Truly Scalable K-Truss and Max-Truss Algorithms for Community Detection in Graphs

Abstract

The notion of k-truss has been introduced a decade ago in social network analysis and security for community detection, as a form of cohesive subgraphs less stringent than a clique (set of pairwise linked nodes), and more selective than a k-core (induced subgraph with minimum degree k). A k-truss is an inclusion-maximal subgraph H in which each edge belongs to at least k − 2 triangles inside H . The truss decomposition establishes, for each edge e, the maximum k for which e belongs to a k-truss. Analogously to the largest clique and to the maximum k-core, the strongest community for k-truss is the max-truss, which corresponds to the k-truss having the maximum k. Even though the computation of truss decomposition and of the max-truss takes polynomial time, on a large scale, it suffers from handling a potentially cubic number of wedges. In this paper, we provide a new algorithm FMT, which advances the state of the art on different sides: lower execution time, lower memory usage, and no need for expensive hardware. We compare FMT experimentally with the most recent state-of-the-art algorithms on a set of large real-world and synthetic networks with over a billion edges. The massive improvement allows FMT to compute the max-truss of networks of tens of billions of edges on a single standard server machine.

On Breaking Truss-Based Communities

“…Let us remark that although \ is not a -truss, it may still contain a smaller -truss. Finally, we recall from [7] a useful property that lets us bound the trussness of a graph by the number of triangles and edges it contains. Theorem 3.7 ( [7]).…”

“…Finally, we recall from [7] a useful property that lets us bound the trussness of a graph by the number of triangles and edges it contains. Theorem 3.7 ( [7]). Given a graph with edges and T triangles, its trussness ( ) is at least T + 2.…”

“…The notion of -truss [6] has been the focus of many works which aim at detecting a maximal -truss for each (e.g., [7]), or a -truss containing certain nodes and/or attributes (e.g., [13,14]). There is also a considerable amount of work on extending the notion oftruss to capture application-specific requirements (e.g., [9,14,22]).…”

“…We also omit the versions of our heuristics that do not employ Truss Update, as they were more than one order of magnitude slower. In our implementations, we used the algorithm of [7] to compute the truss decomposition. Our code is available at https://bitbucket.org/ breakingtruss/kdd2021.…”

“…Proof of Theorem 3.7. The proof of Theorem 1 in [7] shows that ≥ + 2 for any subgraph of with triangles and edges. The claim holds as is a subgraph of itself.…”