Local Structure Theorems for Erdos Renyi Graphs and their Algorithmic Application

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.