Distribution of shortest cycle lengths in random networks

TL;DR

This paper derives analytical expressions for the distribution of shortest cycle lengths in various types of random networks, linking it to shortest path lengths and validating results with simulations.

Contribution

It introduces a novel analytical approach to determine the distribution of shortest cycle lengths in configuration model networks, extending previous work on shortest path lengths.

Findings

Analytical formulas for DSCL in ER, regular, and scale-free networks.

Good agreement between analytical results and simulations.

Calculated mean and variance of DSCL.

Abstract

We present analytical results for the distribution of shortest cycle lengths (DSCL) in random networks. The approach is based on the relation between the DSCL and the distribution of shortest path lengths (DSPL). We apply this approach to configuration model networks, for which analytical results for the DSPL were obtained before. We first calculate the fraction of nodes in the network which reside on at least one cycle. Conditioning on being on a cycle, we provide the DSCL over ensembles of configuration model networks with degree distributions which follow a Poisson distribution (Erdos-R\'enyi network), degenerate distribution (random regular graph) and a power-law distribution (scale-free network). The mean and variance of the DSCL are calculated. The analytical results are found to be in very good agreement with the results of computer simulations.