Related papers: Strong law of large numbers on graphs and groups
This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with…
We give variants of the Krein bound and the absolute bound for graphs with a spectrum similar to that of a strongly regular graph. In particular, we investigate what we call approximately strongly regular graphs. We apply our results to…
De Finetti's classical result of [18] identifying the law of an exchangeable family of random variables as a mixture of i.i.d. laws was extended to structure theorems for more complex notions of exchangeability by Aldous [1,2,3], Hoover…
We consider graphs with vertices of degree 1 or 2 and prove that the numbers of components of sizes 2 to q have a limit normal distribution for any q > 1. The result is also extended to multigraphs.
Sampling is a standard approach in big-graph analytics; the goal is to efficiently estimate the graph properties by consulting a sample of the whole population. A perfect sample is assumed to mirror every property of the whole population.…
We give a simpler proof using automata theory of a recent result of Kapovich, Weidmann and Myasnikov according to which so-called benign graphs of groups preserve decidability of the generalized word problem. These include graphs of groups…
In this work we give precise asymptotic expressions on the probability of the existence of fixed-size components at the threshold of connectivity for random geometric graphs.
The Chirikov standard map family is a one-parameter family of volume-preserving maps exhibiting hyperbolicity on a `large' but noninvariant subset of phase space. Based on this predominant hyperbolicity and numerical experiments, it is…
Given a Gromov-hyperbolic group $G$ endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of $G$. More generally, we obtain a…
This is an introduction to graph theory, from a geometric and analytic viewpoint. A finite graph $X$ is described by its adjacency matrix $d\in M_N(0,1)$, which can be thought of as being a kind of discrete Laplacian, and we first discuss…
Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind,…
We show that a sufficiently large graph of bounded degree can be decomposed into quasi-homogeneous pieces. The result can be viewed as a "finitarization" of the classical Farrell-Varadarajan Ergodic Decomposition Theorem.
For any fixed integer $R \geq 2$ we characterise the typical structure of undirected graphs with vertices $1, ..., n$ and maximum degree $R$, as $n$ tends to infinity. The information is used to prove that such graphs satisfy a labelled…
We prove a law of large numbers in terms of complete convergence of independent random variables taking values in increments of monotone functions, with convergence uniform both in the initial and the final time. The result holds also for…
The law of large numbers is one of the fundamental properties which algorithmically random infinite sequences ought to satisfy. In this paper, we show that the law of large numbers can be effectivized for an arbitrary Schnorr random…
Colouring the vertices of a graph $G$ according to certain conditions can be considered as a random experiment and a discrete random variable $X$ can be defined as the number of vertices having a particular colour in the proper colouring of…
In this paper we prove an extreme value law for a stochastic process obtained by iterating the R\'enyi map $x \mapsto \beta x \pmod 1$, where we assume that $\beta>1$ is an integer. Haiman (2018) derived a recursion formula for the Lebesgue…
We introduce a {\it non-regular} generalisation of the N\"{o}rlund mean, and show its equivalence with a certain moving average. The Abelian and Tauberian theorems establish relations with convergent sequences and certain power series. A…
A method for compression of large graphs and matrices to a block structure is further developed. Szemer\'edi's regularity lemma is used as a generic motivation of the significance of stochastic block models. Another ingredient of the method…
These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…