Induced subgraphs in sparse random graphs with given degree sequence

TL;DR

This paper calculates the probability of specific induced subgraphs within sparse random graphs with a fixed degree sequence, extending results to broader graph models.

Contribution

It provides explicit probability formulas for induced subgraphs in sparse random graphs with given degree sequences, generalizing previous models.

Findings

Probability formulas for induced subgraphs in sparse graphs

Extension to graphs with specified degree sequences

Applicable for degree bounds up to o(n^{1/3})

Abstract

For any $S\subset [n]$, we compute the probability that the subgraph of $\mathcal{G}_{n,d}$ induced by $S$ is a given graph $H$ on the vertex set $S$. The result holds for any $d=o(n^{1/3})$ and is further extended to $\mathcal{G}_{{\bf d}}$, the probability space of random graphs with a given degree sequence $\bf d$.