On the Dimension and Euler characteristic of random graphs

The inductive dimension dim(G) of a finite undirected graph G is a rational number defined inductively as 1 plus the arithmetic mean of the dimensions of the unit spheres dim(S(x)) at vertices x primed by the requirement that the empty graph has dimension −1. We look at the distribution of the random variable dim on the Erdös-Rényi probability space G(n, p), where each of the n(n− 1)/2 edges appears independently with probability 0 ≤ p ≤ 1. We show that the average dimension dn(p) = Ep,n[dim] is a computable polynomial of degree n(n − 1)/2 in p. The explicit formulas allow experimentally to explore limiting laws for the dimension of large graphs. In this context of random graph geometry, we mention explicit formulas for the expectation Ep,n[χ] of the Euler characteristic χ, considered as a random variable on G(n, p). We look experimentally at the statistics of curvature K(v) and local dimension dim(v) = 1 + dim(S(v)) which satisfy χ(G) = ∑ v∈V K(v) and dim(G) = 1 |V | ∑ v∈V dim(v). We also look at the signature functions f(p) = Ep[dim], g(p) = Ep[χ] and matrix values functions Av,w(p) = Covp[dim(v),dim(w)], Bv,w(p) = Cov[K(v),K(w)] on the probability space G(p) of all subgraphs of a host graph G = (V,E) with the same vertex set V , where each edge is turned on with probability p. 1. Dimension and Euler characteristic of graphs The inductive dimension for graphs G = (V,E) is formally close to the MengerUhryson dimension in topology. It was in [7] defined as dim(∅) = −1,dim(G) = 1 + 1 |V | ∑