributed community search, social circle discovery, and querying geosocial groups. We first highlight the challenges posed by the community search problems. We continue the presentation of their principles, methodologies, algorithms, and applications, and give a comprehensive comparison of the state-of-the-art techniques. This tutorial finally concludes by offering future directions for research in this important and growing area.

Searching for local communities is an important research challenge that allows for personalized community discovery and supports advanced data analysis in various complex networks, such as the World Wide Web, social networks, and brain networks. The evolution of these networks over time has motivated several recent studies to identify local communities in temporal networks. Given any query nodes, Community Search aims to find a densely connected subgraph containing query nodes. However, existing community search approaches in temporal networks have two main limitations: (1) they adopt pre-defined subgraph patterns to model communities, which cannot find communities that do not conform to these patterns in real-world networks, and (2) they only use the aggregation of disjoint structural information to measure quality, missing the dynamic of connections and temporal properties. In this paper, we propose a query-driven Temporal Graph Convolutional Network (CS-TGN) that can capture flexible community structures by learning from the ground-truth communities in a data-driven manner. CS-TGN first combines the local query-dependent structure and the global graph embedding in each snapshot of the network and then uses a GRU cell with contextual attention to learn the dynamics of interactions and update node embeddings over time. We demonstrate how this model can be used for interactive community search in an online setting, allowing users to evaluate the found communities and provide feedback. Experiments on real-world temporal graphs with ground-truth communities validate the superior quality of the solutions obtained and the efficiency of our model in both temporal and interactive static settings.

Simplifying social networks via triangle-based cohesive subgraphs

Community search that finds query-dependent communities has been studied on various kinds of graphs. As one instance of community search, intimate-core group (community) search over a weighted graph is to find a connected <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core containing all query nodes with the smallest group weight. However, existing state-of-the-art methods start from the maximal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core to refine an answer, which is practically inefficient for large networks. In this paper, we develop an efficient framework, called <u>l</u> ocal <u>e</u> xploration <u>k</u> -core <u>s</u> earch (LEKS), to find intimate-core groups in graphs. We propose a small-weighted spanning tree to connect query nodes, and then expand the tree level by level to a connected <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core, which is finally refined as an intimate-core group. In addition, to support the intimate group search over large weighted graphs, we develop a weighted-core index (WC-index) and two new WC-index-based algorithms for expansion and refinement phases in LEKS. Specifically, we propose a WC-index-based expansion to efficiently find a candidate graph of intimate-core group, leveraging on a two-level expansion of <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -breadth and 1-depth. We propose two graph removal strategies: coarse-grained refinement is designed for large graphs to delete a batch of nodes in a few iterations; fine-grained refinement is designed for small graphs to remove nodes carefully and achieve high-quality answers. Extensive experiments on real-life networks with ground-truth communities validate the effectiveness and efficiency of our proposed methods.

These proceedings contain the papers selected for presentation at the seventh edition of the ACM SIGSPATIAL Workshop on Location-Based Social Networks (LBSN 2014) which is being held in conjunction with the ACM SIGSPATIAL GIS conference in 2014. Social networks have been prevalent on the Internet and become a hot research topic attracting many professionals from a variety of fields. The advances in location-acquisition and mobile communication technologies empower people to use location data with existing online social networks. The dimension of location helps bridge the gap between the physical world and online social networking services. Furthermore, people in an existing social network can expand their social structure with the new interdependency derived from their locations. As location is one of the most important components of user context, extensive knowledge about an individual's interests, behaviors, and relationships with others can be learned from locations. These kinds of location-embedded and location-driven social structures are known as location-based social networks. The objective of this workshop is to provide professionals, researchers, and technologists with a single forum where they can discuss and share the state-of-the-art of LBSN development and applications, present their ideas and contributions, and set future directions in emerging innovative research for location based social networks. This year program is composed of two tracks of predominantly "social" and "spatial" views on LBSNs.

