26,965 research outputs found
Random subgraphs of finite graphs: I. The scaling window under the triangle condition
We study random subgraphs of an arbitrary finite connected transitive graph
obtained by independently deleting edges with probability .
Let be the number of vertices in , and let be their
degree. We define the critical threshold to be the
value of for which the expected cluster size of a fixed vertex attains the
value , where is fixed and positive. We show that
for any such model, there is a phase transition at analogous to the phase
transition for the random graph, provided that a quantity called the triangle
diagram is sufficiently small at the threshold . In particular, we show
that the largest cluster inside a scaling window of size
|p-p_c|=\Theta(\cn^{-1}V^{-1/3}) is of size , while below
this scaling window, it is much smaller, of order
, with \epsilon=\cn(p_c-p). We also obtain
an upper bound O(\cn(p-p_c)V) for the expected size of the largest cluster
above the window. In addition, we define and analyze the percolation
probability above the window and show that it is of order \Theta(\cn(p-p_c)).
Among the models for which the triangle diagram is small enough to allow us to
draw these conclusions are the random graph, the -cube and certain Hamming
cubes, as well as the spread-out -dimensional torus for
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
Critical random forests
Let denote a random forest on a set of vertices, chosen
uniformly from all forests with edges. Let denote the forest
obtained by conditioning the Erdos-Renyi graph to be acyclic. We
describe scaling limits for the largest components of and , in
the critical window or . Aldous
described a scaling limit for the largest components of within the
critical window in terms of the excursion lengths of a reflected Brownian
motion with time-dependent drift. Our scaling limit for critical random forests
is of a similar nature, but now based on a reflected diffusion whose drift
depends on space as well as on time
Critical random graphs: limiting constructions and distributional properties
We consider the Erdos-Renyi random graph G(n,p) inside the critical window,
where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a previous
paper (arXiv:0903.4730) that considering the connected components of G(n,p) as
a sequence of metric spaces with the graph distance rescaled by n^{-1/3} and
letting n go to infinity yields a non-trivial sequence of limit metric spaces C
= (C_1, C_2, ...). These limit metric spaces can be constructed from certain
random real trees with vertex-identifications. For a single such metric space,
we give here two equivalent constructions, both of which are in terms of more
standard probabilistic objects. The first is a global construction using
Dirichlet random variables and Aldous' Brownian continuum random tree. The
second is a recursive construction from an inhomogeneous Poisson point process
on R_+. These constructions allow us to characterize the distributions of the
masses and lengths in the constituent parts of a limit component when it is
decomposed according to its cycle structure. In particular, this strengthens
results of Luczak, Pittel and Wierman by providing precise distributional
convergence for the lengths of paths between kernel vertices and the length of
a shortest cycle, within any fixed limit component.Comment: 30 pages, 4 figure
Percolation on dense graph sequences
In this paper we determine the percolation threshold for an arbitrary
sequence of dense graphs . Let be the largest eigenvalue of
the adjacency matrix of , and let be the random subgraph of
obtained by keeping each edge independently with probability . We
show that the appearance of a giant component in has a sharp
threshold at . In fact, we prove much more: if
converges to an irreducible limit, then the density of the largest component of
tends to the survival probability of a multi-type branching process
defined in terms of this limit. Here the notions of convergence and limit are
those of Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi. In addition to using
basic properties of convergence, we make heavy use of the methods of
Bollob\'as, Janson and Riordan, who used multi-type branching processes to
study the emergence of a giant component in a very broad family of sparse
inhomogeneous random graphs.Comment: Published in at http://dx.doi.org/10.1214/09-AOP478 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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