An algorithm for counting circuits: application to real-world and random graphs

TL;DR

This paper presents an algorithm that efficiently estimates the number of circuits in both real-world and random graphs, providing insights into their structure and complexity.

Contribution

The paper introduces a novel algorithm capable of estimating circuit counts in graphs, applicable to large real-world networks and sparse random graphs.

Findings

Estimates the number of circuits as a function of length in various graphs.

Provides analytical results for circuit entropy in sparse random graphs.

Demonstrates polynomial-time estimation of large circuit counts in real-world networks.

Abstract

We introduce an algorithm which estimates the number of circuits in a graph as a function of their length. This approach provides analytical results for the typical entropy of circuits in sparse random graphs. When applied to real-world networks, it allows to estimate exponentially large numbers of circuits in polynomial time. We illustrate the method by studying a graph of the Internet structure.