Abstract
We study the pressure of the “edge-triangle model”, which is equivalent to the cumulant generating function of triangles in the Erdös–Rényi random graph. The investigation involves a population dynamics method on finite graphs of increasing volume, as well as a discretization of the graphon variational problem arising in the infinite volume limit. As a result, we locate a curve in the parameter space where a one-step replica symmetry breaking transition occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon with a structure very close to the one of an equi-bipartite graph.
Highlights
Sampling random graphs with prescribed macroscopic properties received considerable attention in recent years
We investigate the sampling from the canonical ensemble by studying the pressure of the edge-triangle model, or equivalently the cumulant generating function μp(α) of triangles in the Erdös–Rényi model with parameter p
We shall collect multiple evidences that the structure of graphs in the canonical ensemble has only two possibilities: it is either the constant graphon describing the Erdös–Rényi graph or it is the graphon describing the 1-step replica symmetric breaking solution
Summary
Sampling random graphs with prescribed macroscopic properties (such as a given density of certain subgraphs) received considerable attention in recent years. From a statistical physics perspective, one can think of two procedures:. – the micro-canonical ensemble, where the sampling is performed with a uniform distribution over the set of all graphs that satisfy the macroscopic constraint exactly;. – the canonical ensemble, where the sampling is done with respect to a larger set of graphs that satisfies the macroscopic constraint only on average
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