193 research outputs found
Detrended fluctuation analysis as a statistical tool to monitor the climate
Detrended fluctuation analysis is used to investigate power law relationship
between the monthly averages of the maximum daily temperatures for different
locations in the western US. On the map created by the power law exponents, we
can distinguish different geographical regions with different power law
exponents. When the power law exponents obtained from the detrended fluctuation
analysis are plotted versus the standard deviation of the temperature
fluctuations, we observe different data points belonging to the different
climates, hence indicating that by observing the long-time trends in the
fluctuations of temperature we can distinguish between different climates.Comment: 8 pages, 4 figures, submitted to JSTA
Width of percolation transition in complex networks
It is known that the critical probability for the percolation transition is
not a sharp threshold, actually it is a region of non-zero width
for systems of finite size. Here we present evidence that for complex networks
, where is the average
length of the percolation cluster, and is the number of nodes in the
network. For Erd\H{o}s-R\'enyi (ER) graphs , while for
scale-free (SF) networks with a degree distribution
and , . We show analytically
and numerically that the \textit{survivability} , which is the
probability of a cluster to survive chemical shells at probability ,
behaves near criticality as . Thus
for probabilities inside the region the behavior of the
system is indistinguishable from that of the critical point
Graph Partitioning Induced Phase Transitions
We study the percolation properties of graph partitioning on random regular
graphs with N vertices of degree . Optimal graph partitioning is directly
related to optimal attack and immunization of complex networks. We find that
for any partitioning process (even if non-optimal) that partitions the graph
into equal sized connected components (clusters), the system undergoes a
percolation phase transition at where is the fraction of
edges removed to partition the graph. For optimal partitioning, at the
percolation threshold, we find where is the size of the
clusters and where is their diameter. Additionally,
we find that undergoes multiple non-percolation transitions for
Fractal dimensions of the Q-state Potts model for the complete and external hulls
Fortuin-Kastelyn clusters in the critical -state Potts model are
conformally invariant fractals. We obtain simulation results for the fractal
dimension of the complete and external (accessible) hulls for Q=1, 2, 3, and 4,
on clusters that wrap around a cylindrical system. We find excellent agreement
between these results and theoretical predictions. We also obtain the
probability distributions of the hull lengths and maximal heights of the
clusters in this geometry and provide a conjecture for their form.Comment: 9 pages 4 figure
Percolation theory applied to measures of fragmentation in social networks
We apply percolation theory to a recently proposed measure of fragmentation
for social networks. The measure is defined as the ratio between the
number of pairs of nodes that are not connected in the fragmented network after
removing a fraction of nodes and the total number of pairs in the original
fully connected network. We compare with the traditional measure used in
percolation theory, , the fraction of nodes in the largest cluster
relative to the total number of nodes. Using both analytical and numerical
methods from percolation, we study Erd\H{o}s-R\'{e}nyi (ER) and scale-free (SF)
networks under various types of node removal strategies. The removal strategies
are: random removal, high degree removal and high betweenness centrality
removal. We find that for a network obtained after removal (all strategies) of
a fraction of nodes above percolation threshold, . For fixed and close to percolation threshold
(), we show that better reflects the actual fragmentation. Close
to , for a given , has a broad distribution and it is
thus possible to improve the fragmentation of the network. We also study and
compare the fragmentation measure and the percolation measure
for a real social network of workplaces linked by the households of the
employees and find similar results.Comment: submitted to PR
On the Aizenman exponent in critical percolation
The probabilities of clusters spanning a hypercube of dimensions two to seven
along one axis of a percolation system under criticality were investigated
numerically. We used a modified Hoshen--Kopelman algorithm combined with
Grassberger's "go with the winner" strategy for the site percolation. We
carried out a finite-size analysis of the data and found that the probabilities
confirm Aizenman's proposal of the multiplicity exponent for dimensions three
to five. A crossover to the mean-field behavior around the upper critical
dimension is also discussed.Comment: 5 pages, 4 figures, 4 table
Self-similarity of complex networks
Complex networks have been studied extensively due to their relevance to many
real systems as diverse as the World-Wide-Web (WWW), the Internet, energy
landscapes, biological and social networks
\cite{ab-review,mendes,vespignani,newman,amaral}. A large number of real
networks are called ``scale-free'' because they show a power-law distribution
of the number of links per node \cite{ab-review,barabasi1999,faloutsos}.
However, it is widely believed that complex networks are not {\it length-scale}
invariant or self-similar. This conclusion originates from the ``small-world''
property of these networks, which implies that the number of nodes increases
exponentially with the ``diameter'' of the network
\cite{erdos,bollobas,milgram,watts}, rather than the power-law relation
expected for a self-similar structure. Nevertheless, here we present a novel
approach to the analysis of such networks, revealing that their structure is
indeed self-similar. This result is achieved by the application of a
renormalization procedure which coarse-grains the system into boxes containing
nodes within a given "size". Concurrently, we identify a power-law relation
between the number of boxes needed to cover the network and the size of the box
defining a finite self-similar exponent. These fundamental properties, which
are shown for the WWW, social, cellular and protein-protein interaction
networks, help to understand the emergence of the scale-free property in
complex networks. They suggest a common self-organization dynamics of diverse
networks at different scales into a critical state and in turn bring together
previously unrelated fields: the statistical physics of complex networks with
renormalization group, fractals and critical phenomena.Comment: 28 pages, 12 figures, more informations at http://www.jamlab.or
Amnestically induced persistence in random walks
We study how the Hurst exponent depends on the fraction of the
total time remembered by non-Markovian random walkers that recall only the
distant past. We find that otherwise nonpersistent random walkers switch to
persistent behavior when inflicted with significant memory loss. Such memory
losses induce the probability density function of the walker's position to
undergo a transition from Gaussian to non-Gaussian. We interpret these findings
of persistence in terms of a breakdown of self-regulation mechanisms and
discuss their possible relevance to some of the burdensome behavioral and
psychological symptoms of Alzheimer's disease and other dementias.Comment: 4 pages, 3 figs, subm. to Phys. Rev. Let
Percolation and Conduction in Restricted Geometries
The finite-size scaling behaviour for percolation and conduction is studied
in two-dimensional triangular-shaped random resistor networks at the
percolation threshold. The numerical simulations are performed using an
efficient star-triangle algorithm. The percolation exponents, linked to the
critical behaviour at corners, are in good agreement with the conformal
results. The conductivity exponent, t', is found to be independent of the shape
of the system. Its value is very close to recent estimates for the surface and
bulk conductivity exponents.Comment: 10 pages, 7 figures, TeX, IOP macros include
Random Walks on a Fluctuating Lattice: A Renormalization Group Approach Applied in One Dimension
We study the problem of a random walk on a lattice in which bonds connecting
nearest neighbor sites open and close randomly in time, a situation often
encountered in fluctuating media. We present a simple renormalization group
technique to solve for the effective diffusive behavior at long times. For
one-dimensional lattices we obtain better quantitative agreement with
simulation data than earlier effective medium results. Our technique works in
principle in any dimension, although the amount of computation required rises
with dimensionality of the lattice.Comment: PostScript file including 2 figures, total 15 pages, 8 other figures
obtainable by mail from D.L. Stei
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