s-Renyi Random Graph Research Articles

Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering

Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs. For an Erd\H{o}s-Renyi random graph, we identify a threshold probability at which with high probability the symmetric edge polytope shares many facet-supporting hyperplanes with that of a complete graph. We also investigate the relationship between the average local clustering, also known as the Watts-Strogatz clustering coefficient, and the number of facets for graphs with either a fixed number of edges or a fixed degree sequence. We use well-known Markov Chain Monte Carlo sampling methods to generate empirical evidence that for a fixed degree sequence, higher average local clustering in a connected graph corresponds to higher facet numbers in the associated symmetric edge polytope.

Local limit theorems for subgraph counts

We introduce a general framework for studying anticoncentration and local limit theorems for random variables, including graph statistics. Our methods involve an interplay between Fourier analysis, decoupling, hypercontractivity of Boolean functions, and transference between ``fixed-size'' and ``independent'' models. We also adapt a notion of ``graph factors'' due to Janson. As a consequence, we derive a local central limit theorem for connected subgraph counts in the Erd\H{o}s-Renyi random graph $G(n,p)$, building on work of Gilmer and Kopparty and of Berkowitz. These results improve an anticoncentration result of Fox, Kwan, and Sauermann and partially answers a question of Fox, Kwan, and Sauermann. We also derive a local limit central limit theorem for induced subgraph counts, as long as $p$ is bounded away from a set of ``problematic'' densities, partially answering a question of Fox, Kwan, and Sauermann. We then prove these restrictions are necessary by exhibiting a disconnected graph for which anticoncentration for subgraph counts at the optimal scale fails for all constant $p$, and finding a graph $H$ for which anticoncentration for induced subgraph counts fails in $G(n,1/2)$. These counterexamples resolve anticoncentration conjectures of Fox, Kwan, and Sauermann in the negative. Finally, we also examine the behavior of counts of $k$-term arithmetic progressions in subsets of $\mathbb{Z}/n\mathbb{Z}$ and deduce a local limit theorem wherein the behavior is Gaussian at a global scale but has nontrivial local oscillations (according to a Ramanujan theta function). These results improve on results of and answer questions of the authors and Berkowitz, and answer a question of Fox, Kwan, and Sauermann.

Stein’s method of exchangeable pairs in multivariate functional approximations

In this paper we develop a framework for multivariate functional approximation by a suitable Gaussian process via an exchangeable pairs coupling that satisfies a suitable approximate linear regression property, thereby building on work by Barbour (1990) and Kasprzak (2020). We demonstrate the applicability of our results by applying it to joint subgraph counts in an Erd\H{o}s-Renyi random graph model on the one hand and to vectors of weighted, degenerate $U$-processes on the other hand. As a concrete instance of the latter class of examples, we provide a bound for the functional approximation of a vector of success runs of different lengths by a suitable Gaussian process which, even in the situation of just a single run, would be outside the scope of the existing theory.

Random matrices have simple spectrum

Let $M_n = (\xi_{ij})_{1 \leq i,j \leq n}$ be a real symmetric random matrix in which the upper-triangular entries $\xi_{ij}, i<j$ and diagonal entries $\xi_{ii}$ are independent. We show that with probability tending to 1, $M_n$ has no repeated eigenvalues. As a corollary, we deduce that the Erd{\H o}s-Renyi random graph has simple spectrum asymptotically almost surely, answering a question of Babai.

Saturation in random graphs

A graph H is Ks-saturated if it is a maximal Ks-free graph, i.e., H contains no clique on s vertices, but the addition of any missing edge creates one. The minimum number of edges in a Ks-saturated graph was determined over 50 years ago by Zykov and independently by Erdi¾?s, Hajnal and Moon. In this paper, we study the random analog of this problem: minimizing the number of edges in a maximal Ks-free subgraph of the Erdi¾?s-Renyi random graph Gn, p. We give asymptotically tight estimates on this minimum, and also provide exact bounds for the related notion of weak saturation in random graphs. Our results reveal some surprising behavior of these parameters. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 169-181, 2017

Large deviations of empirical neighborhood distribution in sparse random graphs

Consider the Erd\H{o}s-Renyi random graph on n vertices where each edge is present independently with probability c/n, with c>0 fixed. For large n, a typical random graph locally behaves like a Galton-Watson tree with Poisson offspring distribution with mean c. Here, we study large deviations from this typical behavior within the framework of the local weak convergence of finite graph sequences. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported on trees. We also establish large deviations for other commonly studied random graph ensembles such as the uniform random graph with given number of edges growing linearly with the number of vertices, or the uniform random graph with given degree sequence. To prove our results, we introduce a new configuration model which allows one to sample uniform random graphs with a given neighborhood distribution, provided the latter is supported on trees. We also introduce a new class of unimodular random trees, which generalizes the usual Galton Watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth. These generalized Galton Watson trees turn out to be useful in the analysis of unimodular random trees and may be considered to be of interest in their own right.