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

The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of edges. What does a typical random graph look like, if drawn from an exponential model subject to such constraints? Will there be a similar phase transition phenomenon (as one varies the parameters) as that which occurs in the unconstrained exponential model? We present some general results for this constrained model and then apply them to get concrete answers in the edge-triangle model with fixed density of edges.Comment: 23 pages, 4 figure

We describe the asymptotic properties of the edge-triangle exponential random graph model as the natural parameters diverge along straight lines. We show that as we continuously vary the slopes of these lines, a typical graph drawn from this model exhibits quantized behavior, jumping from one complete multipartite graph to another, and the jumps happen precisely at the normal lines of a polyhedral set with infinitely many facets. As a result, we provide a complete description of all asymptotic extremal behaviors of the model.Comment: 38 pages, 7 figure

The exponential family of random graphs is one of the most promising class of network models. Dependence between the random edges is defined through certain finite subgraphs, analogous to the use of potential energy to provide dependence between particle states in a grand canonical ensemble of statistical physics. By adjusting the specific values of these subgraph densities, one can analyze the influence of various local features on the global structure of the network. Loosely put, a phase transition occurs when a singularity arises in the limiting free energy density, as it is the generating function for the limiting expectations of all thermodynamic observables. We derive the full phase diagram for a large family of 3-parameter exponential random graph models with attraction and show that they all consist of a first order surface phase transition bordered by a second order critical curve.

Suppose that m drivers each choose a preferred parking space in a linear car park with n spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Classical parking functions correspond to the case m = n; we study here combinatorial and probabilistic aspects of this generalized case.We construct a family of bijections between parking functions PF(m, n) with m cars and n spots and spanning forests F (n + 1, n + 1 − m) with n + 1 vertices and n + 1 − m distinct trees having specified roots. This leads to a bijective correspondence between PF(m, n) and monomial terms in the associated Tutte polynomial of a disjoint union of n − m + 1 complete graphs. We present an identity between the "inversion enumerator" of spanning forests with fixed roots and the "displacement enumerator" of parking functions. The displacement is then related to the number of graphs on n + 1 labeled vertices with a fixed number of edges, where the graph has n + 1 − m disjoint rooted components with specified roots.We investigate various probabilistic properties of a uniform parking function, giving a formula for the law of a single coordinate. As a side result we obtain a recurrence relation for the displacement enumerator. Adapting known results on random linear probes, we further deduce the covariance between two coordinates when m = n.