5,867 research outputs found
Logical limit laws for minor-closed classes of graphs
Let be an addable, minor-closed class of graphs. We prove that
the zero-one law holds in monadic second-order logic (MSO) for the random graph
drawn uniformly at random from all {\em connected} graphs in on
vertices, and the convergence law in MSO holds if we draw uniformly at
random from all graphs in on vertices. We also prove analogues
of these results for the class of graphs embeddable on a fixed surface,
provided we restrict attention to first order logic (FO). Moreover, the
limiting probability that a given FO sentence is satisfied is independent of
the surface . We also prove that the closure of the set of limiting
probabilities is always the finite union of at least two disjoint intervals,
and that it is the same for FO and MSO. For the classes of forests and planar
graphs we are able to determine the closure of the set of limiting
probabilities precisely. For planar graphs it consists of exactly 108
intervals, each of length . Finally, we analyse
examples of non-addable classes where the behaviour is quite different. For
instance, the zero-one law does not hold for the random caterpillar on
vertices, even in FO.Comment: minor changes; accepted for publication by JCT
The Configuration Model for Partially Directed Graphs
The configuration model was originally defined for undirected networks and
has recently been extended to directed networks. Many empirical networks are
however neither undirected nor completely directed, but instead usually
partially directed meaning that certain edges are directed and others are
undirected. In the paper we define a configuration model for such networks
where nodes have in-, out-, and undirected degrees that may be dependent. We
prove conditions under which the resulting degree distributions converge to the
intended degree distributions. The new model is shown to better approximate
several empirical networks compared to undirected and completely directed
networks.Comment: 19 pages, 3 figures, 2 table
The phase transition in inhomogeneous random graphs
We introduce a very general model of an inhomogenous random graph with
independence between the edges, which scales so that the number of edges is
linear in the number of vertices. This scaling corresponds to the p=c/n scaling
for G(n,p) used to study the phase transition; also, it seems to be a property
of many large real-world graphs. Our model includes as special cases many
models previously studied.
We show that under one very weak assumption (that the expected number of
edges is `what it should be'), many properties of the model can be determined,
in particular the critical point of the phase transition, and the size of the
giant component above the transition. We do this by relating our random graphs
to branching processes, which are much easier to analyze.
We also consider other properties of the model, showing, for example, that
when there is a giant component, it is `stable': for a typical random graph, no
matter how we add or delete o(n) edges, the size of the giant component does
not change by more than o(n).Comment: 135 pages; revised and expanded slightly. To appear in Random
Structures and Algorithm
Sparse random graphs with clustering
In 2007 we introduced a general model of sparse random graphs with
independence between the edges. The aim of this paper is to present an
extension of this model in which the edges are far from independent, and to
prove several results about this extension. The basic idea is to construct the
random graph by adding not only edges but also other small graphs. In other
words, we first construct an inhomogeneous random hypergraph with independent
hyperedges, and then replace each hyperedge by a (perhaps complete) graph.
Although flexible enough to produce graphs with significant dependence between
edges, this model is nonetheless mathematically tractable. Indeed, we find the
critical point where a giant component emerges in full generality, in terms of
the norm of a certain integral operator, and relate the size of the giant
component to the survival probability of a certain (non-Poisson) multi-type
branching process. While our main focus is the phase transition, we also study
the degree distribution and the numbers of small subgraphs. We illustrate the
model with a simple special case that produces graphs with power-law degree
sequences with a wide range of degree exponents and clustering coefficients.Comment: 62 pages; minor revisio
Spatial preferential attachment networks: Power laws and clustering coefficients
We define a class of growing networks in which new nodes are given a spatial
position and are connected to existing nodes with a probability mechanism
favoring short distances and high degrees. The competition of preferential
attachment and spatial clustering gives this model a range of interesting
properties. Empirical degree distributions converge to a limit law, which can
be a power law with any exponent . The average clustering coefficient
of the networks converges to a positive limit. Finally, a phase transition
occurs in the global clustering coefficients and empirical distribution of edge
lengths when the power-law exponent crosses the critical value . Our
main tool in the proof of these results is a general weak law of large numbers
in the spirit of Penrose and Yukich.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1006 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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