46 research outputs found
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
Upper bounds for number of removed edges in the Erased Configuration Model
Models for generating simple graphs are important in the study of real-world
complex networks. A well established example of such a model is the erased
configuration model, where each node receives a number of half-edges that are
connected to half-edges of other nodes at random, and then self-loops are
removed and multiple edges are concatenated to make the graph simple. Although
asymptotic results for many properties of this model, such as the limiting
degree distribution, are known, the exact speed of convergence in terms of the
graph sizes remains an open question. We provide a first answer by analyzing
the size dependence of the average number of removed edges in the erased
configuration model. By combining known upper bounds with a Tauberian Theorem
we obtain upper bounds for the number of removed edges, in terms of the size of
the graph. Remarkably, when the degree distribution follows a power-law, we
observe three scaling regimes, depending on the power law exponent. Our results
provide a strong theoretical basis for evaluating finite-size effects in
networks
Cutoff for non-backtracking random walks on sparse random graphs
A finite ergodic Markov chain is said to exhibit cutoff if its distance to
stationarity remains close to 1 over a certain number of iterations and then
abruptly drops to near 0 on a much shorter time scale. Discovered in the
context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now
believed to be rather typical among fast mixing Markov chains. Yet,
establishing it rigorously often requires a challengingly detailed
understanding of the underlying chain. Here we consider non-backtracking random
walks on random graphs with a given degree sequence. Under a general sparsity
condition, we establish the cutoff phenomenon, determine its precise window,
and prove that the (suitably rescaled) cutoff profile approaches a remarkably
simple, universal shape
The largest component in a subcritical random graph with a power law degree distribution
It is shown that in a subcritical random graph with given vertex degrees
satisfying a power law degree distribution with exponent , the
largest component is of order . More precisely, the order of
the largest component is approximatively given by a simple constant times the
largest vertex degree. These results are extended to several other random graph
models with power law degree distributions. This proves a conjecture by
Durrett.Comment: Published in at http://dx.doi.org/10.1214/07-AAP490 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Phase transitions for scaling of structural correlations in directed networks
Analysis of degree-degree dependencies in complex networks, and their impact
on processes on networks requires null models, i.e. models that generate
uncorrelated scale-free networks. Most models to date however show structural
negative dependencies, caused by finite size effects. We analyze the behavior
of these structural negative degree-degree dependencies, using rank based
correlation measures, in the directed Erased Configuration Model. We obtain
expressions for the scaling as a function of the exponents of the
distributions. Moreover, we show that this scaling undergoes a phase
transition, where one region exhibits scaling related to the natural cut-off of
the network while another region has scaling similar to the structural cut-off
for uncorrelated networks. By establishing the speed of convergence of these
structural dependencies we are able to asses statistical significance of
degree-degree dependencies on finite complex networks when compared to networks
generated by the directed Erased Configuration Model
The structure of typical clusters in large sparse random configurations
The initial purpose of this work is to provide a probabilistic explanation of
a recent result on a version of Smoluchowski's coagulation equations in which
the number of aggregations is limited. The latter models the deterministic
evolution of concentrations of particles in a medium where particles coalesce
pairwise as time passes and each particle can only perform a given number of
aggregations. Under appropriate assumptions, the concentrations of particles
converge as time tends to infinity to some measure which bears a striking
resemblance with the distribution of the total population of a Galton-Watson
process started from two ancestors. Roughly speaking, the configuration model
is a stochastic construction which aims at producing a typical graph on a set
of vertices with pre-described degrees. Specifically, one attaches to each
vertex a certain number of stubs, and then join pairwise the stubs uniformly at
random to create edges between vertices. In this work, we use the configuration
model as the stochastic counterpart of Smoluchowski's coagulation equations
with limited aggregations. We establish a hydrodynamical type limit theorem for
the empirical measure of the shapes of clusters in the configuration model when
the number of vertices tends to . The limit is given in terms of the
distribution of a Galton-Watson process started with two ancestors
Degree correlations in scale-free null models
We study the average nearest neighbor degree of vertices with degree
. In many real-world networks with power-law degree distribution
falls off in , a property ascribed to the constraint that any two vertices
are connected by at most one edge. We show that indeed decays in in
three simple random graph null models with power-law degrees: the erased
configuration model, the rank-1 inhomogeneous random graph and the hyperbolic
random graph. We consider the large-network limit when the number of nodes
tends to infinity. We find for all three null models that starts to
decay beyond and then settles on a power law , with the degree exponent.Comment: 21 pages, 4 figure