Identifying sparse and dense sub-graphs in large graphs with a fast algorithm

TL;DR

This paper presents a fast algorithm for identifying small sparse sub-graphs within large dynamic or static graphs by leveraging Estrada-Benzi communicability and Krylov approximation, achieving efficient recovery with low computational complexity.

Contribution

The work introduces a novel, computationally efficient method for sub-graph identification that works in both dynamic and static large graphs, improving speed and accuracy over existing approaches.

Findings

Successful recovery of sparse sub-graphs using the proposed method.

Achieves low computational complexity, O(N n) with multiple backgrounds.

Single background approach also effective with complexity O(n log(n)).

Abstract

Identifying the nodes of small sub-graphs with no a priori information is a hard problem. In this work, we want to find each node of a sparse sub-graph embedded in both dynamic and static background graphs, of larger average degree. We show that exploiting the summability over several background realizations of the Estrada-Benzi communicability and the Krylov approximation of the matrix exponential, it is possible to recover the sub-graph with a fast algorithm with computational complexity O(N n). Relaxing the problem to complete sub-graphs, the same performance is obtained with a single background. The worst case complexity for the single background is O(n log(n)).