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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 aarXiv:1709.09152v2 fatcat:zgpfy23yvbdcngrsavnaomgjqa