Related papers: Strong approximation in random towers of graphs
Strongly regular graphs are regular graphs with a constant number of common neighbours between adjacent vertices, and a constant number of common neighbours between non-adjacent vertices. These graphs have been of great interest over the…
A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on…
A family of random matrices is said to converge strongly to a limiting family of operators if the operator norm of every noncommutative polynomial of the matrices converges to that of the limiting operators. Recent developments surrounding…
We begin with a new analysis of formal words. Let w be a formal word in letters g_1,...,g_k. The word map associated with w maps the permutations s_1,...,s_k in S_n to the permutation obtained by replacing for each i, every occurrence of…
This document is an expanded version of a lecture presented at a conference on "Thin Groups and Superstrong Approximation" held at the Mathematical Sciences Research Institute in February 2012. Superstrong approximation is a criterion on a…
We provide a full and rigorous proof of a theorem attributed to \.Zuk, stating that random groups in the Gromov density model for d > 1/3 have property (T) with high probability. The original paper had numerous gaps, in particular, crucial…
A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and…
We propose a family of models to study the evolution of ties in a network of interacting agents by reinforcement and penalization of their connections according to certain local laws of interaction. The family of stochastic dynamical…
Social studies researchers use graphs to model group activities in social networks. An important property in this context is the centrality of a vertex: the inverse of the average distance to each other vertex. We describe a randomized…
We investigate a slight weakening of the classical property of strong approximation, which we call almost strong approximation, for connected reductive algebraic groups over global fields with respect to special sets of valuations. While…
We develop a new framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world…
For any positive integer $m$, the complete graph on $2^{2m}(2^m+2)$ vertices is decomposed into $2^m+1$ commuting strongly regular graphs, which give rise to a symmetric association scheme of class $2^{m+2}-2$. Furthermore, the…
A family of random matrices $\boldsymbol{X}^N=(X_1^N,\ldots,X_d^N)$ is said to converge strongly to a family of bounded operators $\boldsymbol{x}=(x_1,\ldots,x_d)$ when $\|P(\boldsymbol{X}^N,\boldsymbol{X}^{N*})\|\to\|P(\boldsymbol{x},…
The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets $S$ in its neighborhood has common neighborhood almost as large as in…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus $\mathbb{Z}_m^n$, where $m$ is even, to any fixed graph: we show that the corresponding probability distribution on…
Correlations and other collective phenomena in a schematic model of heterogeneous binary agents (individual spin-glass samples) are considered on the complete graph and also on 2d and 3d regular lattices. The system's stochastic dynamics is…
A wide variety of complex networks (social, biological, information etc.) exhibit local clustering with substantial variation in the clustering coefficient (the probability of neighbors being connected). Existing models of large graphs…
This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse…
Random graphs offer a useful mathematical representation of a variety of real world complex networks. Exponential random graphs, for example, are particularly suited towards generating random graphs constrained to have specified statistical…