1,116 research outputs found
Biased random walks on random graphs
These notes cover one of the topics programmed for the St Petersburg School
in Probability and Statistical Physics of June 2012.
The aim is to review recent mathematical developments in the field of random
walks in random environment. Our main focus will be on directionally transient
and reversible random walks on different types of underlying graph structures,
such as , trees and for .Comment: Survey based one of the topics programmed for the St Petersburg
School in Probability and Statistical Physics of June 2012. 64 pages, 16
figure
Critical scaling in standard biased random walks
The spatial coverage produced by a single discrete-time random walk, with
asymmetric jump probability and non-uniform steps, moving on an
infinite one-dimensional lattice is investigated. Analytical calculations are
complemented with Monte Carlo simulations. We show that, for appropriate step
sizes, the model displays a critical phenomenon, at . Its scaling
properties as well as the main features of the fragmented coverage occurring in
the vicinity of the critical point are shown. In particular, in the limit , the distribution of fragment lengths is scale-free, with nontrivial
exponents. Moreover, the spatial distribution of cracks (unvisited sites)
defines a fractal set over the spanned interval. Thus, from the perspective of
the covered territory, a very rich critical phenomenology is revealed in a
simple one-dimensional standard model.Comment: 4 pages, 4 figure
Topologically biased random walk with application for community finding in networks
We present a new approach of topology biased random walks for undirected
networks. We focus on a one parameter family of biases and by using a formal
analogy with perturbation theory in quantum mechanics we investigate the
features of biased random walks. This analogy is extended through the use of
parametric equations of motion (PEM) to study the features of random walks {\em
vs.} parameter values. Furthermore, we show an analysis of the spectral gap
maximum associated to the value of the second eigenvalue of the transition
matrix related to the relaxation rate to the stationary state. Applications of
these studies allow {\em ad hoc} algorithms for the exploration of complex
networks and their communities.Comment: 8 pages, 7 figure
Localization Transition of Biased Random Walks on Random Networks
We study random walks on large random graphs that are biased towards a
randomly chosen but fixed target node. We show that a critical bias strength
b_c exists such that most walks find the target within a finite time when
b>b_c. For b<b_c, a finite fraction of walks drifts off to infinity before
hitting the target. The phase transition at b=b_c is second order, but finite
size behavior is complex and does not obey the usual finite size scaling
ansatz. By extending rigorous results for biased walks on Galton-Watson trees,
we give the exact analytical value for b_c and verify it by large scale
simulations.Comment: 4 pages, includes 4 figure
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