621,691 research outputs found
Quotient graphs for power graphs
In a previous paper of the first author a procedure was developed for
counting the components of a graph through the knowledge of the components of
its quotient graphs. We apply here that procedure to the proper power graph
of a finite group , finding a formula for the number
of its components which is particularly illuminative when
is a fusion controlled permutation group. We make use of the proper
quotient power graph , the proper order graph
and the proper type graph . We show that
all those graphs are quotient of and demonstrate a strong
link between them dealing with . We find simultaneously
as well as the number of components of
, and
Push is Fast on Sparse Random Graphs
We consider the classical push broadcast process on a large class of sparse
random multigraphs that includes random power law graphs and multigraphs. Our
analysis shows that for every , whp rounds are
sufficient to inform all but an -fraction of the vertices.
It is not hard to see that, e.g. for random power law graphs, the push
process needs whp rounds to inform all vertices. Fountoulakis,
Panagiotou and Sauerwald proved that for random graphs that have power law
degree sequences with , the push-pull protocol needs
to inform all but vertices whp. Our result demonstrates that,
for such random graphs, the pull mechanism does not (asymptotically) improve
the running time. This is surprising as it is known that, on random power law
graphs with , push-pull is exponentially faster than pull
Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution
Even though power-law or close-to-power-law degree distributions are
ubiquitously observed in a great variety of large real networks, the
mathematically satisfactory treatment of random power-law graphs satisfying
basic statistical requirements of realism is still lacking. These requirements
are: sparsity, exchangeability, projectivity, and unbiasedness. The last
requirement states that entropy of the graph ensemble must be maximized under
the degree distribution constraints. Here we prove that the hypersoft
configuration model (HSCM), belonging to the class of random graphs with latent
hyperparameters, also known as inhomogeneous random graphs or -random
graphs, is an ensemble of random power-law graphs that are sparse, unbiased,
and either exchangeable or projective. The proof of their unbiasedness relies
on generalized graphons, and on mapping the problem of maximization of the
normalized Gibbs entropy of a random graph ensemble, to the graphon entropy
maximization problem, showing that the two entropies converge to each other in
the large-graph limit
Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution
Even though power-law or close-to-power-law degree distributions are
ubiquitously observed in a great variety of large real networks, the
mathematically satisfactory treatment of random power-law graphs satisfying
basic statistical requirements of realism is still lacking. These requirements
are: sparsity, exchangeability, projectivity, and unbiasedness. The last
requirement states that entropy of the graph ensemble must be maximized under
the degree distribution constraints. Here we prove that the hypersoft
configuration model (HSCM), belonging to the class of random graphs with latent
hyperparameters, also known as inhomogeneous random graphs or -random
graphs, is an ensemble of random power-law graphs that are sparse, unbiased,
and either exchangeable or projective. The proof of their unbiasedness relies
on generalized graphons, and on mapping the problem of maximization of the
normalized Gibbs entropy of a random graph ensemble, to the graphon entropy
maximization problem, showing that the two entropies converge to each other in
the large-graph limit
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