15 research outputs found
Symmetric motifs in random geometric graphs
We study symmetric motifs in random geometric graphs. Symmetric motifs are
subsets of nodes which have the same adjacencies. These subgraphs are
particularly prevalent in random geometric graphs and appear in the Laplacian
and adjacency spectrum as sharp, distinct peaks, a feature often found in
real-world networks. We look at the probabilities of their appearance and
compare these across parameter space and dimension. We then use the Chen-Stein
method to derive the minimum separation distance in random geometric graphs
which we apply to study symmetric motifs in both the intensive and
thermodynamic limits. In the thermodynamic limit the probability that the
closest nodes are symmetric approaches one, whilst in the intensive limit this
probability depends upon the dimension.Comment: 11 page
Temporal-varying failures of nodes in networks
We consider networks in which random walkers are removed because of the
failure of specific nodes. We interpret the rate of loss as a measure of the
importance of nodes, a notion we denote as failure-centrality. We show that the
degree of the node is not sufficient to determine this measure and that, in a
first approximation, the shortest loops through the node have to be taken into
account. We propose approximations of the failure-centrality which are valid
for temporal-varying failures and we dwell on the possibility of externally
changing the relative importance of nodes in a given network, by exploiting the
interference between the loops of a node and the cycles of the temporal pattern
of failures. In the limit of long failure cycles we show analytically that the
escape in a node is larger than the one estimated from a stochastic failure
with the same failure probability. We test our general formalism in two
real-world networks (air-transportation and e-mail users) and show how
communities lead to deviations from predictions for failures in hubs.Comment: 7 pages, 3 figure
Follow the fugitive: an application of the method of images to open dynamical systems
Borrowing and extending the method of images we introduce a theoretical
framework that greatly simplifies analytical and numerical investigations of
the escape rate in open dynamical systems. As an example, we explicitly derive
the exact size- and position-dependent escape rate in a Markov case for holes
of finite size. Moreover, a general relation between the transfer operators of
closed and corresponding open systems, together with the generating function of
the probability of return to the hole is derived. This relation is then used to
compute the small hole asymptotic behavior, in terms of readily calculable
quantities. As an example we derive logarithmic corrections in the second order
term. Being valid for Markov systems, our framework can find application in
information theory, network theory, quantum Weyl law and via Ulam's method can
be used as an approximation method in more general dynamical systems.Comment: 9 pages, 1 figur
Linear and fractal diffusion coefficients in a family of one dimensional chaotic maps
We analyse deterministic diffusion in a simple, one-dimensional setting
consisting of a family of four parameter dependent, chaotic maps defined over
the real line. When iterated under these maps, a probability density function
spreads out and one can define a diffusion coefficient. We look at how the
diffusion coefficient varies across the family of maps and under parameter
variation. Using a technique by which Taylor-Green-Kubo formulae are evaluated
in terms of generalised Takagi functions, we derive exact, fully analytical
expressions for the diffusion coefficients. Typically, for simple maps these
quantities are fractal functions of control parameters. However, our family of
four maps exhibits both fractal and linear behavior. We explain these different
structures by looking at the topology of the Markov partitions and the ergodic
properties of the maps.Comment: 21 pages, 19 figure
Dependence of chaotic diffusion on the size and position of holes
A particle driven by deterministic chaos and moving in a spatially extended
environment can exhibit normal diffusion, with its mean square displacement
growing proportional to the time. Here we consider the dependence of the
diffusion coefficient on the size and the position of areas of phase space
linking spatial regions (`holes') in a class of simple one-dimensional,
periodically lifted maps. The parameter dependent diffusion coefficient can be
obtained analytically via a Taylor-Green-Kubo formula in terms of a functional
recursion relation. We find that the diffusion coefficient varies
non-monotonically with the size of a hole and its position, which implies that
a diffusion coefficient can increase by making the hole smaller. We derive
analytic formulas for small holes in terms of periodic orbits covered by the
holes. The asymptotic regimes that we observe show deviations from the standard
stochastic random walk approximation. The escape rate of the corresponding open
system is also calculated. The resulting parameter dependencies are compared
with the ones for the diffusion coefficient and explained in terms of periodic
orbits.Comment: 12 pages, 5 figure
Capturing correlations in chaotic diffusion by approximation methods
We investigate three different methods for systematically approximating the
diffusion coefficient of a deterministic random walk on the line which contains
dynamical correlations that change irregularly under parameter variation.
Capturing these correlations by incorporating higher order terms, all schemes
converge to the analytically exact result. Two of these methods are based on
expanding the Taylor-Green-Kubo formula for diffusion, whilst the third method
approximates Markov partitions and transition matrices by using the escape rate
theory of chaotic diffusion. We check the practicability of the different
methods by working them out analytically and numerically for a simple
one-dimensional map, study their convergence and critically discuss their
usefulness in identifying a possible fractal instability of parameter-dependent
diffusion, in case of dynamics where exact results for the diffusion
coefficient are not available.Comment: 11 pages, 5 figure
Sparre-Andersen theorem with spatiotemporal correlations
The Sparre-Andersen theorem is a remarkable result in one-dimensional random
walk theory concerning the universality of the ubiquitous first-passage-time
distribution. It states that the probability distribution of the
number of steps needed for a walker starting at the origin to land on the
positive semi-axes does not depend on the details of the distribution for the
jumps of the walker, provided this distribution is symmetric and continuous,
where in particular for large number of steps . On
the other hand, there are many physical situations in which the time spent by
the walker in doing one step depends on the length of the step and the interest
concentrates on the time needed for a return, not on the number of steps. Here
we modify the Sparre-Andersen proof to deal with such cases, in rather general
situations in which the time variable correlates with the step variable. As an
example we present a natural process in 2D that shows deviations from normal
scaling are present for the first-passage-time distribution on a semi plane.Comment: 6 pages. Minor modifications in accordance with the published versio
Counting geodesic paths in 1-D VANETs
In the IEEE 802.11p standard addressing vehicular communications, Basic
Safety Messages (BSMs) can be bundled together and relayed as to increase the
effective communication range of transmitting vehicles. This process forms a
vehicular ad hoc network (VANET) for the dissemination of safety information.
The number of "shortest multihop paths" (or geodesics) connecting two network
nodes is an important statistic which can be used to enhance throughput,
validate threat events, protect against collusion attacks, infer location
information, and also limit redundant broadcasts thus reducing interference. To
this end, we analytically calculate for the first time the mean and variance of
the number of geodesics in 1D VANETs.Comment: 11 pages, 5 figure