40,918 research outputs found
Kinetic growth walks on complex networks
Kinetically grown self-avoiding walks on various types of generalized random
networks have been studied. Networks with short- and long-tailed degree
distributions were considered (, degree or connectivity), including
scale-free networks with . The long-range behaviour of
self-avoiding walks on random networks is found to be determined by finite-size
effects. The mean self-intersection length of non-reversal random walks, ,
scales as a power of the system size $N$: $ \sim N^{\beta}$, with an
exponent $\beta = 0.5$ for short-tailed degree distributions and $\beta < 0.5$
for scale-free networks with $\gamma < 3$. The mean attrition length of kinetic
growth walks, , scales as , with an exponent
which depends on the lowest degree in the network. Results of
approximate probabilistic calculations are supported by those derived from
simulations of various kinds of networks. The efficiency of kinetic growth
walks to explore networks is largely reduced by inhomogeneity in the degree
distribution, as happens for scale-free networks.Comment: 10 pages, 8 figure
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
Rotated multifractal network generator
The recently introduced multifractal network generator (MFNG), has been shown
to provide a simple and flexible tool for creating random graphs with very
diverse features. The MFNG is based on multifractal measures embedded in 2d,
leading also to isolated nodes, whose number is relatively low for realistic
cases, but may become dominant in the limiting case of infinitely large network
sizes. Here we discuss the relation between this effect and the information
dimension for the 1d projection of the link probability measure (LPM), and
argue that the node isolation can be avoided by a simple transformation of the
LPM based on rotation.Comment: Accepted for publication in JSTA
Algebraic Aspects of Conditional Independence and Graphical Models
This chapter of the forthcoming Handbook of Graphical Models contains an
overview of basic theorems and techniques from algebraic geometry and how they
can be applied to the study of conditional independence and graphical models.
It also introduces binomial ideals and some ideas from real algebraic geometry.
When random variables are discrete or Gaussian, tools from computational
algebraic geometry can be used to understand implications between conditional
independence statements. This is accomplished by computing primary
decompositions of conditional independence ideals. As examples the chapter
presents in detail the graphical model of a four cycle and the intersection
axiom, a certain implication of conditional independence statements. Another
important problem in the area is to determine all constraints on a graphical
model, for example, equations determined by trek separation. The full set of
equality constraints can be determined by computing the model's vanishing
ideal. The chapter illustrates these techniques and ideas with examples from
the literature and provides references for further reading.Comment: 20 pages, 1 figur
Core percolation on complex networks
As a fundamental structural transition in complex networks, core percolation
is related to a wide range of important problems. Yet, previous theoretical
studies of core percolation have been focusing on the classical
Erd\H{o}s-R\'enyi random networks with Poisson degree distribution, which are
quite unlike many real-world networks with scale-free or fat-tailed degree
distributions. Here we show that core percolation can be analytically studied
for complex networks with arbitrary degree distributions. We derive the
condition for core percolation and find that purely scale-free networks have no
core for any degree exponents. We show that for undirected networks if core
percolation occurs then it is always continuous while for directed networks it
becomes discontinuous when the in- and out-degree distributions are different.
We also apply our theory to real-world directed networks and find,
surprisingly, that they often have much larger core sizes as compared to random
models. These findings would help us better understand the interesting
interplay between the structural and dynamical properties of complex networks.Comment: 17 pages, 6 figure
Simple random walk on distance-regular graphs
A survey is presented of known results concerning simple random walk on the
class of distance-regular graphs. One of the highlights is that electric
resistance and hitting times between points can be explicitly calculated and
given strong bounds for, which leads in turn to bounds on cover times, mixing
times, etc. Also discussed are harmonic functions, moments of hitting and cover
times, the Green's function, and the cutoff phenomenon. The main goal of the
paper is to present these graphs as a natural setting in which to study simple
random walk, and to stimulate further research in the field
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