22,612 research outputs found
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
Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees
In this paper, we analyze the limiting spectral distribution of the adjacency
matrix of a random graph ensemble, proposed by Chung and Lu, in which a given
expected degree sequence
is prescribed on the ensemble. Let if there is an edge
between the nodes and zero otherwise, and consider the normalized
random adjacency matrix of the graph ensemble: . The empirical spectral distribution
of denoted by is the empirical
measure putting a mass at each of the real eigenvalues of the
symmetric matrix . Under some technical conditions on the
expected degree sequence, we show that with probability one,
converges weakly to a deterministic
distribution . Furthermore, we fully characterize this
distribution by providing explicit expressions for the moments of
. We apply our results to well-known degree distributions,
such as power-law and exponential. The asymptotic expressions of the spectral
moments in each case provide significant insights about the bulk behavior of
the eigenvalue spectrum
Spectra of "Real-World" Graphs: Beyond the Semi-Circle Law
Many natural and social systems develop complex networks, that are usually
modelled as random graphs. The eigenvalue spectrum of these graphs provides
information about their structural properties. While the semi-circle law is
known to describe the spectral density of uncorrelated random graphs, much less
is known about the eigenvalues of real-world graphs, describing such complex
systems as the Internet, metabolic pathways, networks of power stations,
scientific collaborations or movie actors, which are inherently correlated and
usually very sparse. An important limitation in addressing the spectra of these
systems is that the numerical determination of the spectra for systems with
more than a few thousand nodes is prohibitively time and memory consuming.
Making use of recent advances in algorithms for spectral characterization, here
we develop new methods to determine the eigenvalues of networks comparable in
size to real systems, obtaining several surprising results on the spectra of
adjacency matrices corresponding to models of real-world graphs. We find that
when the number of links grows as the number of nodes, the spectral density of
uncorrelated random graphs does not converge to the semi-circle law.
Furthermore, the spectral densities of real-world graphs have specific features
depending on the details of the corresponding models. In particular, scale-free
graphs develop a triangle-like spectral density with a power law tail, while
small-world graphs have a complex spectral density function consisting of
several sharp peaks. These and further results indicate that the spectra of
correlated graphs represent a practical tool for graph classification and can
provide useful insight into the relevant structural properties of real
networks.Comment: 14 pages, 9 figures (corrected typos, added references) accepted for
Phys. Rev.
Testing goodness-of-fit of random graph models
Random graphs are matrices with independent 0, 1 elements with probabilities
determined by a small number of parameters. One of the oldest model is the
Rasch model where the odds are ratios of positive numbers scaling the rows and
columns. Later Persi Diaconis with his coworkers rediscovered the model for
symmetric matrices and called the model beta. Here we give goodnes-of-fit tests
for the model and extend the model to a version of the block model introduced
by Holland, Laskey, and Leinhard
Numerical Investigation of Graph Spectra and Information Interpretability of Eigenvalues
We undertake an extensive numerical investigation of the graph spectra of
thousands regular graphs, a set of random Erd\"os-R\'enyi graphs, the two most
popular types of complex networks and an evolving genetic network by using
novel conceptual and experimental tools. Our objective in so doing is to
contribute to an understanding of the meaning of the Eigenvalues of a graph
relative to its topological and information-theoretic properties. We introduce
a technique for identifying the most informative Eigenvalues of evolving
networks by comparing graph spectra behavior to their algorithmic complexity.
We suggest that extending techniques can be used to further investigate the
behavior of evolving biological networks. In the extended version of this paper
we apply these techniques to seven tissue specific regulatory networks as
static example and network of a na\"ive pluripotent immune cell in the process
of differentiating towards a Th17 cell as evolving example, finding the most
and least informative Eigenvalues at every stage.Comment: Forthcoming in 3rd International Work-Conference on Bioinformatics
and Biomedical Engineering (IWBBIO), Lecture Notes in Bioinformatics, 201
Synchronization in random networks with given expected degree sequences
Synchronization in random networks with given expected degree sequences is studied. We also investigate in details the synchronization in networks whose topology is described by classical random graphs, power-law random graphs and hybrid graphs when N goes to infinity. In particular, we show that random graphs almost surely synchronize. We also show that adding small number of global edges to a local graph makes the corresponding hybrid graph to synchroniz
Hierarchy and assortativity as new tools for affinity investigation: the case of the TBA aptamer-ligand complex
Aptamers are single stranded DNA, RNA or peptide sequences having the ability
to bind a variety of specific targets (proteins, molecules as well as ions).
Therefore, aptamer production and selection for therapeutic and diagnostic
applications is very challenging. Usually they are in vitro generated, but,
recently, computational approaches have been developed for the in silico
selection, with a higher affinity for the specific target. Anyway, the
mechanism of aptamer-ligand formation is not completely clear, and not obvious
to predict. This paper aims to develop a computational model able to describe
aptamer-ligand affinity performance by using the topological structure of the
corresponding graphs, assessed by means of numerical tools such as the
conventional degree distribution, but also the rank-degree distribution
(hierarchy) and the node assortativity. Calculations are applied to the
thrombin binding aptamer (TBA), and the TBA-thrombin complex, produced in the
presence of Na+ or K+. The topological analysis reveals different affinity
performances between the macromolecules in the presence of the two cations, as
expected by previous investigations in literature. These results nominate the
graph topological analysis as a novel theoretical tool for testing affinity.
Otherwise, starting from the graphs, an electrical network can be obtained by
using the specific electrical properties of amino acids and nucleobases.
Therefore, a further analysis concerns with the electrical response, which
reveals that the resistance sensitively depends on the presence of sodium or
potassium thus posing resistance as a crucial physical parameter for testing
affinity.Comment: 12 pages, 5 figure
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