285 research outputs found
The number of matchings in random graphs
We study matchings on sparse random graphs by means of the cavity method. We
first show how the method reproduces several known results about maximum and
perfect matchings in regular and Erdos-Renyi random graphs. Our main new result
is the computation of the entropy, i.e. the leading order of the logarithm of
the number of solutions, of matchings with a given size. We derive both an
algorithm to compute this entropy for an arbitrary graph with a girth that
diverges in the large size limit, and an analytic result for the entropy in
regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical
Mechanic
The spread of fire on a random multigraph
We study a model for the destruction of a random network by fire. Suppose
that we are given a multigraph of minimum degree at least 2 having real-valued
edge-lengths. We pick a uniform point from along the length and set it alight;
the edges of the multigraph burn at speed 1. If the fire reaches a vertex of
degree 2, the fire gets directly passed on to the neighbouring edge; a vertex
of degree at least 3, however, passes the fire either to all of its neighbours
or none, each with probability . If the fire goes out before the whole
network is burnt, we again set fire to a uniform point. We are interested in
the number of fires which must be set in order to burn the whole network, and
the number of points which are burnt from two different directions. We analyse
these quantities for a random multigraph having vertices of degree 3 and
vertices of degree 4, where as ,
with i.i.d. standard exponential edge-lengths. Depending on whether or , we prove that as these
quantities converge jointly in distribution when suitably rescaled to either a
pair of constants or to (complicated) functionals of Brownian motion.
We use our analysis of this model to make progress towards a conjecture of
Aronson, Frieze and Pittel concerning the number of vertices which remain
unmatched when we use the Karp-Sipser algorithm to find a matching on the
Erd\H{o}s-R\'enyi random graph.Comment: 42 page
Matchings on infinite graphs
Elek and Lippner (2010) showed that the convergence of a sequence of
bounded-degree graphs implies the existence of a limit for the proportion of
vertices covered by a maximum matching. We provide a characterization of the
limiting parameter via a local recursion defined directly on the limit of the
graph sequence. Interestingly, the recursion may admit multiple solutions,
implying non-trivial long-range dependencies between the covered vertices. We
overcome this lack of correlation decay by introducing a perturbative parameter
(temperature), which we let progressively go to zero. This allows us to
uniquely identify the correct solution. In the important case where the graph
limit is a unimodular Galton-Watson tree, the recursion simplifies into a
distributional equation that can be solved explicitly, leading to a new
asymptotic formula that considerably extends the well-known one by Karp and
Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page
The rank of diluted random graphs
We investigate the rank of the adjacency matrix of large diluted random
graphs: for a sequence of graphs converging locally to a
Galton--Watson tree (GWT), we provide an explicit formula for the
asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating
function of . In the first part, we show that the adjacency
operator associated with is always self-adjoint; we analyze the associated
spectral measure at the root and characterize the distribution of its atomic
mass at 0. In the second part, we establish a sufficient condition on
for the expectation of this atomic mass to be precisely the normalized limit of
the dimension of the kernel of the adjacency matrices of . Our
proofs borrow ideas from analysis of algorithms, functional analysis, random
matrix theory and statistical physics.Comment: Published in at http://dx.doi.org/10.1214/10-AOP567 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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