99 research outputs found
The probability that a random multigraph is simple
Consider a random multigraph G* with given vertex degrees d_1,...,d_n,
contructed by the configuration model. We show that, asymptotically for a
sequence of such multigraphs with the number of edges (d_1+...+d_n)/2 tending
to infinity, the probability that the multigraph is simple stays away from 0 if
and only if \sum d_i^2=O(\sum d_i). This was previously known only under extra
assumtions on the maximum degree. We also give an asymptotic formula for this
probability, extending previous results by several authors.Comment: 24 page
Mixing times of random walks on dynamic configuration models
The mixing time of a random walk, with or without backtracking, on a random
graph generated according to the configuration model on vertices, is known
to be of order . In this paper we investigate what happens when the
random graph becomes {\em dynamic}, namely, at each unit of time a fraction
of the edges is randomly rewired. Under mild conditions on the
degree sequence, guaranteeing that the graph is locally tree-like, we show that
for every the -mixing time of random walk
without backtracking grows like
as , provided
that . The latter condition
corresponds to a regime of fast enough graph dynamics. Our proof is based on a
randomised stopping time argument, in combination with coupling techniques and
combinatorial estimates. The stopping time of interest is the first time that
the walk moves along an edge that was rewired before, which turns out to be
close to a strong stationary time.Comment: 23 pages, 6 figures. Previous version contained a mistake in one of
the proofs. In this version we look at nonbacktracking random walk instead of
simple random wal
On Bootstrap Percolation in Living Neural Networks
Recent experimental studies of living neural networks reveal that their
global activation induced by electrical stimulation can be explained using the
concept of bootstrap percolation on a directed random network. The experiment
consists in activating externally an initial random fraction of the neurons and
observe the process of firing until its equilibrium. The final portion of
neurons that are active depends in a non linear way on the initial fraction.
The main result of this paper is a theorem which enables us to find the
asymptotic of final proportion of the fired neurons in the case of random
directed graphs with given node degrees as the model for interacting network.
This gives a rigorous mathematical proof of a phenomena observed by physicists
in neural networks
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