Cycle statistics in complex networks and Ihara's zeta function

Abstract

Network representations are popular tools for characterizing and visualizing patterns of interaction between the microconstituents of large, complex synthetic, social, or biological systems. They reduce the full complexity of such systems to topological properties of their associated graphs, which are more amenable to analysis. In particular, the cyclic structure of complex networks is receiving increasing attention, since the presence of cycles affects strongly the behavior of processes supported by these networks. In this paper, we survey the analysis of cyclic properties of networks, and in particular the use of Ihara’s zeta function for counting cycles in networks.