# Counting cliques and cycles in scale-free inhomogeneous random graphs

## TL;DR

This paper analyzes the growth of small cliques and cycles in scale-free networks modeled as inhomogeneous random graphs with infinite-variance weights, providing asymptotic formulas for their expected counts.

## Contribution

It introduces exact integral expressions and asymptotic analysis for counting cliques and cycles in scale-free inhomogeneous random graphs with infinite-variance weights.

## Key findings

- Asymptotic growth rates for the number of cliques and cycles are derived.
- Exact integral formulas for counts of cliques and cycles are provided.
- Cycle asymptotics utilize the theory of circulant matrices.

## Abstract

Scale-free networks contain many small cliques and cycles. We model such networks as inhomogeneous random graphs with regularly varying infinite-variance weights. For these models, the number of cliques and cycles have exact integral expressions amenable to asymptotic analysis. We obtain various asymptotic descriptions for how the average number of cliques and cycles, of any size, grow with the network size. For the cycle asymptotics we invoke the theory of circulant matrices.