24 research outputs found
Contact processes on random graphs with power law degree distributions have critical value 0
If we consider the contact process with infection rate on a random
graph on vertices with power law degree distributions, mean field
calculations suggest that the critical value of the infection rate
is positive if the power . Physicists seem to regard this as an
established fact, since the result has recently been generalized to bipartite
graphs by G\'{o}mez-Garde\~{n}es et al. [Proc. Natl. Acad. Sci. USA 105 (2008)
1399--1404]. Here, we show that the critical value is zero for any
value of , and the contact process starting from all vertices
infected, with a probability tending to 1 as , maintains a positive
density of infected sites for time at least for any
. Using the last result, together with the contact process duality,
we can establish the existence of a quasi-stationary distribution in which a
randomly chosen vertex is occupied with probability . It is
expected that as . Here we
show that , and so for . Thus
even though the graph is locally tree-like, does not take the mean
field critical value .Comment: Published in at http://dx.doi.org/10.1214/09-AOP471 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Asymptotic behavior of Aldous' gossip process
Aldous [(2007) Preprint] defined a gossip process in which space is a
discrete torus, and the state of the process at time is the set
of individuals who know the information. Information spreads from a site to its
nearest neighbors at rate 1/4 each and at rate to a site chosen
at random from the torus. We will be interested in the case in which
, where the long range transmission significantly accelerates the
time at which everyone knows the information. We prove three results that
precisely describe the spread of information in a slightly simplified model on
the real torus. The time until everyone knows the information is asymptotically
. If is the fraction of the
population who know the information at time and is small
then, for large , the time until reaches is
, where is a
random variable determined by the early spread of the information. The value of
at time is almost a deterministic function
which satisfies an odd looking integro-differential equation. The last
result confirms a heuristic calculation of Aldous.Comment: Published in at http://dx.doi.org/10.1214/10-AAP750 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Jigsaw percolation: What social networks can collaboratively solve a puzzle?
We introduce a new kind of percolation on finite graphs called jigsaw
percolation. This model attempts to capture networks of people who innovate by
merging ideas and who solve problems by piecing together solutions. Each person
in a social network has a unique piece of a jigsaw puzzle. Acquainted people
with compatible puzzle pieces merge their puzzle pieces. More generally, groups
of people with merged puzzle pieces merge if the groups know one another and
have a pair of compatible puzzle pieces. The social network solves the puzzle
if it eventually merges all the puzzle pieces. For an Erd\H{o}s-R\'{e}nyi
social network with vertices and edge probability , we define the
critical value for a connected puzzle graph to be the for which
the chance of solving the puzzle equals . We prove that for the -cycle
(ring) puzzle, , and for an arbitrary connected puzzle
graph with bounded maximum degree, and for
any . Surprisingly, with probability tending to 1 as the network size
increases to infinity, social networks with a power-law degree distribution
cannot solve any bounded-degree puzzle. This model suggests a mechanism for
recent empirical claims that innovation increases with social density, and it
might begin to show what social networks stifle creativity and what networks
collectively innovate.Comment: Published at http://dx.doi.org/10.1214/14-AAP1041 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org