The Most Persistent Soft-Clique in a Set of Sampled Graphs

TL;DR

This paper introduces the most persistent soft-clique concept to identify densely connected vertex subsets that appear across multiple noisy graph instances, using novel optimization formulations and demonstrating effectiveness on synthetic and real social network data.

Contribution

It presents a new measure of clique-ness and formulates the persistent soft-clique problem as two equivalent optimization problems solved via a partial Lagrangian method.

Findings

Successfully identifies soft cliques in noisy graph data.

Effective on both synthetic and real social network datasets.

Robust to random noise and unreliable observations.

Abstract

When searching for characteristic subpatterns in potentially noisy graph data, it appears self-evident that having multiple observations would be better than having just one. However, it turns out that the inconsistencies introduced when different graph instances have different edge sets pose a serious challenge. In this work we address this challenge for the problem of finding maximum weighted cliques. We introduce the concept of most persistent soft-clique. This is subset of vertices, that 1) is almost fully or at least densely connected, 2) occurs in all or almost all graph instances, and 3) has the maximum weight. We present a measure of clique-ness, that essentially counts the number of edge missing to make a subset of vertices into a clique. With this measure, we show that the problem of finding the most persistent soft-clique problem can be cast either as: a) a max-min two person game optimization problem, or b) a min-min soft margin optimization problem. Both formulations lead to the same solution when using a partial Lagrangian method to solve the optimization problems. By experiments on synthetic data and on real social network data, we show that the proposed method is able to reliably find soft cliques in graph data, even if that is distorted by random noise or unreliable observations.