41,822 research outputs found
Uniform generation of random graphs with power-law degree sequences
We give a linear-time algorithm that approximately uniformly generates a
random simple graph with a power-law degree sequence whose exponent is at least
2.8811. While sampling graphs with power-law degree sequence of exponent at
least 3 is fairly easy, and many samplers work efficiently in this case, the
problem becomes dramatically more difficult when the exponent drops below 3;
ours is the first provably practicable sampler for this case. We also show that
with an appropriate rejection scheme, our algorithm can be tuned into an exact
uniform sampler. The running time of the exact sampler is O(n^{2.107}) with
high probability, and O(n^{4.081}) in expectation.Comment: 50 page
The mixing time of the switch Markov chains: a unified approach
Since 1997 a considerable effort has been spent to study the mixing time of
switch Markov chains on the realizations of graphic degree sequences of simple
graphs. Several results were proved on rapidly mixing Markov chains on
unconstrained, bipartite, and directed sequences, using different mechanisms.
The aim of this paper is to unify these approaches. We will illustrate the
strength of the unified method by showing that on any -stable family of
unconstrained/bipartite/directed degree sequences the switch Markov chain is
rapidly mixing. This is a common generalization of every known result that
shows the rapid mixing nature of the switch Markov chain on a region of degree
sequences. Two applications of this general result will be presented. One is an
almost uniform sampler for power-law degree sequences with exponent
. The other one shows that the switch Markov chain on the
degree sequence of an Erd\H{o}s-R\'enyi random graph is asymptotically
almost surely rapidly mixing if is bounded away from 0 and 1 by at least
.Comment: Clarification
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