Listing dense subgraphs in large graphs plays a key task in varieties of network analysis applications like community detection. Clique, as the densest model, has been widely investigated. However, in practice, communities rarely form as cliques for various reasons, e.g., data noise. Therefore, $k$-plex, -- graph with each vertex adjacent to all but at most $k$ vertices, is introduced as a relaxed version of clique. Often, to better simulate cohesive communities, an emphasis is placed on connected $k$-plexes with small $k$. In this paper, we continue the research line of listing all maximal $k$-plexes and maximal $k$-plexes of prescribed size. Our first contribution is algorithm \emph{ListPlex} that lists all maximal $k$-plexes in $O^*(\gamma^D)$ time for each constant $k$, where $\gamma$ is a value related to $k$ but strictly smaller than 2, and $D$ is the degeneracy of the graph that is far less than the vertex number $n$ in real-word graphs. Compared to the trivial bound of $2^n$, the improvement is significant, and our bound is better than all previously known results. In practice, we further use several techniques to accelerate listing $k$-plexes of a given size, such as structural-based prune rules, cache-efficient data structures, and parallel techniques. All these together result in a very practical algorithm. Empirical results show that our approach outperforms the state-of-the-art solutions by up to orders of magnitude.
翻译:在大图表中列出密度浓密的子图在网络分析应用程序的种类中扮演着关键任务,比如社区检测。 Clique, 作为最稠密的模型, 已被广泛调查。 但是, 在实际中, 社区由于各种原因, 例如数据噪音等, 很少形成 clique 。 因此, $k$- plex, -- 以每个顶点相邻, 但最多为 $k$ 的图表, 被引入一个宽松的球形版。 通常, 要更好地模拟具有凝聚力的社区, 重点是以小美元连接的 $k$- 。 在本文中, 我们继续列出所有最大 $k$- lex 和 最大 $- lex- lex 的双倍。 我们的第一个贡献是算算法 \ emph{ListexPlex}, 将所有最大 $k$( $) 的双倍的双倍值作为宽松的版本。 $\ gammamamamamamaine to code lax liver rudeal rudeal rudeal rude, le lex lex lex lex lex levels a list list levels a levels ex listers list list list levels fol ex levels list lexn lex lex lex lex lex lex lex lex lex lex lex lex lex lex legs legs legs levels levels lesters legs lex lex exfol lex lex lex lex lex lex lex exfs lex lex lex lex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex