Phase Transitions in Edge-Weighted Exponential Random Graphs: Near-Degeneracy and Universality

Conventionally used exponential random graphs cannot directly model weighted networks as the underlying probability space consists of simple graphs only. Since many substantively important networks are weighted, this limitation is especially problematic. We extend the existing exponential framework by proposing a generic common distribution for the edge weights. Minimal assumptions are placed on the distribution, that is, it is non-degenerate and supported on the unit interval. By doing so, we recognize the essential properties associated with near-degeneracy and universality in edge-weighted exponential random graphs.Comment: 15 pages, 4 figures. This article extends arXiv:1607.04084, which derives general formulas for the normalization constant and characterizes phase transitions in exponential random graphs with uniformly distributed edge weights. The present article places minimal assumptions on the edge-weight distribution, thereby recognizing essential properties associated with near-degeneracy and universalit

Exponential Random Graphs and a Generalization of Parking Functions

Random graphs are a powerful tool in the analysis of modern networks. Exponential random graph models provide a framework that allows one to encode desirable subgraph features directly into the probability measure. Using the theory of graph limits pioneered by Borgs et. al. as a foundation, we build upon the work of Chatterjee & Diaconis and Radin & Yin. We add complexity to the previously studied models by considering exponential random graph models with edge-weights coming from a generic distribution satisfying mild assumptions. In particular, we show that a large family of two-parameter, edge-weighted exponential random graphs display a phase transtion and identify the limiting behavior of such graphs in the dual space provided by the Legendre-Fenchel transform. For finite systems, we analyze the mixing time of exponential random graph models. The mixing time of unweighted exponential random graphs was studied by Bhamidi, Bresler, and Sly. We extend upon the work of Levin, Luczak, and Peres by studying the Glauber dynamics of a certain vertex-weighted exponential random graph model on the complete graph. Specifically, we identify regions of the parameter space where the mixing time is Θ(n log n) and where it is exponentially slow. Toward the end of this work, we take a drastic turn in a different direction by studying a generalization of parking functions that we call interval parking functions. Parking functions are a classical combinatorial object dating back to the work of Konheim and Weiss in the 1960s. Among other things, we explore the connections that bioutcomes of interval parking functions have to various partial orders on the symmetric group on n letters including the (left) weak order, (strong) Bruhat order, and the bubble-sorting order

Interval parking functions

Interval parking functions (IPFs) are a generalization of ordinary parking functions in which each car is willing to park only in a fixed interval of spaces. Each interval parking function can be expressed as a pair (a, b), where a is a parking function and b is a dual parking function. We say that a pair of permutations (x,y) is reachable if there is an IPF (a,b) such that x, y are the outcomes of a, b, respectively, as parking functions. Reachability is reflexive and antisymmetric, but not in general transitive. We prove that its transitive closure, the pseudoreachability order, is precisely the bubble-sorting order on the symmetric group Sn, which can be expressed in terms of the normal form of a permutation in the sense of du Cloux; in particular, it is isomorphic to the product of chains of lengths 2,...,n. It is thus seen to be a special case of Armstrong’s sorting order, which lies between the Bruhat and (left) weak orders