69,951 research outputs found
Epidemics on random intersection graphs
In this paper we consider a model for the spread of a stochastic SIR
(Susceptible Infectious Recovered) epidemic on a network of
individuals described by a random intersection graph. Individuals belong to a
random number of cliques, each of random size, and infection can be transmitted
between two individuals if and only if there is a clique they both belong to.
Both the clique sizes and the number of cliques an individual belongs to follow
mixed Poisson distributions. An infinite-type branching process approximation
(with type being given by the length of an individual's infectious period) for
the early stages of an epidemic is developed and made fully rigorous by proving
an associated limit theorem as the population size tends to infinity. This
leads to a threshold parameter , so that in a large population an epidemic
with few initial infectives can give rise to a large outbreak if and only if
. A functional equation for the survival probability of the
approximating infinite-type branching process is determined; if , this
equation has no nonzero solution, while if , it is shown to have
precisely one nonzero solution. A law of large numbers for the size of such a
large outbreak is proved by exploiting a single-type branching process that
approximates the size of the susceptibility set of a typical individual.Comment: Published in at http://dx.doi.org/10.1214/13-AAP942 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Moment-based parameter estimation in binomial random intersection graph models
Binomial random intersection graphs can be used as parsimonious statistical
models of large and sparse networks, with one parameter for the average degree
and another for transitivity, the tendency of neighbours of a node to be
connected. This paper discusses the estimation of these parameters from a
single observed instance of the graph, using moment estimators based on
observed degrees and frequencies of 2-stars and triangles. The observed data
set is assumed to be a subgraph induced by a set of nodes sampled from
the full set of nodes. We prove the consistency of the proposed estimators
by showing that the relative estimation error is small with high probability
for . As a byproduct, our analysis confirms that the
empirical transitivity coefficient of the graph is with high probability close
to the theoretical clustering coefficient of the model.Comment: 15 pages, 6 figure
Triangle-Intersecting Families of Graphs
A family of graphs F is said to be triangle-intersecting if for any two
graphs G,H in F, the intersection of G and H contains a triangle. A conjecture
of Simonovits and Sos from 1976 states that the largest triangle-intersecting
families of graphs on a fixed set of n vertices are those obtained by fixing a
specific triangle and taking all graphs containing it, resulting in a family of
size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations
(for example, we prove that the same is true of odd-cycle-intersecting
families, and we obtain best possible bounds on the size of the family under
different, not necessarily uniform, measures). We also obtain stability
results, showing that almost-largest triangle-intersecting families have
approximately the same structure.Comment: 43 page
Random subcube intersection graphs I: cliques and covering
We study random subcube intersection graphs, that is, graphs obtained by
selecting a random collection of subcubes of a fixed hypercube to serve
as the vertices of the graph, and setting an edge between a pair of subcubes if
their intersection is non-empty. Our motivation for considering such graphs is
to model `random compatibility' between vertices in a large network. For both
of the models considered in this paper, we determine the thresholds for
covering the underlying hypercube and for the appearance of s-cliques. In
addition we pose some open problems.Comment: 38 pages, 1 figur
On Topological Properties of Wireless Sensor Networks under the q-Composite Key Predistribution Scheme with On/Off Channels
The q-composite key predistribution scheme [1] is used prevalently for secure
communications in large-scale wireless sensor networks (WSNs). Prior work
[2]-[4] explores topological properties of WSNs employing the q-composite
scheme for q = 1 with unreliable communication links modeled as independent
on/off channels. In this paper, we investigate topological properties related
to the node degree in WSNs operating under the q-composite scheme and the
on/off channel model. Our results apply to general q and are stronger than
those reported for the node degree in prior work even for the case of q being
1. Specifically, we show that the number of nodes with certain degree
asymptotically converges in distribution to a Poisson random variable, present
the asymptotic probability distribution for the minimum degree of the network,
and establish the asymptotically exact probability for the property that the
minimum degree is at least an arbitrary value. Numerical experiments confirm
the validity of our analytical findings.Comment: Best Student Paper Finalist in IEEE International Symposium on
Information Theory (ISIT) 201
- …