Local Structure Theorems for Erdos Renyi Graphs and their Algorithmic Application

TL;DR

This paper investigates local structural properties of sparse Erdős-Rényi graphs, showing they are almost surely nowhere dense with bounded local treewidth, and applies these findings to develop efficient algorithms for subgraph isomorphism.

Contribution

The paper provides new proofs that sparse Erdős-Rényi graphs have bounded expansion and local treewidth, enabling efficient algorithms for certain graph problems.

Findings

Sparse Erdős-Rényi graphs are almost surely nowhere dense.

They have bounded local treewidth for certain parameters.

Efficient algorithms are developed for subgraph isomorphism problems.

Abstract

We analyze some local properties of sparse Erdos-Renyi graphs, where $d(n)/n$ is the edge probability. In particular we study the behavior of very short paths. For $d(n)=n^{o(1)}$ we show that $G(n,d(n)/n)$ has asymptotically almost surely (a.a.s.~) bounded local treewidth and therefore is a.a.s.~nowhere dense. We also discover a new and simpler proof that $G(n,d/n)$ has a.a.s.~bounded expansion for constant~$d$. The local structure of sparse Erdos-Renyi Gaphs is very special: The $r$-neighborhood of a vertex is a tree with some additional edges, where the probability that there are $m$ additional edges decreases with~$m$. This implies efficient algorithms for subgraph isomorphism, in particular for finding subgraphs with small diameter. Finally we note that experiments suggest that preferential attachment graphs might have similar properties after deleting a small number of vertices.