Algorithm for counting large directed loops

TL;DR

This paper introduces a Belief-Propagation algorithm designed to efficiently count large loops in directed networks, enabling analysis of complex network structures beyond small-scale enumeration.

Contribution

The paper presents a novel Belief-Propagation method for counting large loops in directed networks, validated against exhaustive methods and applicable to large-scale networks.

Findings

The algorithm accurately estimates loop counts in large directed networks.

It effectively compares loop structures between real and randomized networks.

The method outperforms traditional counting approaches in scalability.

Abstract

We derive a Belief-Propagation algorithm for counting large loops in a directed network. We evaluate the distribution of the number of small loops in a directed random network with given degree sequence. We apply the algorithm to a few characteristic directed networks of various network sizes and loop structures and compare the algorithm with exhaustive counting results when possible. The algorithm is adequate in estimating loop counts for large directed networks and can be used to compare the loop structure of directed networks and their randomized counterparts.