Query Optimal k-Plex Based Community in Graphs

Abstract

Community search problem, which is to find good communities given a set of query nodes in a graph, has attracted increasing research interest recently. Though various measurement models have been proposed to define and solve community search problem. Few of them could define a community concisely and have good quality of query results. They either involve additional constraints for modeling communities, such as size and diameter, or suffer from the free rider effect, i.e., include irrelevant subgraphs. In this paper, we propose a new k-plex based community model for community search. We show that our model not only is simple and clear, but also meets with basic requirements of defining a community search problem. We formulate the maximum k-plex community query (MCKPQ) problem, that is, given a set of query nodes Q, searching for optimal k-plex containing Q. We prove that MCKPQ is NP-hard, and it is hard to approximate in any constant factor. We first give exact solutions. Then, we propose an efficient branch-and-bound (B&B) method and design an effective upper bound function and a pruning strategy. Furthermore, we optimize the basic B&B by fast candidate generation. We also give a fast heuristic solution, which produces high-quality results in practice. The effectiveness of our model of community and the efficiency of our methods are verified by elaborate experiments.

A survey of community search over big graphs

“…In this section, we survey the CS solutions [45,205,195,187,125] using k-clique model. We can divide them into two groups, where the first group [45,205,195,187] focuses on simple graphs, while the second group [125] is developed for attributed graphs.…”

“…In this section, we survey the CS solutions [45,205,195,187,125] using k-clique model. We can divide them into two groups, where the first group [45,205,195,187] focuses on simple graphs, while the second group [125] is developed for attributed graphs. In the first group, the first one [45] uses quasi-clique model, the second one [205] adopts k-clique model, and the last two [195,187] are based on k-plex model.…”

“…We can divide them into two groups, where the first group [45,205,195,187] focuses on simple graphs, while the second group [125] is developed for attributed graphs. In the first group, the first one [45] uses quasi-clique model, the second one [205] adopts k-clique model, and the last two [195,187] are based on k-plex model. However, to our best knowledge, there is no systematic study to compare the goodness of different k-clique based models in real-life applications, which is crucial for researchers and practitioners to choose desirable models in practice.…”

“…Simple graphs Attributed graphs Keyword Location Temporal Influence (weight) Profile k-core [175,46,15,66] (P. 1, 2, 3, 4, 5) [61,58] (P. 6) [60,65,185,221] (P. 7,8,9) [129] (P. 10) [127,128,30,215,21,126] (P. 12,13) [31] (P. 14) k-truss [98,6,101] (P. 15,16) [102] (P. 17) -- [216] (P. 18) k-clique [45,205,195,187] (P. 19,20,21,22) -- [125] Example 2 Let us reconsider the graph G in Fig. 2(a).…”

“…Obviously, this method is too expensive and impractical for large graphs. To alleviate this issue, Wang et al developed a more advanced method based on the branch-and-bound paradigm with some effective pruning criteria and a heuristic method which performs fast but has no theoretical guarantee [187]. We skip the details due to space limitation.…”