67 research outputs found
An algorithm to prescribe the configuration of a finite graph
We provide algorithms involving edge slides, for a connected simple graph to
evolve in a finite number of steps to another connected simple graph in a
prescribed configuration, and for the regularization of such a graph by the
minimization of an appropriate energy functional
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
Uniform generation of spanning regular subgraphs of a dense graph
Let be a graph on vertices and let \ber{H_n} denote the
complement of . Suppose that is the maximum degree of
\ber{H_n}. We analyse three algorithms for sampling -regular subgraphs
(-factors) of . This is equivalent to uniformly sampling -regular
graphs which avoid a set E(\ber{H_n}) of forbidden edges. Here is a
positive integer which may depend on .
Two of these algorithms produce a uniformly random -factor of in
expected runtime which is linear in and low-degree polynomial in and
. The first algorithm applies when . This
improves on an earlier algorithm by the first author, which required constant
and at most a linear number of edges in \ber{H_n}. The second algorithm
applies when is regular and , adapting an approach
developed by the first author together with Wormald. The third algorithm is a
simplification of the second, and produces an approximately uniform -factor
of in time . Here the output distribution differs from uniform by
in total variation distance, provided that
Sampling Hypergraphs with Given Degrees
There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm for sampling bipartite graphs with given degrees, uniformly or nearly uniformly, in (expected) polynomial time. The expected runtime of the hypergraph sampling algorithm depends on the (expected) runtime of the bipartite graph sampling algorithm , and the probability that a uniformly random bipartite graph with given degrees corresponds to a simple hypergraph. We give some conditions on the hypergraph degree sequence which guarantee that this probability is bounded below by a constant
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
- …