4,136 research outputs found
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
Stability for large forbidden subgraphs
We extend the classical stability theorem of Erdos and Simonovits for
forbidden graphs of logarithmic order.Comment: Some polishing. Updated reference
Spread-out percolation in R^d
Let be either or the points of a Poisson process in of
intensity 1. Given parameters and , join each pair of points of
within distance independently with probability . This is the simplest
case of a `spread-out' percolation model studied by Penrose, who showed that,
as , the average degree of the corresponding random graph at the
percolation threshold tends to 1, i.e., the percolation threshold and the
threshold for criticality of the naturally associated branching process
approach one another. Here we show that this result follows immediately from of
a general result of the authors on inhomogeneous random graphs.Comment: 9 pages. Title changed. Minor changes to text, including updated
references to [3]. To appear in Random Structures and Algorithm
A note on the Harris-Kesten Theorem
Recently, a short proof of the Harris-Kesten result that the critical
probability for bond percolation in the planar square lattice is 1/2 was given,
using a sharp threshold result of Friedgut and Kalai. Here we point out that a
key part of this proof may be replaced by an argument of Russo from 1982, using
his approximate zero-one law in place of the Friedgut-Kalai result. Russo's
paper gave a new proof of the Harris-Kesten Theorem that seems to have received
little attention.Comment: 4 pages; author list changed, acknowledgement adde
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
Distances in random graphs with finite variance degrees
In this paper we study a random graph with nodes, where node has
degree and are i.i.d. with \prob(D_j\leq x)=F(x). We
assume that for some and some constant
. This graph model is a variant of the so-called configuration model, and
includes heavy tail degrees with finite variance.
The minimal number of edges between two arbitrary connected nodes, also known
as the graph distance or the hopcount, is investigated when . We
prove that the graph distance grows like , when the base of the
logarithm equals \nu=\expec[D_j(D_j -1)]/\expec[D_j]>1. This confirms the
heuristic argument of Newman, Strogatz and Watts \cite{NSW00}. In addition, the
random fluctuations around this asymptotic mean are
characterized and shown to be uniformly bounded. In particular, we show
convergence in distribution of the centered graph distance along exponentially
growing subsequences.Comment: 40 pages, 2 figure
Counting connected hypergraphs via the probabilistic method
In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number
of connected graphs on with edges, whenever and the nullity
tend to infinity. Asymptotic formulae for the number of connected
-uniform hypergraphs on with edges and so nullity
were proved by Karo\'nski and \L uczak for the case ,
and Behrisch, Coja-Oghlan and Kang for . Here we prove such a
formula for any fixed, and any satisfying and
as . This leaves open only the (much simpler) case
, which we will consider in future work. ( arXiv:1511.04739 )
Our approach is probabilistic. Let denote the random -uniform
hypergraph on in which each edge is present independently with
probability . Let and be the numbers of vertices and edges in
the largest component of . We prove a local limit theorem giving an
asymptotic formula for the probability that and take any given pair
of values within the `typical' range, for any in the supercritical
regime, i.e., when where
and ; our enumerative result then follows
easily.
Taking as a starting point the recent joint central limit theorem for
and , we use smoothing techniques to show that `nearby' pairs of values
arise with about the same probability, leading to the local limit theorem.
Behrisch et al used similar ideas in a very different way, that does not seem
to work in our setting.
Independently, Sato and Wormald have recently proved the special case ,
with an additional restriction on . They use complementary, more enumerative
methods, which seem to have a more limited scope, but to give additional
information when they do work.Comment: Expanded; asymptotics clarified - no significant mathematical
changes. 67 pages (including appendix
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
Hitting time results for Maker-Breaker games
We study Maker-Breaker games played on the edge set of a random graph.
Specifically, we consider the random graph process and analyze the first time
in a typical random graph process that Maker starts having a winning strategy
for his final graph to admit some property \mP. We focus on three natural
properties for Maker's graph, namely being -vertex-connected, admitting a
perfect matching, and being Hamiltonian. We prove the following optimal hitting
time results: with high probability Maker wins the -vertex connectivity game
exactly at the time the random graph process first reaches minimum degree ;
with high probability Maker wins the perfect matching game exactly at the time
the random graph process first reaches minimum degree ; with high
probability Maker wins the Hamiltonicity game exactly at the time the random
graph process first reaches minimum degree . The latter two statements
settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page
The minimal density of triangles in tripartite graphs
We determine the minimal density of triangles in a tripartite graph with
prescribed edge densities. This extends a previous result of Bondy, Shen,
Thomass\'e and Thomassen characterizing those edge densities guaranteeing the
existence of a triangle in a tripartite graph.
To be precise we show that a suitably weighted copy of the graph formed by
deleting a certain 9-cycle from has minimal triangle density among
all weighted tripartite graphs with prescribed edge densities.Comment: 44 pages including 12 page appendix of C++ cod
- …