On the Shannon Capacity of Probabilistic Graphs

Let G = (V, E) be a graph with vertex set V and edge set E, and let P be a probability distribution on V. The nth normal power of G is the graph Gn = (Vn,En), where Vn is the set of sequences of length n from V, and two sequences xn, yn ∈ Vn are connected if xi = yi, or {xi, yi} ∈ E for every i = 1, ..., n. Let Gn(P, ϵ) denote the subgraph induced in Gn by the "(P, ϵ)-typical sequences," i.e., the sequences in which the frequencies of elements of V are within ϵ of their P-probabilities. We give an upper bound for the logarithm of the independence number α(Gn(P, ϵ)) of the graph Gn(P, ϵ) for large n and small ϵ. Our bound λ(G, P) is a probabilistic analogue of the logarithm of Lovász′s bound for the Shannon capacity of graphs. Paralleling Lovász′s paper, we prove, using recent results from the theory of antiblocking pairs of polyhedra, that λ(G, P) is additive with respect to the product of probability distributions and two kinds of graph products. We also investigate the relationship of the functional λ(G, P) with other functionals defined on probabilistic graphs : graph entropy and complementary graph entropy.