5 research outputs found
Duality in inhomogeneous random graphs, and the cut metric
The classical random graph model satisfies a `duality
principle', in that removing the giant component from a supercritical instance
of the model leaves (essentially) a subcritical instance. Such principles have
been proved for various models; they are useful since it is often much easier
to study the subcritical model than to directly study small components in the
supercritical model. Here we prove a duality principle of this type for a very
general class of random graphs with independence between the edges, defined by
convergence of the matrices of edge probabilities in the cut metric.Comment: 13 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
Random minimum spanning tree and dense graph limits
A theorem of Frieze from 1985 asserts that the total weight of the minimum
spanning tree of the complete graph whose edges get independent weights
from the distribution converges to Ap\'ery's constant in
probability, as . We generalize this result to sequences of graphs
that converge to a graphon . Further, we allow the weights of the
edges to be drawn from different distributions (subject to moderate
conditions). The limiting total weight of the minimum spanning tree
is expressed in terms of a certain branching process defined on , which was
studied previously by Bollob\'as, Janson and Riordan in connection with the
giant component in inhomogeneous random graphs.Comment: 20 pages, 1 figur
Susceptibility in inhomogeneous random graphs
We study the susceptibility, i.e., the mean size of the component containing
a random vertex, in a general model of inhomogeneous random graphs. This is one
of the fundamental quantities associated to (percolation) phase transitions; in
practice one of its main uses is that it often gives a way of determining the
critical point by solving certain linear equations. Here we relate the
susceptibility of suitable random graphs to a quantity associated to the
corresponding branching process, and study both quantities in various natural
examples.Comment: 51 page