809 research outputs found
Horizontal Visibility graphs generated by type-I intermittency
The type-I intermittency route to (or out of) chaos is investigated within
the Horizontal Visibility graph theory. For that purpose, we address the
trajectories generated by unimodal maps close to an inverse tangent bifurcation
and construct, according to the Horizontal Visibility algorithm, their
associated graphs. We show how the alternation of laminar episodes and chaotic
bursts has a fingerprint in the resulting graph structure. Accordingly, we
derive a phenomenological theory that predicts quantitative values of several
network parameters. In particular, we predict that the characteristic power law
scaling of the mean length of laminar trend sizes is fully inherited in the
variance of the graph degree distribution, in good agreement with the numerics.
We also report numerical evidence on how the characteristic power-law scaling
of the Lyapunov exponent as a function of the distance to the tangent
bifurcation is inherited in the graph by an analogous scaling of the block
entropy over the degree distribution. Furthermore, we are able to recast the
full set of HV graphs generated by intermittent dynamics into a renormalization
group framework, where the fixed points of its graph-theoretical RG flow
account for the different types of dynamics. We also establish that the
nontrivial fixed point of this flow coincides with the tangency condition and
that the corresponding invariant graph exhibit extremal entropic properties.Comment: 8 figure
Exploring the randomness of Directed Acyclic Networks
The feed-forward relationship naturally observed in time-dependent processes
and in a diverse number of real systems -such as some food-webs and electronic
and neural wiring- can be described in terms of so-called directed acyclic
graphs (DAGs). An important ingredient of the analysis of such networks is a
proper comparison of their observed architecture against an ensemble of
randomized graphs, thereby quantifying the {\em randomness} of the real systems
with respect to suitable null models. This approximation is particularly
relevant when the finite size and/or large connectivity of real systems make
inadequate a comparison with the predictions obtained from the so-called {\em
configuration model}. In this paper we analyze four methods of DAG
randomization as defined by the desired combination of topological invariants
(directed and undirected degree sequence and component distributions) aimed to
be preserved. A highly ordered DAG, called \textit{snake}-graph and a
Erd\:os-R\'enyi DAG were used to validate the performance of the algorithms.
Finally, three real case studies, namely, the \textit{C. elegans} cell lineage
network, a PhD student-advisor network and the Milgram's citation network were
analyzed using each randomization method. Results show how the interpretation
of degree-degree relations in DAGs respect to their randomized ensembles depend
on the topological invariants imposed. In general, real DAGs provide disordered
values, lower than the expected by chance when the directedness of the links is
not preserved in the randomization process. Conversely, if the direction of the
links is conserved throughout the randomization process, disorder indicators
are close to the obtained from the null-model ensemble, although some
deviations are observed.Comment: 13 pages, 5 figures and 5 table
Walk entropies on graphs
Entropies based on walks on graphs and on their line-graphs are defined. They are based on the summation over diagonal and off-diagonal elements of the thermal Greenâs function of a graph also known as the communicability. The walk entropies are strongly related to the walk regularity of graphs and line-graphs. They are not biased by the graph size and have significantly better correlation with the inverse participation ratio of the eigenmodes of the adjacency matrix than other graph entropies. The temperature dependence of the walk entropies is also discussed. In particular, the walk entropy of graphs is shown to be non-monotonic for regular but non-walk-regular graphs in contrast to non-regular graphs
Nonlocal multi-trace sources and bulk entanglement in holographic conformal field theories
We consider CFT states defined by adding nonlocal multi-trace sources to the
Euclidean path integral defining the vacuum state. For holographic theories, we
argue that these states correspond to states in the gravitational theory with a
good semiclassical description but with a more general structure of bulk
entanglement than states defined from single-trace sources. We show that at
leading order in large N, the entanglement entropies for any such state are
precisely the same as those of another state defined by appropriate
single-trace effective sources; thus, if the leading order entanglement
entropies are geometrical for the single-trace states of a CFT, they are
geometrical for all the multi-trace states as well. Next, we consider the
perturbative calculation of 1/N corrections to the CFT entanglement entropies,
demonstrating that these show qualitatively different features, including
non-analyticity in the sources and/or divergences in the naive perturbative
expansion. These features are consistent with the expectation that the 1/N
corrections include contributions from bulk entanglement on the gravity side.
Finally, we investigate the dynamical constraints on the bulk geometry and the
quantum state of the bulk fields which must be satisfied so that the entropies
can be reproduced via the quantum-corrected Ryu-Takayanagi formula.Comment: 60 pages + appendices, 7 figures; v2: minor additions, published
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Approximate entropy of network parameters
We study the notion of approximate entropy within the framework of network
theory. Approximate entropy is an uncertainty measure originally proposed in
the context of dynamical systems and time series. We firstly define a purely
structural entropy obtained by computing the approximate entropy of the so
called slide sequence. This is a surrogate of the degree sequence and it is
suggested by the frequency partition of a graph. We examine this quantity for
standard scale-free and Erd\H{o}s-R\'enyi networks. By using classical results
of Pincus, we show that our entropy measure converges with network size to a
certain binary Shannon entropy. On a second step, with specific attention to
networks generated by dynamical processes, we investigate approximate entropy
of horizontal visibility graphs. Visibility graphs permit to naturally
associate to a network the notion of temporal correlations, therefore providing
the measure a dynamical garment. We show that approximate entropy distinguishes
visibility graphs generated by processes with different complexity. The result
probes to a greater extent these networks for the study of dynamical systems.
Applications to certain biological data arising in cancer genomics are finally
considered in the light of both approaches.Comment: 11 pages, 5 EPS figure
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