A Fast Algorithm to Compute Maximum k-Plexes in Social Network Analysis

A clique model is one of the most important techniques on the cohesive subgraph detection; however, its applications are rather limited due to restrictive conditions of the model. Hence much research resorts to k-plex a graph in which any vertex is adjacent to all but at most k vertices which is a relaxation model of the clique. In this paper, we study the maximum k-plex problem and propose a fast algorithm to compute maximum k-plexes by exploiting structural properties of the problem. In an n-vertex graph, the algorithm computes optimal solutions in cn time for a constant c < 2 depending only on k. To the best of our knowledge, this is the first algorithm that breaks the trivial theoretical bound of 2 for each k ≥ 3. We also provide experimental results over multiple real-world social network instances in support. Introduction In computational social networks, finding a large cohesive subgraph is an extensively studied topic with a large number of applications. Clique is one of the earliest and most commonly used models in the field of cohesive subgraphs detection. A clique is a graph with an edge between any pair of vertices, which can be regarded as the most cohesive graph. The MAXIMUM CLIQUE problem, to find a clique of maximum size in a graph, is a fundamental problem in graph algorithms not only having great applications in social networks but also finding applications in ad hoc wireless networks (Chen, Liestman, and Liu 2004), data mining (Washio and Motoda 2003), biochemistry and genomics (Butenko and Wilhelm 2006), and many others. Due to its overly restrictive (Alba 1973) and modeling disadvantages (Freeman 1992), the clique has been challenged by many practical problems. Alternative approaches were suggested that essentially relaxed the definition of cliques. Researchers have relaxed a variety of clique properties including familiarity, reachability, and robustness (Balasundaram, Butenko, and Hicks 2011). In graph theoretic terms, these properties correspond to vertex degree, path length, and connectivity respectively. This paper focuses on a relaxation model of clique by relaxing its familiarty restriction known as a k-plex (Seidman and Foster 1978). A simple undirected graph with n vertices is a k-plex if the degree of each vertex of the graph is at least n − k. When k = 1, a 1-plex is a clique. In the MAXIMUM k-PLEX problem, we aim to find a maximum vertex subset S of a given graph such that the subgraph G[S] induced by S is a k-plex. The applications and research on k-plex receive growing attention such as using k-plex to analyze social networks of terrorists (Krebs 2002), clustering and partitioning of graph-based data using k-plex (Du et al. 2007; Newman 2001), etc. Note that the complement graph of a k-plex is a graph of maximum degree at most k− 1. To find a maximum k-plex in a graph G is equivalent to find a maximum induced subgraph of degree bounded by k − 1 in the complement graph of G. The later problem is also known as the k′-BOUNDED-DEGREE VERTEX DELETION problem (to make the degree of a graph at most k′ by deleting a minimum number of vertices). k′-BOUNDED-DEGREE VERTEX DELETION itself also has many applications in serval areas (Fellows et al. 2011; Xiao 2015). The NP-completeness of MAXIMUM k-PLEX and k′BOUNDED-DEGREE VERTEX DELETION problems with each fixed k ≥ 1 (or k′ ≥ 0) was established many years ago (Lewis and Yannakakis 1980). For MAXIMUM 1-PLEX, known as MAXIMUM CLIQUE or MAXIMUM INDEPENDENT SET in the complement graph, it is a fundamental problem in exact exponential algorithms and it can be solved in O∗(1.1996n) time (Xiao and Nagamochi 2013) in an nvertex graph. For k = 2, MAXIMUM 2-PLEX can be solved in O∗(1.3656n) time (Xiao and Kou 2016). A simple bruteforce algorithm for MAXIMUM k-PLEX by enumerating and checking all vertex subsets of the graph runs in 2n time. We are not aware of any algorithm faster than the trivial exponential bound 2 for any k ≥ 3. In parallel, Balasundaram et al. (2011) gave an integer programming formulation and designed a branch-and-cut algorithm to solve MAXIMUM k-PLEX exactly. McClosky et al. (2012) derived a new upper bound on the cardinality of k-plexes and adapted some clique combinatorial algorithms to find maximum k-plexes, both of heuristic and exact nature. Moser et al. (2012) gave an exact algorithm with better experimental results. All the above exact algorithms run in 2n time theoretically. Our Contributions. This paper contributes to the k-plex literature both from theory and practice. We investigate several structural properties of MAXIMUM k-PLEX, most of which are related to the lower bound and will be used to prune the search branches in our algorithm. Based on these properties, we design a branch-and-search algorithm for MAXIMUM k-PLEX and analyze its running time bound in a theoretical way. We prove Theorem 1. Some values of σk for small k are shown in Table 1. Theorem 1. MAXIMUM k-PLEX can be solved in σ kn time, where σk < 2 is a value related to k.