2,237 research outputs found
Clustering in complex networks. II. Percolation properties
The percolation properties of clustered networks are analyzed in detail. In
the case of weak clustering, we present an analytical approach that allows to
find the critical threshold and the size of the giant component. Numerical
simulations confirm the accuracy of our results. In more general terms, we show
that weak clustering hinders the onset of the giant component whereas strong
clustering favors its appearance. This is a direct consequence of the
differences in the -core structure of the networks, which are found to be
totally different depending on the level of clustering. An empirical analysis
of a real social network confirms our predictions.Comment: Updated reference lis
Analytical results for stochastically growing networks: connection to the zero range process
We introduce a stochastic model of growing networks where both, the number of
new nodes which joins the network and the number of connections, vary
stochastically. We provide an exact mapping between this model and zero range
process, and use this mapping to derive an analytical solution of degree
distribution for any given evolution rule. One can also use this mapping to
infer about a possible evolution rule for a given network. We demonstrate this
for protein-protein interaction (PPI) network for Saccharomyces Cerevisiae.Comment: 4+ pages, revtex, 3 eps figure
Maximum size of reverse-free sets of permutations
Two words have a reverse if they have the same pair of distinct letters on
the same pair of positions, but in reversed order. A set of words no two of
which have a reverse is said to be reverse-free. Let F(n,k) be the maximum size
of a reverse-free set of words from [n]^k where no letter repeats within a
word. We show the following lower and upper bounds in the case n >= k: F(n,k)
\in n^k k^{-k/2 + O(k/log k)}. As a consequence of the lower bound, a set of
n-permutations each two having a reverse has size at most n^{n/2 + O(n/log n)}.Comment: 10 page
Comparison of Ising magnet on directed versus undirected Erdos-Renyi and scale-free network
Scale-free networks are a recently developed approach to model the
interactions found in complex natural and man-made systems. Such networks
exhibit a power-law distribution of node link (degree) frequencies n(k) in
which a small number of highly connected nodes predominate over a much greater
number of sparsely connected ones. In contrast, in an Erdos-Renyi network each
of N sites is connected to every site with a low probability p (of the orde r
of 1/N). Then the number k of neighbors will fluctuate according to a Poisson
distribution. One can instead assume that each site selects exactly k neighbors
among the other sites. Here we compare in both cases the usual network with the
directed network, when site A selects site B as a neighbor, and then B
influences A but A does not influence B. As we change from undirected to
directed scale-free networks, the spontaneous magnetization vanishes after an
equilibration time following an Arrhenius law, while the directed ER networks
have a positive Curie temperature.Comment: 10 pages including all figures, for Int. J, Mod. Phys. C 1
Statistical Analysis of Airport Network of China
Through the study of airport network of China (ANC), composed of 128 airports
(nodes) and 1165 flights (edges), we show the topological structure of ANC
conveys two characteristics of small worlds, a short average path length
(2.067) and a high degree of clustering (0.733). The cumulative degree
distributions of both directed and undirected ANC obey two-regime power laws
with different exponents, i.e., the so-called Double Pareto Law. In-degrees and
out-degrees of each airport have positive correlations, whereas the undirected
degrees of adjacent airports have significant linear anticorrelations. It is
demonstrated both weekly and daily cumulative distributions of flight weights
(frequencies) of ANC have power-law tails. Besides, the weight of any given
flight is proportional to the degrees of both airports at the two ends of that
flight. It is also shown the diameter of each sub-cluster (consisting of an
airport and all those airports to which it is linked) is inversely proportional
to its density of connectivity. Efficiency of ANC and of its sub-clusters are
measured through a simple definition. In terms of that, the efficiency of ANC's
sub-clusters increases as the density of connectivity does. ANC is found to
have an efficiency of 0.484.Comment: 6 Pages, 5 figure
On graphs with a large chromatic number containing no small odd cycles
In this paper, we present the lower bounds for the number of vertices in a
graph with a large chromatic number containing no small odd cycles
An analysis of the fixation probability of a mutant on special classes of non-directed graphs
There is a growing interest in the study of evolutionary dynamics on populations with some non-homogeneous structure. In this paper we follow the model of Lieberman et al. (Lieberman et al. 2005 Nature 433, 312–316) of evolutionary dynamics on a graph. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. We further demonstrate that finding similar solutions on graphs outside these classes is far more complex. Finally, we investigate our chosen classes numerically and discuss a number of features of the graphs; for example, we find the fixation probabilities for different initial starting positions and observe that average fixation probabilities are always increased for advantageous mutants as compared with those of unstructured populations
Two-dimensional gauge theories of the symmetric group S(n) and branched n-coverings of Riemann surfaces in the large-n limit
Branched n-coverings of Riemann surfaces are described by a 2d lattice gauge
theory of the symmetric group S(n) defined on a cell discretization of the
surface. We study the theory in the large-n limit, and we find a rich phase
diagram with first and second order transition lines. The various phases are
characterized by different connectivity properties of the covering surface. We
point out some interesting connections with the theory of random walks on group
manifolds and with random graph theory.Comment: Talk presented at the "Light-cone physics: particles and strings",
Trento, Italy, September 200
On a Problem of Sidon in Additive Number Theory and on Some Related Problems Addendum
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135303/1/jlms0208.pd
Unified model for network dynamics exhibiting nonextensive statistics
We introduce a dynamical network model which unifies a number of network
families which are individually known to exhibit -exponential degree
distributions. The present model dynamics incorporates static (non-growing)
self-organizing networks, preferentially growing networks, and (preferentially)
rewiring networks. Further, it exhibits a natural random graph limit. The
proposed model generalizes network dynamics to rewiring and growth modes which
depend on internal topology as well as on a metric imposed by the space they
are embedded in. In all of the networks emerging from the presented model we
find q-exponential degree distributions over a large parameter space. We
comment on the parameter dependence of the corresponding entropic index q for
the degree distributions, and on the behavior of the clustering coefficients
and neighboring connectivity distributions.Comment: 11 pages 8 fig
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