Convergent sequences of sparse graphs: A large deviations approach

In this paper we introduce a new notion of convergence of sparse graphs which we call Large Deviations or LD-convergence and which is based on the theory of large deviations. The notion is introduced by ”decorating” the nodes of the graph with random uniform i.i.d. weights and constructing random measures on [0, 1] and [0, 1]2 based on the decoration of nodes and edges. A graph sequence is defined to be converging if the corresponding sequence of random measures satisfies the Large Deviations Principle with respect to the topology of weak convergence on bounded measures on [0, 1]d, d = 1, 2. We then establish that LD-convergence implies several previous notions of convergence, namely so-called right-convergence, left-convergence, and partition-convergence. The corresponding large deviation rate function can be interpreted as the limit object of the sparse graph sequence. In particular, we can express the limiting free energies in terms of this limit object. Finally, we establish several previously unknown relationships between the formerly defined notions of convergence. In particular, we show that partition-convergence does not imply left or right-convergence, and that right-convergence does not imply partitionconvergence. ∗Microsoft Research New England; e-mail: [email protected]. †Microsoft Research New England; e-mail: [email protected]. ‡MIT; e-mail: [email protected] supported by the NSF grants CMMI-1031332.