10 research outputs found
The k-core and branching processes
The k-core of a graph G is the maximal subgraph of G having minimum degree at
least k. In 1996, Pittel, Spencer and Wormald found the threshold
for the emergence of a non-trivial k-core in the random graph ,
and the asymptotic size of the k-core above the threshold. We give a new proof
of this result using a local coupling of the graph to a suitable branching
process. This proof extends to a general model of inhomogeneous random graphs
with independence between the edges. As an example, we study the k-core in a
certain power-law or `scale-free' graph with a parameter c controlling the
overall density of edges. For each k at least 3, we find the threshold value of
c at which the k-core emerges, and the fraction of vertices in the k-core when
c is \epsilon above the threshold. In contrast to , this
fraction tends to 0 as \epsilon tends to 0.Comment: 30 pages, 1 figure. Minor revisions. To appear in Combinatorics,
Probability and Computin
Susceptibility in inhomogeneous random graphs
We study the susceptibility, i.e., the mean size of the component containing
a random vertex, in a general model of inhomogeneous random graphs. This is one
of the fundamental quantities associated to (percolation) phase transitions; in
practice one of its main uses is that it often gives a way of determining the
critical point by solving certain linear equations. Here we relate the
susceptibility of suitable random graphs to a quantity associated to the
corresponding branching process, and study both quantities in various natural
examples.Comment: 51 page
Scaling limits and universality: Critical percolation on weighted graphs converging to an graphon
We develop a general universality technique for establishing metric scaling
limits of critical random discrete structures exhibiting mean-field behavior
that requires four ingredients: (i) from the barely subcritical regime to the
critical window, components merge approximately like the multiplicative
coalescent, (ii) asymptotics of the susceptibility functions are the same as
that of the Erdos-Renyi random graph, (iii) asymptotic negligibility of the
maximal component size and the diameter in the barely subcritical regime, and
(iv) macroscopic averaging of distances between vertices in the barely
subcritical regime.
As an application of the general universality theorem, we establish, under
some regularity conditions, the critical percolation scaling limit of graphs
that converge, in a suitable topology, to an graphon. In particular, we
define a notion of the critical window in this setting. The assumption
ensures that the model is in the Erdos-Renyi universality class and that the
scaling limit is Brownian. Our results do not assume any specific functional
form for the graphon. As a consequence of our results on graphons, we obtain
the metric scaling limit for Aldous-Pittel's RGIV model [9] inside the critical
window.
Our universality principle has applications in a number of other problems
including in the study of noise sensitivity of critical random graphs [52]. In
[10], we use our universality theorem to establish the metric scaling limit of
critical bounded size rules. Our method should yield the critical metric
scaling limit of Rucinski and Wormald's random graph process with degree
restrictions [56] provided an additional technical condition about the barely
subcritical behavior of this model can be proved.Comment: 65 pages, 1 figure, the universality principle (Theorem 3.4) from
arXiv:1411.3417 has now been included in this paper. v2: minor change
The phase transition in inhomogeneous random graphs
We introduce a very general model of an inhomogenous random graph with
independence between the edges, which scales so that the number of edges is
linear in the number of vertices. This scaling corresponds to the p=c/n scaling
for G(n,p) used to study the phase transition; also, it seems to be a property
of many large real-world graphs. Our model includes as special cases many
models previously studied.
We show that under one very weak assumption (that the expected number of
edges is `what it should be'), many properties of the model can be determined,
in particular the critical point of the phase transition, and the size of the
giant component above the transition. We do this by relating our random graphs
to branching processes, which are much easier to analyze.
We also consider other properties of the model, showing, for example, that
when there is a giant component, it is `stable': for a typical random graph, no
matter how we add or delete o(n) edges, the size of the giant component does
not change by more than o(n).Comment: 135 pages; revised and expanded slightly. To appear in Random
Structures and Algorithm
Rigorous Result for the CHKNS Random Graph Model
We study the phase transition in a random graph in which vertices and edges are added at constant rates. Two recent papers in Physical Review E by Callaway, Hopcroft, Kleinberg, Newman, and Strogatz, and Dorogovstev, Mendes, and Samukhin have computed the critical value of this model, shown that the fraction of vertices in finite clusters is infinitely differentiable at the critical value, and that in the subcritical phase the cluster size distribution has a polynomial decay rate with a continuously varying power. Here we sketch rigorous proofs for the first and third results and a new estimates about connectivity probabilities at the critical value
Rigorous Result for the CHKNS Random Graph Model
We study the phase transition in a random graph in which vertices and edges are added at constant rates. Two recent papers in Physical Review E by Callaway, Hopcroft, Kleinberg, Newman, and Strogatz, and Dorogovstev, Mendes, and Samukhin have computed the critical value of this model, shown that the fraction of vertices in finite clusters is infinitely differentiable at the critical value, and that in the subcritical phase the cluster size distribution has a polynomial decay rate with a continuously varying power. Here we sketch rigorous proofs for the first and third results and a new estimates about connectivity probabilities at the critical value