115 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
A Fast and Accurate Nonlinear Spectral Method for Image Recognition and Registration
This article addresses the problem of two- and higher dimensional pattern
matching, i.e. the identification of instances of a template within a larger
signal space, which is a form of registration. Unlike traditional correlation,
we aim at obtaining more selective matchings by considering more strict
comparisons of gray-level intensity. In order to achieve fast matching, a
nonlinear thresholded version of the fast Fourier transform is applied to a
gray-level decomposition of the original 2D image. The potential of the method
is substantiated with respect to real data involving the selective
identification of neuronal cell bodies in gray-level images.Comment: 4 pages, 3 figure
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
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''
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
Bailout Embeddings, Targeting of KAM Orbits, and the Control of Hamiltonian Chaos
We present a novel technique, which we term bailout embedding, that can be
used to target orbits having particular properties out of all orbits in a flow
or map. We explicitly construct a bailout embedding for Hamiltonian systems so
as to target KAM orbits. We show how the bailout dynamics is able to lock onto
extremely small KAM islands in an ergodic sea.Comment: 3 figures, 9 subpanel
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
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
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