14 research outputs found
On the spectral distribution of large weighted random regular graphs
McKay proved that the limiting spectral measures of the ensembles of
-regular graphs with vertices converge to Kesten's measure as
. In this paper we explore the case of weighted graphs. More
precisely, given a large -regular graph we assign random weights, drawn from
some distribution , to its edges. We study the relationship
between and the associated limiting spectral distribution
obtained by averaging over the weighted graphs. Among other results, we
establish the existence of a unique `eigendistribution', i.e., a weight
distribution such that the associated limiting spectral
distribution is a rescaling of . Initial investigations suggested
that the eigendistribution was the semi-circle distribution, which by Wigner's
Law is the limiting spectral measure for real symmetric matrices. We prove this
is not the case, though the deviation between the eigendistribution and the
semi-circular density is small (the first seven moments agree, and the
difference in each higher moment is ). Our analysis uses
combinatorial results about closed acyclic walks in large trees, which may be
of independent interest.Comment: Version 1.0, 19 page
Joint Vertex Degrees in an Inhomogeneous Random Graph Model
In a random graph, counts for the number of vertices with given degrees will
typically be dependent. We show via a multivariate normal and a Poisson process
approximation that, for graphs which have independent edges, with a possibly
inhomogeneous distribution, only when the degrees are large can we reasonably
approximate the joint counts as independent. The proofs are based on Stein's
method and the Stein-Chen method with a new size-biased coupling for such
inhomogeneous random graphs, and hence bounds on distributional distance are
obtained. Finally we illustrate that apparent (pseudo-) power-law type
behaviour can arise in such inhomogeneous networks despite not actually
following a power-law degree distribution.Comment: 30 pages, 9 figure
Global information from local observations of the noisy voter model on a graph
We observe the outcome of the discrete time noisy voter model at a single
vertex of a graph. We show that certain pairs of graphs can be distinguished by
the frequency of repetitions in the sequence of observations. We prove that
this statistic is asymptotically normal and that it distinguishes between
(asymptotically) almost all pairs of finite graphs. We conjecture that the
noisy voter model distinguishes between any two graphs other than stars
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Combinatorics and Probability
For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices