Circuits in random graphs: from local trees to global loops

TL;DR

This paper analyzes the structure of circuits and loops in random regular graphs, providing analytic results for infinite graphs and computational enumeration for finite graphs, impacting the cavity method's validity.

Contribution

It offers new analytic formulas for circuits in infinite graphs and a counting algorithm for finite graphs, linking local tree-like structures to global loops.

Findings

Analytic results agree with existing exact solutions.

Finite N enumeration reveals behavior of circuits in finite graphs.

Implications for the cavity approach's validity.

Abstract

We compute the number of circuits and of loops with multiple crossings in random regular graphs. We discuss the importance of this issue for the validity of the cavity approach. On the one side we obtain analytic results for the infinite volume limit in agreement with existing exact results. On the other side we implement a counting algorithm, enumerate circuits at finite N and draw some general conclusions about the finite N behavior of the circuits.