Statistics of cycles in large networks

TL;DR

This paper introduces a Markov Chain Monte Carlo method to analyze cycle length distributions in large networks, revealing how cycle properties relate to network growth rules and system size.

Contribution

It develops a novel MCMC sampling approach for cycle lengths and links cycle exponents to local growth rules, not just degree distributions.

Findings

Mean cycle length grows algebraically with network size.

Cycle exponent varies with local growth rules, not degree exponent.

Example: Internet at Autonomous Systems level has cycle exponent ~0.76.

Abstract

We present a Markov Chain Monte Carlo method for sampling cycle length in large graphs. Cycles are treated as microstates of a system with many degrees of freedom. Cycle length corresponds to energy such that the length histogram is obtained as the density of states from Metropolis sampling. In many growing networks, mean cycle length increases algebraically with system size. The cycle exponent $\alpha$ is characteristic of the local growth rules and not determined by the degree exponent $\gamma$. For example, $\alpha=0.76(4)$ for the Internet at the Autonomous Systems level.