37,189 research outputs found
Percolation in invariant Poisson graphs with i.i.d. degrees
Let each point of a homogeneous Poisson process in R^d independently be
equipped with a random number of stubs (half-edges) according to a given
probability distribution mu on the positive integers. We consider
translation-invariant schemes for perfectly matching the stubs to obtain a
simple graph with degree distribution mu. Leaving aside degenerate cases, we
prove that for any mu there exist schemes that give only finite components as
well as schemes that give infinite components. For a particular matching scheme
that is a natural extension of Gale-Shapley stable marriage, we give sufficient
conditions on mu for the absence and presence of infinite components
Tailored graph ensembles as proxies or null models for real networks I: tools for quantifying structure
We study the tailoring of structured random graph ensembles to real networks,
with the objective of generating precise and practical mathematical tools for
quantifying and comparing network topologies macroscopically, beyond the level
of degree statistics. Our family of ensembles can produce graphs with any
prescribed degree distribution and any degree-degree correlation function, its
control parameters can be calculated fully analytically, and as a result we can
calculate (asymptotically) formulae for entropies and complexities, and for
information-theoretic distances between networks, expressed directly and
explicitly in terms of their measured degree distribution and degree
correlations.Comment: 25 pages, 3 figure
A simple Havel-Hakimi type algorithm to realize graphical degree sequences of directed graphs
One of the simplest ways to decide whether a given finite sequence of
positive integers can arise as the degree sequence of a simple graph is the
greedy algorithm of Havel and Hakimi. This note extends their approach to
directed graphs. It also studies cases of some simple forbidden edge-sets.
Finally, it proves a result which is useful to design an MCMC algorithm to find
random realizations of prescribed directed degree sequences.Comment: 11 pages, 1 figure submitted to "The Electronic Journal of
Combinatorics
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