Title: Cascades on correlated and modular networks
Speaker: Prof. J.P. Gleeson, University of Limerick, Ireland
Abstract:
An analytical approach to determining the mean avalanche size in a
broad class of dynamical models on random networks is introduced.
Previous results on percolation transitions and epidemic sizes are
shown to be special cases of the method. The time-dependence of
cascades and extensions to networks with community structure or
degree-degree correlations are discussed. Analytical results for the
rate of spread of innovations (Watts, 2002) in a modular network and
for the size of k-cores (Dorogotsev et al, 2006) in networks with
degree-degree correlations are confirmed with numerical simulations.
The dynamics of cascades are strongly dependent upon the topological
structure of the underlying network and on the details of how the
cascade spreads among the nodes of the network. In the class of
examples considered here, each node of the network can be in one of
two states, either active (also termed damaged or infected) or
inactive (undamaged or susceptible), with nodes updating their states
depending on the number and state of the node's immediate neighbors
in the (undirected) network. Networks are chosen from an ensemble of
graphs with specified degree distribution (i.e. using the
configuration model (Newman et al, 2001)), and both synchronous and
asynchronous updating may be considered. We show that for a class of
such models the average cascade size may be determined analytically
(averages being taken over an ensemble of realizations) (Gleeson and
Cahalane, 2007). This basic model is also extended to networks with
strong community structure or with degree-degree correlations.
Previous results on percolation and k-core sizes are shown to be
special cases of our general approach.
References:
S.N. Dorogovtsev, A.V. Goltsev, and J.F.F. Mendes (2006): k-Core
organization of complex networks. Phys. Rev. Lett., 96, 040601.
J.P. Gleeson and D.J. Cahalane (2007): Seed size strongly affects
cascades on random networks. Phys. Rev. E., 75, 056103.
M.E.J. Newman, S.H. Strogatz, and D.J. Watts (2001): Random graphs
with arbitrary degree distributions and their applications. Phys.
Rev. E, 64, 026118.
D.J. Watts (2002): A simple mode of global cascades on random
networks. Proc. Nat. Acad. Sci. 99, 5766.