7,923 research outputs found
Percolation on random graphs with a fixed degree sequence
We consider bond percolation on random graphs with given degrees and bounded average degree. In particular, we consider the order of the largest component after the random deletion of the edges of such a random graph. We give a rough characterization of those degree distributions for which bond percolation with high probability leaves a component of linear order, known usually as a giant component. We show that essentially the critical condition has to do with the tail of the degree distribution. Our proof makes use of recent technique which is based on the switching method and avoids the use of the classic configuration model on degree sequences that have a limiting distribution. Thus our results hold for sparse degree sequences without the usual restrictions that accompany the configuration model.The research was also supported by the EPSRC, grant no. EP/M009408/1.Postprint (author's final draft
The phase transition in the configuration model
Let be a random graph with a given degree sequence , such as a
random -regular graph where is fixed and . We study
the percolation phase transition on such graphs , i.e., the emergence as
increases of a unique giant component in the random subgraph obtained by
keeping edges independently with probability . More generally, we study the
emergence of a giant component in itself as varies. We show that a
single method can be used to prove very precise results below, inside and above
the `scaling window' of the phase transition, matching many of the known
results for the much simpler model . This method is a natural extension
of that used by Bollobas and the author to study , itself based on work
of Aldous and of Nachmias and Peres; the calculations are significantly more
involved in the present setting.Comment: 37 page
Processes on Unimodular Random Networks
We investigate unimodular random networks. Our motivations include their
characterization via reversibility of an associated random walk and their
similarities to unimodular quasi-transitive graphs. We extend various theorems
concerning random walks, percolation, spanning forests, and amenability from
the known context of unimodular quasi-transitive graphs to the more general
context of unimodular random networks. We give properties of a trace associated
to unimodular random networks with applications to stochastic comparison of
continuous-time random walk.Comment: 66 pages; 3rd version corrects formula (4.4) -- the published version
is incorrect --, as well as a minor error in the proof of Proposition 4.10;
4th version corrects proof of Proposition 7.1; 5th version corrects proof of
Theorem 5.1; 6th version makes a few more minor correction
Percolation on sparse random graphs with given degree sequence
We study the two most common types of percolation process on a sparse random
graph with a given degree sequence. Namely, we examine first a bond percolation
process where the edges of the graph are retained with probability p and
afterwards we focus on site percolation where the vertices are retained with
probability p. We establish critical values for p above which a giant component
emerges in both cases. Moreover, we show that in fact these coincide. As a
special case, our results apply to power law random graphs. We obtain rigorous
proofs for formulas derived by several physicists for such graphs.Comment: 20 page
Percolation on nonunimodular transitive graphs
We extend some of the fundamental results about percolation on unimodular
nonamenable graphs to nonunimodular graphs. We show that they cannot have
infinitely many infinite clusters at critical Bernoulli percolation. In the
case of heavy clusters, this result has already been established, but it also
follows from one of our results. We give a general necessary condition for
nonunimodular graphs to have a phase with infinitely many heavy clusters. We
present an invariant spanning tree with on some nonunimodular graph.
Such trees cannot exist for nonamenable unimodular graphs. We show a new way of
constructing nonunimodular graphs that have properties more peculiar than the
ones previously known.Comment: Published at http://dx.doi.org/10.1214/009117906000000494 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On giant components and treewidth in the layers model
Given an undirected -vertex graph and an integer , let
denote the random vertex induced subgraph of generated by ordering
according to a random permutation and including in those
vertices with at most of their neighbors preceding them in this order.
The distribution of subgraphs sampled in this manner is called the \emph{layers
model with parameter} . The layers model has found applications in studying
-degenerate subgraphs, the design of algorithms for the maximum
independent set problem, and in bootstrap percolation.
In the current work we expand the study of structural properties of the
layers model.
We prove that there are -regular graphs for which with high
probability has a connected component of size . Moreover,
this connected component has treewidth . This lower bound on the
treewidth extends to many other random graph models. In contrast, is
known to be a forest (hence of treewidth~1), and we establish that if is of
bounded degree then with high probability the largest connected component in
is of size . We also consider the infinite two-dimensional
grid, for which we prove that the first four layers contain a unique infinite
connected component with probability
Maximal entropy random networks with given degree distribution
Using a maximum entropy principle to assign a statistical weight to any
graph, we introduce a model of random graphs with arbitrary degree distribution
in the framework of standard statistical mechanics. We compute the free energy
and the distribution of connected components. We determine the size of the
percolation cluster above the percolation threshold. The conditional degree
distribution on the percolation cluster is also given. We briefly present the
analogous discussion for oriented graphs, giving for example the percolation
criterion.Comment: 22 pages, LateX, no figur
Universality for critical heavy-tailed network models: Metric structure of maximal components
We study limits of the largest connected components (viewed as metric spaces)
obtained by critical percolation on uniformly chosen graphs and configuration
models with heavy-tailed degrees. For rank-one inhomogeneous random graphs,
such results were derived by Bhamidi, van der Hofstad, Sen [Probab. Theory
Relat. Fields 2018]. We develop general principles under which the identical
scaling limits as the rank-one case can be obtained. Of independent interest,
we derive refined asymptotics for various susceptibility functions and the
maximal diameter in the barely subcritical regime.Comment: Final published version. 47 pages, 6 figure
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