158,397 research outputs found
Percolation on sparse random graphs with given degree sequence
We study the two most common types of percolation process on a sparse random
graph with a given degree sequence. Namely, we examine first a bond percolation
process where the edges of the graph are retained with probability p and
afterwards we focus on site percolation where the vertices are retained with
probability p. We establish critical values for p above which a giant component
emerges in both cases. Moreover, we show that in fact these coincide. As a
special case, our results apply to power law random graphs. We obtain rigorous
proofs for formulas derived by several physicists for such graphs.Comment: 20 page
Susceptibility of random graphs with given vertex degrees
We study the susceptibility, i.e., the mean cluster size, in random graphs
with given vertex degrees. We show, under weak assumptions, that the
susceptibility converges to the expected cluster size in the corresponding
branching process. In the supercritical case, a corresponding result holds for
the modified susceptibility ignoring the giant component and the expected size
of a finite cluster in the branching process; this is proved using a duality
theorem.
The critical behaviour is studied. Examples are given where the critical
exponents differ on the subcritical and supercritical sides.Comment: 25 page
The component sizes of a critical random graph with given degree sequence
Consider a critical random multigraph with vertices
constructed by the configuration model such that its vertex degrees are
independent random variables with the same distribution (criticality
means that the second moment of is finite and equals twice its first
moment). We specify the scaling limits of the ordered sequence of component
sizes of as tends to infinity in different cases. When
has finite third moment, the components sizes rescaled by
converge to the excursion lengths of a Brownian motion with parabolic drift
above past minima, whereas when is a power law distribution with exponent
, the components sizes rescaled by converge to the excursion lengths of a certain nontrivial
drifted process with independent increments above past minima. We deduce the
asymptotic behavior of the component sizes of a critical random simple graph
when has finite third moment.Comment: Published in at http://dx.doi.org/10.1214/13-AAP985 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)
Although the ``scale-free'' literature is large and growing, it gives neither
a precise definition of scale-free graphs nor rigorous proofs of many of their
claimed properties. In fact, it is easily shown that the existing theory has
many inherent contradictions and verifiably false claims. In this paper, we
propose a new, mathematically precise, and structural definition of the extent
to which a graph is scale-free, and prove a series of results that recover many
of the claimed properties while suggesting the potential for a rich and
interesting theory. With this definition, scale-free (or its opposite,
scale-rich) is closely related to other structural graph properties such as
various notions of self-similarity (or respectively, self-dissimilarity).
Scale-free graphs are also shown to be the likely outcome of random
construction processes, consistent with the heuristic definitions implicit in
existing random graph approaches. Our approach clarifies much of the confusion
surrounding the sensational qualitative claims in the scale-free literature,
and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in Internet
Mathematics (2005
Percolation by cumulative merging and phase transition for the contact process on random graphs
Given a weighted graph, we introduce a partition of its vertex set such that
the distance between any two clusters is bounded from below by a power of the
minimum weight of both clusters. This partition is obtained by recursively
merging smaller clusters and cumulating their weights. For several classical
random weighted graphs, we show that there exists a phase transition regarding
the existence of an infinite cluster.
The motivation for introducing this partition arises from a connection with
the contact process as it roughly describes the geometry of the sets where the
process survives for a long time. We give a sufficient condition on a graph to
ensure that the contact process has a non trivial phase transition in terms of
the existence of an infinite cluster. As an application, we prove that the
contact process admits a sub-critical phase on d-dimensional random geometric
graphs and on random Delaunay triangulations. To the best of our knowledge,
these are the first examples of graphs with unbounded degrees where the
critical parameter is shown to be strictly positive.Comment: 50 pages, many figure
Nonuniform random geometric graphs with location-dependent radii
We propose a distribution-free approach to the study of random geometric
graphs. The distribution of vertices follows a Poisson point process with
intensity function , where , and is a
probability density function on . A vertex located at
connects via directed edges to other vertices that are within a cut-off
distance . We prove strong law results for (i) the critical cut-off
function so that almost surely, the graph does not contain any node with
out-degree zero for sufficiently large and (ii) the maximum and minimum
vertex degrees. We also provide a characterization of the cut-off function for
which the number of nodes with out-degree zero converges in distribution to a
Poisson random variable. We illustrate this result for a class of densities
with compact support that have at most polynomial rates of decay to zero.
Finally, we state a sufficient condition for an enhanced version of the above
graph to be almost surely connected eventually.Comment: Published in at http://dx.doi.org/10.1214/11-AAP823 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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