Abstract. We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite node limit. We then determine the optimizing graphs for small triangle density and a range of edge density, though we can only prove they are local, not global, maxima of the entropy density. With this assumption we then prove that the resulting entropy density must lose its analyticity in various regimes. In particular this implies the existence of a phase transition between distinct heterogeneous multipartite phases at low triangle density, and a phase transition between these phases and the disordered phase at high triangle density.

We derive the full phase diagram for a large family of two-parameter exponential random graph models, each containing a first order transition curve ending in a critical point.Comment: Published in at http://dx.doi.org/10.1214/12-AAP907 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

No abstract

No abstract

Abstract. We prove that the asymptotic entropy of large simple graphs, as a function of fixed edge and triangle densities, is nondifferentiable along a certain curve.

We prove for a large class of tilings that, given a finite tile set, if it is possible to tile Euclidean n-space with isometric copies of this set, then there is a tiling with the 'local isomorphism property'.

No abstract

We study the asymptotics of large, simple, labeled graphs constrained by the densities of edges and of k-star subgraphs, k ≥ 2 fixed. We prove that under such constraints graphs are "multipodal": asymptotically in the number of vertices there is a partition of the vertices into M < ∞ subsets V 1 , V 2 , . . . , V M , and a set of well-defined probabilities g ij of an edge between any v i ∈ V i and v j ∈ V j . For 2 ≤ k ≤ 30 we determine the phase space: the combinations of edge and k-star densities achievable asymptotically. For these models there are special points on the boundary of the phase space with nonunique asymptotic (graphon) structure; for the 2-star model we prove that the nonuniqueness extends to entropy maximizers in the interior of the phase space.