Eigenvalues of random graphs with cycles

TL;DR

This paper presents a new, more intuitive approach to understanding the eigenvalues of random graphs with cycles by relating them to cycle weights using the method of moments, focusing on graphs with short cycles and circulant structures.

Contribution

It introduces a cycle-based perspective on spectral properties of random graphs, simplifying analysis compared to traditional free probability and cavity methods.

Findings

Derived relations between eigenvalues and cycle weights.

Analyzed spectral properties of graphs with short cycles.

Explored eigenvalues of circulant directed graphs.

Abstract

Networks are often studied using the eigenvalues of their adjacency matrix, a powerful mathematical tool with a wide range of applications. Since in real systems the exact graph structure is not known, researchers resort to random graphs to obtain eigenvalue properties from known structural features. However, this theory is far from intuitive and often requires training of free probability, cavity methods or a strong familiarity with probability theory. In this note we offer a different perspective on this field by focusing on the cycles in a graph. We use the so-called method of moments to obtain relation between eigenvalues and cycle weights and then we obtain spectral properties of random graphs with cyclic motifs. We use it to explore properties of the eigenvalues of adjacency matrices of graphs with short cycles and of circulant directed graphs. Although our result is not as powerful as the some of the existing methods, they are nevertheless useful and far easier to understand.