Where Graph Topology Matters: The Robust Subgraph Problem

TL;DR

This paper introduces the robust subgraph problem, aiming to find highly resilient local subgraphs within large networks by considering topology and robustness, and proposes heuristic algorithms with experimental validation.

Contribution

It formulates the novel RLS-PROBLEM focusing on local robustness, proves its NP-hardness, and develops heuristic algorithms for practical solutions.

Findings

Heuristic algorithms outperform densest subgraph methods in robustness.

Robust subgraphs can be found at lower densities than densest subgraphs.

Experiments on real-world data validate the effectiveness of the proposed methods.

Abstract

Robustness is a critical measure of the resilience of large networked systems, such as transportation and communication networks. Most prior works focus on the global robustness of a given graph at large, e.g., by measuring its overall vulnerability to external attacks or random failures. In this paper, we turn attention to local robustness and pose a novel problem in the lines of subgraph mining: given a large graph, how can we find its most robust local subgraph (RLS)? We define a robust subgraph as a subset of nodes with high communicability among them, and formulate the RLS-PROBLEM of finding a subgraph of given size with maximum robustness in the host graph. Our formulation is related to the recently proposed general framework for the densest subgraph problem, however differs from it substantially in that besides the number of edges in the subgraph, robustness also concerns with the placement of edges, i.e., the subgraph topology. We show that the RLS-PROBLEM is NP-hard and propose two heuristic algorithms based on top-down and bottom-up search strategies. Further, we present modifications of our algorithms to handle three practical variants of the RLS-PROBLEM. Experiments on synthetic and real-world graphs demonstrate that we find subgraphs with larger robustness than the densest subgraphs even at lower densities, suggesting that the existing approaches are not suitable for the new problem setting.