Characterizing cycle structure in complex networks

TL;DR

This paper introduces a new matrix and index to analyze cycle structures in networks, revealing that cycle ratio provides unique insights and improves identification of vital nodes compared to traditional metrics.

Contribution

It proposes the cycle number matrix and cycle ratio index, offering novel tools for analyzing cycle structures and node importance in complex networks.

Findings

Cycle ratio provides distinct node rankings from traditional metrics.

Cycle ratio effectively identifies vital nodes for network robustness.

Cycle ratio outperforms other benchmarks in spreading and synchronization tasks.

Abstract

Cycle is the simplest structure that brings redundant paths in network connectivity and feedback effects in network dynamics. Focusing on cycle structure, this paper defines a new matrix, named cycle number matrix, to represent cycle information of a network, and an index, named cycle ratio, to quantify the node importance. Experiments on real networks suggest that cycle ratio contains rich information in addition to well-known benchmark indices, for example, the node rankings by cycle ratio are largely different from rankings by degree, H-index, coreness, betweenness and articulation ranking, while the rankings by degree, H-index, coreness are very similar to each other. Extensive experiments on identifying vital nodes that maintain network connectivity, facilitate network synchronization and maximize the early reach of spreading show that cycle ratio is competitive to betweenness and overall better than other benchmarks. We believe the in-depth analyses on cycle structure may yield novel insights, metrics, models and algorithms for network science.