1,010 research outputs found
Shortest-weight paths in random regular graphs
Consider a random regular graph with degree and of size . Assign to
each edge an i.i.d. exponential random variable with mean one. In this paper we
establish a precise asymptotic expression for the maximum number of edges on
the shortest-weight paths between a fixed vertex and all the other vertices, as
well as between any pair of vertices. Namely, for any fixed , we show
that the longest of these shortest-weight paths has about
edges where is the unique solution of the equation , for .Comment: 20 pages. arXiv admin note: text overlap with arXiv:1112.633
The winner takes it all
We study competing first passage percolation on graphs generated by the
configuration model. At time 0, vertex 1 and vertex 2 are infected with the
type 1 and the type 2 infection, respectively, and an uninfected vertex then
becomes type 1 (2) infected at rate () times the number
of edges connecting it to a type 1 (2) infected neighbor. Our main result is
that, if the degree distribution is a power-law with exponent ,
then, as the number of vertices tends to infinity and with high probability,
one of the infection types will occupy all but a finite number of vertices.
Furthermore, which one of the infections wins is random and both infections
have a positive probability of winning regardless of the values of
and . The picture is similar with multiple starting points for the
infections
The diameter of weighted random graphs
In this paper we study the impact of random exponential edge weights on the
distances in a random graph and, in particular, on its diameter. Our main
result consists of a precise asymptotic expression for the maximal weight of
the shortest weight paths between all vertices (the weighted diameter) of
sparse random graphs, when the edge weights are i.i.d. exponential random
variables.Comment: Published at http://dx.doi.org/10.1214/14-AAP1034 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Fixed speed competition on the configuration model with infinite variance degrees: unequal speeds
We study competition of two spreading colors starting from single sources on
the configuration model with i.i.d. degrees following a power-law distribution
with exponent tau in (2,3). In this model two colors spread with a fixed but
not necessarily equal speed on the unweighted random graph. We show that if the
speeds are not equal, then the faster color paints almost all vertices, while
the slower color can paint only a random subpolynomial fraction of the
vertices. We investigate the case when the speeds are equal and typical
distances in a follow-up paper.Comment: 44 pages, 9 picture
Coexistence of competing first passage percolation on hyperbolic graphs
We study a natural growth process with competition, which was recently
introduced to analyze MDLA, a challenging model for the growth of an aggregate
by diffusing particles. The growth process consists of two first-passage
percolation processes and , spreading with
rates and respectively, on a graph . starts
from a single vertex at the origin , while the initial configuration of
consists of infinitely many \emph{seeds} distributed
according to a product of Bernoulli measures of parameter on
. starts spreading from time 0, while each
seed of only starts spreading after it has been reached by
either or . A fundamental question in this
model, and in growth processes with competition in general, is whether the two
processes coexist (i.e., both produce infinite clusters) with positive
probability. We show that this is the case when is vertex transitive,
non-amenable and hyperbolic, in particular, for any there is a
such that for all the two
processes coexist with positive probability. This is the first non-trivial
instance where coexistence is established for this model. We also show that
produces an infinite cluster almost surely for any
positive , establishing fundamental differences with the behavior
of such processes on .Comment: 53 pages, 13 figure
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