143 research outputs found
Spatial preferential attachment networks: Power laws and clustering coefficients
We define a class of growing networks in which new nodes are given a spatial
position and are connected to existing nodes with a probability mechanism
favoring short distances and high degrees. The competition of preferential
attachment and spatial clustering gives this model a range of interesting
properties. Empirical degree distributions converge to a limit law, which can
be a power law with any exponent . The average clustering coefficient
of the networks converges to a positive limit. Finally, a phase transition
occurs in the global clustering coefficients and empirical distribution of edge
lengths when the power-law exponent crosses the critical value . Our
main tool in the proof of these results is a general weak law of large numbers
in the spirit of Penrose and Yukich.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1006 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Random networks with sublinear preferential attachment: The giant component
We study a dynamical random network model in which at every construction step
a new vertex is introduced and attached to every existing vertex independently
with a probability proportional to a concave function f of its current degree.
We give a criterion for the existence of a giant component, which is both
necessary and sufficient, and which becomes explicit when f is linear.
Otherwise it allows the derivation of explicit necessary and sufficient
conditions, which are often fairly close. We give an explicit criterion to
decide whether the giant component is robust under random removal of edges. We
also determine asymptotically the size of the giant component and the empirical
distribution of component sizes in terms of the survival probability and size
distribution of a multitype branching random walk associated with f.Comment: Published in at http://dx.doi.org/10.1214/11-AOP697 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On the most visited sites of planar Brownian motion
Let (B_t : t > 0) be a planar Brownian motion and define gauge functions
for . If we show that
almost surely there exists a point x in the plane such that , but if almost surely simultaneously for all . This resolves a longstanding open
problem posed by S.,J. Taylor in 1986
Emergence of condensation in Kingman's model of selection and mutation
We describe the onset of condensation in the simple model for the balance
between selection and mutation given by Kingman in terms of a scaling limit
theorem. Loosely speaking, this shows that the wave moving towards genes of
maximal fitness has the shape of a gamma distribution. We conjecture that this
wave shape is a universal phenomenon that can also be found in a variety of
more complex models, well beyond the genetics context, and provide some further
evidence for this
The spread of infections on evolving scale-free networks
We study the contact process on a class of evolving scale-free networks,
where each node updates its connections at independent random times. We give a
rigorous mathematical proof that there is a transition between a phase where
for all infection rates the infection survives for a long time, at least
exponential in the network size, and a phase where for sufficiently small
infection rates extinction occurs quickly, at most like the square root of the
network size. The phase transition occurs when the power-law exponent crosses
the value four. This behaviour is in contrast to that of the contact process on
the corresponding static model, where there is no phase transition, as well as
that of a classical mean-field approximation, which has a phase transition at
power-law exponent three. The new observation behind our result is that
temporal variability of networks can simultaneously increase the rate at which
the infection spreads in the network, and decrease the time which the infection
spends in metastable states.Comment: 17 pages, 1 figur
Large deviation principles for empirical measures of colored random graphs
For any finite colored graph we define the empirical neighborhood measure,
which counts the number of vertices of a given color connected to a given
number of vertices of each color, and the empirical pair measure, which counts
the number of edges connecting each pair of colors. For a class of models of
sparse colored random graphs, we prove large deviation principles for these
empirical measures in the weak topology. The rate functions governing our large
deviation principles can be expressed explicitly in terms of relative
entropies. We derive a large deviation principle for the degree distribution of
Erd\H{o}s--R\'{e}nyi graphs near criticality.Comment: Published in at http://dx.doi.org/10.1214/09-AAP647 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Ageing in the parabolic Anderson model
The parabolic Anderson model is the Cauchy problem for the heat equation with
a random potential. We consider this model in a setting which is continuous in
time and discrete in space, and focus on time-constant, independent and
identically distributed potentials with polynomial tails at infinity. We are
concerned with the long-term temporal dynamics of this system. Our main result
is that the periods, in which the profile of the solutions remains nearly
constant, are increasing linearly over time, a phenomenon known as ageing. We
describe this phenomenon in the weak sense, by looking at the asymptotic
probability of a change in a given time window, and in the strong sense, by
identifying the almost sure upper envelope for the process of the time
remaining until the next change of profile. We also prove functional scaling
limit theorems for profile and growth rate of the solution of the parabolic
Anderson model.Comment: 43 pages, 4 figure
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