5,754 research outputs found
The structure and function of complex networks
Inspired by empirical studies of networked systems such as the Internet,
social networks, and biological networks, researchers have in recent years
developed a variety of techniques and models to help us understand or predict
the behavior of these systems. Here we review developments in this field,
including such concepts as the small-world effect, degree distributions,
clustering, network correlations, random graph models, models of network growth
and preferential attachment, and dynamical processes taking place on networks.Comment: Review article, 58 pages, 16 figures, 3 tables, 429 references,
published in SIAM Review (2003
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
Extreme value theory, Poisson-Dirichlet distributions and FPP on random networks
We study first passage percolation on the configuration model (CM) having
power-law degrees with exponent . To this end we equip the edges
with exponential weights. We derive the distributional limit of the minimal
weight of a path between typical vertices in the network and the number of
edges on the minimal weight path, which can be computed in terms of the
Poisson-Dirichlet distribution. We explicitly describe these limits via the
construction of an infinite limiting object describing the FPP problem in the
densely connected core of the network. We consider two separate cases, namely,
the {\it original CM}, in which each edge, regardless of its multiplicity,
receives an independent exponential weight, as well as the {\it erased CM}, for
which there is an independent exponential weight between any pair of direct
neighbors. While the results are qualitatively similar, surprisingly the
limiting random variables are quite different.
Our results imply that the flow carrying properties of the network are
markedly different from either the mean-field setting or the locally tree-like
setting, which occurs as , and for which the hopcount between typical
vertices scales as . In our setting the hopcount is tight and has an
explicit limiting distribution, showing that one can transfer information
remarkably quickly between different vertices in the network. This efficiency
has a down side in that such networks are remarkably fragile to directed
attacks. These results continue a general program by the authors to obtain a
complete picture of how random disorder changes the inherent geometry of
various random network models
Degree distribution of shortest path trees and bias of network sampling algorithms
In this article, we explicitly derive the limiting degree distribution of the
shortest path tree from a single source on various random network models with
edge weights. We determine the asymptotics of the degree distribution for large
degrees of this tree and compare it to the degree distribution of the original
graph. We perform this analysis for the complete graph with edge weights that
are powers of exponential random variables (weak disorder in the stochastic
mean-field model of distance), as well as on the configuration model with
edge-weights drawn according to any continuous distribution. In the latter, the
focus is on settings where the degrees obey a power law, and we show that the
shortest path tree again obeys a power law with the same degree power-law
exponent. We also consider random -regular graphs for large , and show
that the degree distribution of the shortest path tree is closely related to
the shortest path tree for the stochastic mean-field model of distance. We use
our results to shed light on an empirically observed bias in network sampling
methods. This is part of a general program initiated in previous works by
Bhamidi, van der Hofstad and Hooghiemstra [Ann. Appl. Probab. 20 (2010)
1907-1965], [Combin. Probab. Comput. 20 (2011) 683-707], [Adv. in Appl. Probab.
42 (2010) 706-738] of analyzing the effect of attaching random edge lengths on
the geometry of random network models.Comment: Published at http://dx.doi.org/10.1214/14-AAP1036 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Temporal Networks
A great variety of systems in nature, society and technology -- from the web
of sexual contacts to the Internet, from the nervous system to power grids --
can be modeled as graphs of vertices coupled by edges. The network structure,
describing how the graph is wired, helps us understand, predict and optimize
the behavior of dynamical systems. In many cases, however, the edges are not
continuously active. As an example, in networks of communication via email,
text messages, or phone calls, edges represent sequences of instantaneous or
practically instantaneous contacts. In some cases, edges are active for
non-negligible periods of time: e.g., the proximity patterns of inpatients at
hospitals can be represented by a graph where an edge between two individuals
is on throughout the time they are at the same ward. Like network topology, the
temporal structure of edge activations can affect dynamics of systems
interacting through the network, from disease contagion on the network of
patients to information diffusion over an e-mail network. In this review, we
present the emergent field of temporal networks, and discuss methods for
analyzing topological and temporal structure and models for elucidating their
relation to the behavior of dynamical systems. In the light of traditional
network theory, one can see this framework as moving the information of when
things happen from the dynamical system on the network, to the network itself.
Since fundamental properties, such as the transitivity of edges, do not
necessarily hold in temporal networks, many of these methods need to be quite
different from those for static networks
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