4,056 research outputs found
Effects of network topology, transmission delays, and refractoriness on the response of coupled excitable systems to a stochastic stimulus
We study the effects of network topology on the response of networks of
coupled discrete excitable systems to an external stochastic stimulus. We
extend recent results that characterize the response in terms of spectral
properties of the adjacency matrix by allowing distributions in the
transmission delays and in the number of refractory states, and by developing a
nonperturbative approximation to the steady state network response. We confirm
our theoretical results with numerical simulations. We find that the steady
state response amplitude is inversely proportional to the duration of
refractoriness, which reduces the maximum attainable dynamic range. We also
find that transmission delays alter the time required to reach steady state.
Importantly, neither delays nor refractoriness impact the general prediction
that criticality and maximum dynamic range occur when the largest eigenvalue of
the adjacency matrix is unity
Multifractal properties of growing networks
We introduce a new family of models for growing networks. In these networks
new edges are attached preferentially to vertices with higher number of
connections, and new vertices are created by already existing ones, inheriting
part of their parent's connections. We show that combination of these two
features produces multifractal degree distributions, where degree is the number
of connections of a vertex. An exact multifractal distribution is found for a
nontrivial model of this class. The distribution tends to a power-law one, , in the infinite network limit.
Nevertheless, for finite networks's sizes, because of multifractality, attempts
to interpret the distribution as a scale-free would result in an ambiguous
value of the exponent .Comment: 7 pages epltex, 1 figur
Ordinary Percolation with Discontinuous Transitions
Percolation on a one-dimensional lattice and fractals such as the Sierpinski
gasket is typically considered to be trivial because they percolate only at
full bond density. By dressing up such lattices with small-world bonds, a novel
percolation transition with explosive cluster growth can emerge at a nontrivial
critical point. There, the usual order parameter, describing the probability of
any node to be part of the largest cluster, jumps instantly to a finite value.
Here, we provide a simple example of this transition in form of a small-world
network consisting of a one-dimensional lattice combined with a hierarchy of
long-range bonds that reveals many features of the transition in a
mathematically rigorous manner.Comment: RevTex, 5 pages, 4 eps-figs, and Mathematica Notebook as Supplement
included. Final version, with several corrections and improvements. For
related work, see http://www.physics.emory.edu/faculty/boettcher
An alternative approach to determining average distance in a class of scale-free modular networks
Various real-life networks of current interest are simultaneously scale-free
and modular. Here we study analytically the average distance in a class of
deterministically growing scale-free modular networks. By virtue of the
recursive relations derived from the self-similar structure of the networks, we
compute rigorously this important quantity, obtaining an explicit closed-form
solution, which recovers the previous result and is corroborated by extensive
numerical calculations. The obtained exact expression shows that the average
distance scales logarithmically with the number of nodes in the networks,
indicating an existence of small-world behavior. We present that this
small-world phenomenon comes from the peculiar architecture of the network
family.Comment: Submitted for publicactio
- …