9,272 research outputs found
Percolation and Connectivity in the Intrinsically Secure Communications Graph
The ability to exchange secret information is critical to many commercial,
governmental, and military networks. The intrinsically secure communications
graph (iS-graph) is a random graph which describes the connections that can be
securely established over a large-scale network, by exploiting the physical
properties of the wireless medium. This paper aims to characterize the global
properties of the iS-graph in terms of: (i) percolation on the infinite plane,
and (ii) full connectivity on a finite region. First, for the Poisson iS-graph
defined on the infinite plane, the existence of a phase transition is proven,
whereby an unbounded component of connected nodes suddenly arises as the
density of legitimate nodes is increased. This shows that long-range secure
communication is still possible in the presence of eavesdroppers. Second, full
connectivity on a finite region of the Poisson iS-graph is considered. The
exact asymptotic behavior of full connectivity in the limit of a large density
of legitimate nodes is characterized. Then, simple, explicit expressions are
derived in order to closely approximate the probability of full connectivity
for a finite density of legitimate nodes. The results help clarify how the
presence of eavesdroppers can compromise long-range secure communication.Comment: Submitted for journal publicatio
The statistical geometry of scale-free random trees
The properties of scale-free random trees are investigated using both
preconditioning on non-extinction and fixed size averages, in order to study
the thermodynamic limit. The scaling form of volume probability is found, the
connectivity dimensions are determined and compared with other exponents which
describe the growth. The (local) spectral dimension is also determined, through
the study of the massless limit of the Gaussian model on such trees.Comment: 21 pages, 2 figures, revtex4, minor changes (published version
Continuum percolation of polydisperse hyperspheres in infinite dimensions
We analyze the critical connectivity of systems of penetrable -dimensional
spheres having size distributions in terms of weighed random geometrical
graphs, in which vertex coordinates correspond to random positions of the
sphere centers and edges are formed between any two overlapping spheres. Edge
weights naturally arise from the different radii of two overlapping spheres.
For the case in which the spheres have bounded size distributions, we show that
clusters of connected spheres are tree-like for and they
contain no closed loops. In this case, we find that the mean cluster size
diverges at the percolation threshold density ,
independently of the particular size distribution. We also show that the mean
number of overlaps for a particle at criticality is smaller than unity,
while only for spheres with fixed radii. We explain these
features by showing that in the large dimensionality limit the critical
connectivity is dominated by the spheres with the largest size. Assuming that
closed loops can be neglected also for unbounded radii distributions, we find
that the asymptotic critical threshold for systems of spheres with radii
following a lognormal distribution is no longer universal, and that it can be
smaller than for .Comment: 11 pages, 5 figure
Damaging 2D Quantum Gravity
We investigate numerically the behaviour of damage spreading in a Kauffman
cellular automaton with quenched rules on a dynamical graph, which is
equivalent to coupling the model to discretized 2D gravity. The model is
interesting from the cellular automaton point of view as it lies midway between
a fully quenched automaton with fixed rules and fixed connectivity and a
(soluble) fully annealed automaton with varying rules and varying connectivity.
In addition, we simulate the automaton on a fixed graph coming from a
2D gravity simulation as a means of exploring the graph geometry.Comment: 6 pages, COLO-HEP-332;LPTHE-Orsay-93-5
Wireless Secrecy in Large-Scale Networks
The ability to exchange secret information is critical to many commercial,
governmental, and military networks. The intrinsically secure communications
graph (iS-graph) is a random graph which describes the connections that can be
securely established over a large-scale network, by exploiting the physical
properties of the wireless medium. This paper provides an overview of the main
properties of this new class of random graphs. We first analyze the local
properties of the iS-graph, namely the degree distributions and their
dependence on fading, target secrecy rate, and eavesdropper collusion. To
mitigate the effect of the eavesdroppers, we propose two techniques that
improve secure connectivity. Then, we analyze the global properties of the
iS-graph, namely percolation on the infinite plane, and full connectivity on a
finite region. These results help clarify how the presence of eavesdroppers can
compromise secure communication in a large-scale network.Comment: To appear: Proc. IEEE Information Theory and Applications Workshop
(ITA'11), San Diego, CA, Feb. 2011, pp. 1-10, Invited Pape
Remarks on Bootstrap Percolation in Metric Networks
We examine bootstrap percolation in d-dimensional, directed metric graphs in
the context of recent measurements of firing dynamics in 2D neuronal cultures.
There are two regimes, depending on the graph size N. Large metric graphs are
ignited by the occurrence of critical nuclei, which initially occupy an
infinitesimal fraction, f_* -> 0, of the graph and then explode throughout a
finite fraction. Smaller metric graphs are effectively random in the sense that
their ignition requires the initial ignition of a finite, unlocalized fraction
of the graph, f_* >0. The crossover between the two regimes is at a size N_*
which scales exponentially with the connectivity range \lambda like_* \sim
\exp\lambda^d. The neuronal cultures are finite metric graphs of size N \simeq
10^5-10^6, which, for the parameters of the experiment, is effectively random
since N<< N_*. This explains the seeming contradiction in the observed finite
f_* in these cultures. Finally, we discuss the dynamics of the firing front
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