Scalable probabilistic truss decomposition using central limit theorem and H-index

Abstract

Truss decomposition is a popular notion of hierarchical dense substructures in graphs. In a nutshell, k -truss is the largest subgraph in which every edge is contained in at least k triangles. Truss decomposition aims to compute k -trusses for each possible value of k . There are many works that study truss decomposition in deterministic graphs. However, in probabilistic graphs, truss decomposition is significantly more challenging and has received much less attention; state-of-the-art approaches do not scale well to large probabilistic graphs. Finding the tail probabilities of the number of triangles that contain each edge is a critical challenge of those approaches. This is achieved using dynamic programming which has quadratic run-time and thus not scalable to real large networks which, quite commonly, can have edges contained in many triangles (in the millions). To address this challenge, we employ a special version of the Central Limit Theorem (CLT) to obtain the tail probabilities efficiently. Based on our CLT approach we propose a peeling algorithm for truss decomposition that scales to large probabilistic graphs and offers significant improvement over state-of-the-art. We also design a second method which progressively tightens the estimate of the truss value of each edge and is based on h -index computation. In contrast to our CLT-based approach, our h -index algorithm (1) is progressive by allowing the user to see near-results along the way, (2) does not sacrifice the exactness of final result, and (3) achieves all these while processing only one edge and its immediate neighbors at a time, thus resulting in smaller memory footprint. We perform extensive experiments to show the scalability of both of our proposed algorithms.

Scaling Up Structural Clustering to Large Probabilistic Graphs Using Lyapunov Central Limit Theorem

“…From the techniques point of view, we utilize the Lyapunov CLT which has not been used before for the problem of clustering probabilistic graphs. Lyapunov CLT has been used in [8,12,11,10] for problems unrelated to clustering, such as core and truss decomposition, non-linear system control, and further applications relevant to social science and economics. The modelling of those problems in terms of Lyapunov CLT does not bear any similarity to our approach.…”