In the last two decades, the issue of Influence Maximization (IM) in traditional online social networks has been extensively studied since it was proposed. It is to find a seed set which has maximum influence spread under a specific network transmission model. However, in real life, the information can be spread not only through online social networks, but also between neighbors who are close to each other in the physical world. Location-Based Social Network (LBSN) is a new type of social network which is emerging increasingly nowadays. In a LBSN, users can not only make friends, but also share the events they participate in at different locations by checking in. In this paper, we aim to study the IM in LBSNs, where we consider both the influence of online and offline interactions. A two-layer network model and an information propagation model are proposed. Also, we formalize the IM problem in LBSNs and present an algorithm obtaining an approximation factor of \((1 - 1/e - \epsilon \)) in near-linear expected time. The experimental results show that the algorithm is efficient meanwhile offering strong theoretical guarantees.

Trust-aware location recommendation in location-based social networks: A graph-based approach

Location-based social network (LBSN) is a highly used social network at present. Detecting the community structures in LBSN plays a significant role in analyzing the network comprehensively. Modularity optimization is a traditional social network community detection method which has excellent performance. Whereas, resolution limit is an inherent problem of it. Some existing community detection methods for LBSN fail to take account of the full network information. To solve above problems, a network weighting scheme is proposed in this paper. Firstly, we combine a network embedding method that named walk2friends with this paper. In view of the shortcoming of walk2friends, we proposes a collaborative filtering method based on social relationship to improve it. Then, the learned similarity is obtained by improved walk2friends and the structural similarity is obtained by a random walk method. These two similarities are fused and then transformed into edge weight. Finally, the undirected unweighted graph is preprocessed as an undirected weighted graph. Experimental results on real networks show that when using modularity optimization method for optimization the weighting scheme not only can avoid the resolution limit but also applicable to LBSN . The mined community is more cohesive than the existing work on LBSN.

The Maximum Clique Problem (MCP) is one of the most important combinatorial optimization problems that has many practical applications such as community search in social networks. Since the MCP is known to be NP-hard, much effort has been devoted to the development of metaheuristic algorithms to find a high quality clique (solution) within reasonable running times. The Multi-start k-opt Local Search incorporating k-opt local search (MKLS) is well known as a simple and effective metaheuristic for MCP. However it takes long time to search the high-quality solution for difficult massive graphs such as real world social networks, because the search space is too large. In the case of applying metaheuristic algorithms for massive sparse graphs, adequate process such as reduction process is necessary to focus on promising search space. In this paper, we present a Multi-start k-opt Local Search with graph Reduction process (MKLS-R), for solving the maximum clique problem on massive graphs. MKLS-R is evaluated on difficult massive graphs of Network-Repository graphs. The experimental results showed that the graph reduction process in MKLS-R contributes to the improvement of the search performance of MKLS for the difficult massive graphs.

Friend recommendation is an important real-world application in Location-based Social Networks (LBSN), helping users discover potential friends and enhance their overall happiness. LBSN mainly comprises two distinct data structures: spatio-temporal data for human mobility and graph data for social networks. These two data structures make it challenging to model the complex relationships between them, which are essential for comprehensively understanding users’ lives. Previous studies have either modeled user trajectories and social networks separately or used classical simple graph-based methods, where a simple edge links only two nodes, failing to capture the multiple relationships inherent in LBSN. Furthermore, most studies have relied on Euclidean space to train their graph models, which could result in significant distortion because of tree-like social network data structure. To address these limitations, we propose a novel heterogeneous LBSN hypergraph that represents user check-in records and continuous trajectories—comprising multiple Points of Interest (POI)—as hyperedges, enabling the representation of complex spatio-temporal relationships. This approach enables us to link multiple nodes of different types by hyperedges and use hyperbolic spaces to create more efficient graph representations. Additionally, we devise a new type-specific attention mechanism for our Heterogeneous Hyperbolic Hypergraph Neural Network (H 3 GNN), which is end-to-end trainable and employs supervised contrastive learning to learn hypergraph node embeddings for the subsequent friend recommendation task with the help of hyperbolic space. Finally, our model H 3 GNN achieves better results than existing methods on six real-world city datasets, and our ablation studies demonstrate the effectiveness of each component. Additionally, our experiments indicate that H 3 GNN requires less data storage and training time compared to previous methods.