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Sparse graph limits, entropy maximization and transitive graphs

In this paper we describe a triple correspondence between graph limits, information theory and group theory. We put forward a new graph limit concept called log-convergence that is closely connected to dense graph limits but its main applications are in the study of sparse graph sequences. We present an information theoretic limit concept for $k$-tuples of random variables that is based on the entropy maximization problem for joint distributions of random variables where a system of marginal distributions is prescribed. We give a fruitful correspondence between the two limit concepts that has a group theoretic nature. Our applications are in graph theory and information theory. We shows that if $H$ is a bipartite graph, $P_1$ is the edge and $t$ is the homomorphism density function then the supremum of $\log t(H,G)/\log t(P_1,G)$ in the set of all graphs $G$ is the same as in the set of graphs that are both edge and vertex transitive. This result gives a group theoretic approach to Sidorenko's famous conjecture. We obtain information theoretic inequalities regarding the entropy maximization problem. We investigate the limits of sparse random graphs and discuss quasi-randomness in our framework.