Characterizing cycle structure in complex networks

Abstract

AbstractA cycle is the simplest structure that brings redundant paths in network connectivity and feedback effects in network dynamics. An in-depth understanding of which cycles are important and what role they play on network structure and dynamics, however, is still lacking. In this paper, we define the cycle number matrix, a matrix enclosing the information about cycles in a network, and the cycle ratio, an index that quantifies node importance. Experiments on real networks suggest that cycle ratio contains rich information in addition to well-known benchmark indices. For example, node rankings by cycle ratio are largely different from rankings by degree, H-index, and coreness, which are very similar indices. Numerical experiments on identifying vital nodes for network connectivity and synchronization and maximizing the early reach of spreading show that the cycle ratio performs overall better than other benchmarks. Finally, we highlight a significant difference between the distribution of shorter cycles in real and model networks. We believe our in-depth analyses on cycle structure may yield insights, metrics, models, and algorithms for network science.