133,049 research outputs found
Dynamic concentration of the triangle-free process
The triangle-free process begins with an empty graph on n vertices and
iteratively adds edges chosen uniformly at random subject to the constraint
that no triangle is formed. We determine the asymptotic number of edges in the
maximal triangle-free graph at which the triangle-free process terminates. We
also bound the independence number of this graph, which gives an improved lower
bound on the Ramsey numbers R(3,t): we show R(3,t) > (1-o(1)) t^2 / (4 log t),
which is within a 4+o(1) factor of the best known upper bound. Our improvement
on previous analyses of this process exploits the self-correcting nature of key
statistics of the process. Furthermore, we determine which bounded size
subgraphs are likely to appear in the maximal triangle-free graph produced by
the triangle-free process: they are precisely those triangle-free graphs with
density at most 2.Comment: 75 pages, 1 figur
Saturating the random graph with an independent family of small range
Motivated by Keisler's order, a far-reaching program of understanding basic
model-theoretic structure through the lens of regular ultrapowers, we prove
that for a class of regular filters on , , the
fact that P(I)/\de has little freedom (as measured by the fact that any
maximal antichain is of size , or even countable) does not prevent
extending to an ultrafilter on which saturates ultrapowers of the
random graph. "Saturates" means that M^I/\de_1 is -saturated
whenever M is a model of the theory of the random graph. This was known to be
true for stable theories, and false for non-simple and non-low theories. This
result and the techniques introduced in the proof have catalyzed the authors'
subsequent work on Keisler's order for simple unstable theories. The
introduction, which includes a part written for model theorists and a part
written for set theorists, discusses our current program and related results.Comment: 14 page
Maximal Entropy Random Walk: solvable cases of dynamics
We focus on the study of dynamics of two kinds of random walk: generic random
walk (GRW) and maximal entropy random walk (MERW) on two model networks: Cayley
trees and ladder graphs. The stationary probability distribution for MERW is
given by the squared components of the eigenvector associated with the largest
eigenvalue \lambda_0 of the adjacency matrix of a graph, while the dynamics of
the probability distribution approaching to the stationary state depends on the
second largest eigenvalue \lambda_1.
Firstly, we give analytic solutions for Cayley trees with arbitrary branching
number, root degree, and number of generations. We determine three regimes of a
tree structure that result in different statics and dynamics of MERW, which are
due to strongly, critically, and weakly branched roots. We show how the
relaxation times, generically shorter for MERW than for GRW, scale with the
graph size.
Secondly, we give numerical results for ladder graphs with symmetric defects.
MERW shows a clear exponential growth of the relaxation time with the size of
defective regions, which indicates trapping of a particle within highly
entropic intact region and its escaping that resembles quantum tunneling
through a potential barrier. GRW shows standard diffusive dependence
irrespective of the defects.Comment: 13 pages, 6 figures, 24th Marian Smoluchowski Symposium on
Statistical Physics (Zakopane, Poland, September 17-22, 2011
The jamming constant of uniform random graphs
By constructing jointly a random graph and an associated exploration process, we define the dynamics of a “parking process” on a class of uniform random graphs as a measure-valued Markov process, representing the empirical degree distribution of non-explored nodes. We then establish a functional law of large numbers for this process as the number of vertices grows to infinity, allowing us to assess the jamming constant of the considered random graphs, i.e. the size of the maximal independent set discovered by the exploration algorithm. This technique, which can be applied to any uniform random graph with a given–possibly unbounded–degree distribution, can be seen as a generalization in the space of measures, of the differential equation method introduced by Wormald.Fil: Bermolen, Paola. Universidad de la República; UruguayFil: Jonckheere, Matthieu Thimothy Samson. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Moyal, Pascal. Northwestern University; Estados Unidos. Universite de Technologie de Compiegne; Franci
Essential edges in Poisson random hypergraphs
Consider a random hypergraph on a set of N vertices in which, for k between 1
and N, a Poisson(N beta_k) number of hyperedges is scattered randomly over all
subsets of size k. We collapse the hypergraph by running the following
algorithm to exhaustion: pick a vertex having a 1-edge and remove it; collapse
the hyperedges over that vertex onto their remaining vertices; repeat until
there are no 1-edges left. We call the vertices removed in this process
"identifiable". Also any hyperedge all of whose vertices are removed is called
"identifiable". We say that a hyperedge is "essential" if its removal prior to
collapse would have reduced the number of identifiable vertices. The limiting
proportions, as N tends to infinity, of identifiable vertices and hyperedges
were obtained by Darling and Norris. In this paper, we establish the limiting
proportion of essential hyperedges. We also discuss, in the case of a random
graph, the relation of essential edges to the 2-core of the graph, the maximal
sub-graph with minimal vertex degree 2.Comment: 12 pages, 3 figures. Revised version with minor
corrections/clarifications and slightly expanded introductio
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