11,127 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
Robustness of scale-free spatial networks
A growing family of random graphs is called robust if it retains a giant
component after percolation with arbitrary positive retention probability. We
study robustness for graphs, in which new vertices are given a spatial position
on the -dimensional torus and are connected to existing vertices with a
probability favouring short spatial distances and high degrees. In this model
of a scale-free network with clustering we can independently tune the power law
exponent of the degree distribution and the rate at which the
connection probability decreases with the distance of two vertices. We show
that the network is robust if , but fails to be robust if
. In the case of one-dimensional space we also show that the network is
not robust if . This implies that robustness of a
scale-free network depends not only on its power-law exponent but also on its
clustering features. Other than the classical models of scale-free networks our
model is not locally tree-like, and hence we need to develop novel methods for
its study, including, for example, a surprising application of the
BK-inequality.Comment: 34 pages, 4 figure
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
Odd Khovanov homology
We describe an invariant of links in the three-sphere which is closely
related to Khovanov's Jones polynomial homology. Our construction replaces the
symmetric algebra appearing in Khovanov's definition with an exterior algebra.
The two invariants have the same reduction modulo 2, but differ over the
rationals. There is a reduced version which is a link invariant whose graded
Euler characteristic is the normalized Jones polynomial.Comment: 16 pages, 12 figure
Kinematic and dynamic vortices in a thin film driven by an applied current and magnetic field
Using a Ginzburg-Landau model, we study the vortex behavior of a rectangular
thin film superconductor subjected to an applied current fed into a portion of
the sides and an applied magnetic field directed orthogonal to the film.
Through a center manifold reduction we develop a rigorous bifurcation theory
for the appearance of periodic solutions in certain parameter regimes near the
normal state. The leading order dynamics yield in particular a motion law for
kinematic vortices moving up and down the center line of the sample. We also
present computations that reveal the co-existence and periodic evolution of
kinematic and magnetic vortices
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