Invariant random graphs with iid degrees in a general geography

Let D be a non-negative integer-valued random variable and let G = (V, E) be an infinite transitive finite-degree graph. Continuing the work of Deijfen and Meester [5] and Deijfen and Jonasson [4], we seek an Aut(G)-invariant random graph model with V as vertex set, iid degrees distributed as D and finite mean connections (i.e. the sum of the edge lengths in the graph metric of G of a given vertex has finite expectation). It is shown that if G has either polynomial growth or rapid growth, then such a random graph model exists if and only if E[D R(D)] < ∞. Here R(n) is the smallest possible radius of a combinatorial ball containing more than n vertices. With rapid growth we mean that the number of vertices in a ball of radius n is of at least order exp(nc) for some c > 0. All known transitive graphs have either polynomial or rapid growth. It is believed that no other growth rates are possible. When G has rapid growth, the result holds also when the degrees form an arbitrary invariant process. A counter-example shows that this is not the case when G grows polynomially. For this case, we provide other, quite sharp, conditions under which the stronger statement does and does not hold respectively. Our work simplifies and generalizes the results for G = Z in [4] and proves e.g. that with G = Zd, there exists an invariant model with finite mean connections if and only if E[D(d+1)/d] < ∞. When G has exponential Chalmers University of Technology Göteborg University [email protected] growth, e.g. when G is a regular tree, the condition becomes E[D log D] < ∞.