102 research outputs found
Relatively Coherent Sets as a Hierarchical Partition Method
Finite time coherent sets [8] have recently been defined by a measure based
objective function describing the degree that sets hold together, along with a
Frobenius-Perron transfer operator method to produce optimally coherent sets.
Here we present an extension to generalize the concept to hierarchially defined
relatively coherent sets based on adjusting the finite time coherent sets to
use relative mesure restricted to sets which are developed iteratively and
hierarchically in a tree of partitions. Several examples help clarify the
meaning and expectation of the techniques, as they are the nonautonomous double
gyre, the standard map, an idealized stratospheric flow, and empirical data
from the Mexico Gulf during the 2010 oil spill. Also for sake of analysis of
computational complexity, we include an appendic concerning the computational
complexity of developing the Ulam-Galerkin matrix extimates of the
Frobenius-Perron operator centrally used here
Estimating good discrete partitions from observed data: symbolic false nearest neighbors
A symbolic analysis of observed time series data requires making a discrete
partition of a continuous state space containing observations of the dynamics.
A particular kind of partition, called ``generating'', preserves all dynamical
information of a deterministic map in the symbolic representation, but such
partitions are not obvious beyond one dimension, and existing methods to find
them require significant knowledge of the dynamical evolution operator or the
spectrum of unstable periodic orbits. We introduce a statistic and algorithm to
refine empirical partitions for symbolic state reconstruction. This method
optimizes an essential property of a generating partition: avoiding topological
degeneracies. It requires only the observed time series and is sensible even in
the presence of noise when no truly generating partition is possible. Because
of its resemblance to a geometrical statistic frequently used for
reconstructing valid time-delay embeddings, we call the algorithm ``symbolic
false nearest neighbors''
Transport in networks with multiple sources and sinks
We investigate the electrical current and flow (number of parallel paths)
between two sets of n sources and n sinks in complex networks. We derive
analytical formulas for the average current and flow as a function of n. We
show that for small n, increasing n improves the total transport in the
network, while for large n bottlenecks begin to form. For the case of flow,
this leads to an optimal n* above which the transport is less efficient. For
current, the typical decrease in the length of the connecting paths for large n
compensates for the effect of the bottlenecks. We also derive an expression for
the average flow as a function of n under the common limitation that transport
takes place between specific pairs of sources and sinks
Intermittent exploration on a scale-free network
We study an intermittent random walk on a random network of scale-free degree
distribution. The walk is a combination of simple random walks of duration
and random long-range jumps. While the time the walker needs to cover all
the nodes increases with , the corresponding time for the edges displays a
non monotonic behavior with a minimum for some nontrivial value of . This
is a heterogeneity-induced effect that is not observed in homogeneous
small-world networks. The optimal increases with the degree of
assortativity in the network. Depending on the nature of degree correlations
and the elapsed time the walker finds an over/under-estimate of the degree
distribution exponent.Comment: 12 pages, 3 figures, 1 table, published versio
Portraits of Complex Networks
We propose a method for characterizing large complex networks by introducing
a new matrix structure, unique for a given network, which encodes structural
information; provides useful visualization, even for very large networks; and
allows for rigorous statistical comparison between networks. Dynamic processes
such as percolation can be visualized using animations. Applications to graph
theory are discussed, as are generalizations to weighted networks, real-world
network similarity testing, and applicability to the graph isomorphism problem.Comment: 6 pages, 9 figure
Learning about knowledge: A complex network approach
This article describes an approach to modeling knowledge acquisition in terms
of walks along complex networks. Each subset of knowledge is represented as a
node, and relations between such knowledge are expressed as edges. Two types of
edges are considered, corresponding to free and conditional transitions. The
latter case implies that a node can only be reached after visiting previously a
set of nodes (the required conditions). The process of knowledge acquisition
can then be simulated by considering the number of nodes visited as a single
agent moves along the network, starting from its lowest layer. It is shown that
hierarchical networks, i.e. networks composed of successive interconnected
layers, arise naturally as a consequence of compositions of the prerequisite
relationships between the nodes. In order to avoid deadlocks, i.e. unreachable
nodes, the subnetwork in each layer is assumed to be a connected component.
Several configurations of such hierarchical knowledge networks are simulated
and the performance of the moving agent quantified in terms of the percentage
of visited nodes after each movement. The Barab\'asi-Albert and random models
are considered for the layer and interconnecting subnetworks. Although all
subnetworks in each realization have the same number of nodes, several
interconnectivities, defined by the average node degree of the interconnection
networks, have been considered. Two visiting strategies are investigated:
random choice among the existing edges and preferential choice to so far
untracked edges. A series of interesting results are obtained, including the
identification of a series of plateaux of knowledge stagnation in the case of
the preferential movements strategy in presence of conditional edges.Comment: 18 pages, 19 figure
Anomalous behavior of trapping on a fractal scale-free network
It is known that the heterogeneity of scale-free networks helps enhancing the
efficiency of trapping processes performed on them. In this paper, we show that
transport efficiency is much lower in a fractal scale-free network than in
non-fractal networks. To this end, we examine a simple random walk with a fixed
trap at a given position on a fractal scale-free network. We calculate
analytically the mean first-passage time (MFPT) as a measure of the efficiency
for the trapping process, and obtain a closed-form expression for MFPT, which
agrees with direct numerical calculations. We find that, in the limit of a
large network order , the MFPT behaves superlinearly as with an exponent 3/2 much larger than 1, which is in sharp contrast
to the scaling with , previously obtained
for non-fractal scale-free networks. Our results indicate that the degree
distribution of scale-free networks is not sufficient to characterize trapping
processes taking place on them. Since various real-world networks are
simultaneously scale-free and fractal, our results may shed light on the
understanding of trapping processes running on real-life systems.Comment: 6 pages, 5 figures; Definitive version accepted for publication in
EPL (Europhysics Letters
Fractal and Transfractal Recursive Scale-Free Nets
We explore the concepts of self-similarity, dimensionality, and
(multi)scaling in a new family of recursive scale-free nets that yield
themselves to exact analysis through renormalization techniques. All nets in
this family are self-similar and some are fractals - possessing a finite
fractal dimension - while others are small world (their diameter grows
logarithmically with their size) and are infinite-dimensional. We show how a
useful measure of "transfinite" dimension may be defined and applied to the
small world nets. Concerning multiscaling, we show how first-passage time for
diffusion and resistance between hub (the most connected nodes) scale
differently than for other nodes. Despite the different scalings, the Einstein
relation between diffusion and conductivity holds separately for hubs and
nodes. The transfinite exponents of small world nets obey Einstein relations
analogous to those in fractal nets.Comment: Includes small revisions and references added as result of readers'
feedbac
- …