Statistics of Cycles: How Loopy is your Network?

TL;DR

This paper investigates the distribution of cycle lengths in large networks, demonstrating that it can serve as an ergodic estimator and characterizing it with parameters that distinguish different network types.

Contribution

It introduces a novel approach to analyze cycle distributions in large networks, including an exact counting method and a Monte-Carlo sampling algorithm for broad network families.

Findings

Cycle distribution peaks around h* ~ N^a

Exponent a relates to degree exponent g in scale-free networks

a increases with network size in scale-free models

Abstract

We study the distribution of cycles of length h in large networks (of size N>>1) and find it to be an excellent ergodic estimator, even in the extreme inhomogeneous case of scale-free networks. The distribution is sharply peaked around a characteristic cycle length, h* ~ N^a. Our results suggest that h* and the exponent a might usefully characterize broad families of networks. In addition to an exact counting of cycles in hierarchical nets, we present a Monte-Carlo sampling algorithm for approximately locating h* and reliably determining a. Our empirical results indicate that for small random scale-free nets of degree exponent g, a=1/(g-1), and a grows as the nets become larger.