74,163 research outputs found
Principles of statistical mechanics of random networks
We develop a statistical mechanics approach for random networks with
uncorrelated vertices. We construct equilibrium statistical ensembles of such
networks and obtain their partition functions and main characteristics. We find
simple dynamical construction procedures that produce equilibrium uncorrelated
random graphs with an arbitrary degree distribution. In particular, we show
that in equilibrium uncorrelated networks, fat-tailed degree distributions may
exist only starting from some critical average number of connections of a
vertex, in a phase with a condensate of edges.Comment: 14 pages, an extended versio
Statistical mechanics of complex networks
Complex networks describe a wide range of systems in nature and society, much
quoted examples including the cell, a network of chemicals linked by chemical
reactions, or the Internet, a network of routers and computers connected by
physical links. While traditionally these systems were modeled as random
graphs, it is increasingly recognized that the topology and evolution of real
networks is governed by robust organizing principles. Here we review the recent
advances in the field of complex networks, focusing on the statistical
mechanics of network topology and dynamics. After reviewing the empirical data
that motivated the recent interest in networks, we discuss the main models and
analytical tools, covering random graphs, small-world and scale-free networks,
as well as the interplay between topology and the network's robustness against
failures and attacks.Comment: 54 pages, submitted to Reviews of Modern Physic
A Network Theoretical Approach to Real-World Problems: Application of the K-Core Algorithm to Various Systems
The study of complex networks is, at its core, an exploration of the mechanisms that control the world in which we live at every scale, from particles no bigger than a grain of sand and amino acids that comprise proteins, to social networks, ecosystems, and even countries. Indeed, we find that, regardless of the physical size of the network\u27s components, we may apply principles of complex network theory, thermodynamics, and statistical mechanics to not only better understand these specific networks, but to formulate theories which may be applied to problems on a more general level. This thesis explores several networks at vastly different scales, ranging from the microscopic (amino acids and frictional packed particles) to the macroscopic (human subjects asked to view a set of videos) to the massive (real ecosystems and the financial ecosystem (Haldane 2011, May 2008) of stocks in the S&P500 stock index). The networks are discussed in chronological order of analysis. We begin with a review of k-core theory, including its applications to certain dynamical systems, as this is an important concept to understand for the next two sections. A discussion of the network structure (specifically, a k-shell decomposition) of both ecological and financial dynamic networks, and the implications of this structure for determining a network\u27s tipping point of collapse, follows. Third, this same k-shell structure is examined for networks of frictional particles approaching a jamming transition, where it is seen that the jamming transition is a k-core transition given by random network theory. Lastly comes a thermodynamical examination of human eye-tracking networks built from data of subjects asked to watch the commercials of the 2014 Super Bowl Game; we determine, using a Maximum Entropy approach, that the collective behavior of this small sample can be used to predict population-wide preferences. The behavior of all of these networks are explained using aspects of network theoretical and statistical mechanics frameworks and can be extended beyond the specific networks analyzed herein
Organic Design of Massively Distributed Systems: A Complex Networks Perspective
The vision of Organic Computing addresses challenges that arise in the design
of future information systems that are comprised of numerous, heterogeneous,
resource-constrained and error-prone components or devices. Here, the notion
organic particularly highlights the idea that, in order to be manageable, such
systems should exhibit self-organization, self-adaptation and self-healing
characteristics similar to those of biological systems. In recent years, the
principles underlying many of the interesting characteristics of natural
systems have been investigated from the perspective of complex systems science,
particularly using the conceptual framework of statistical physics and
statistical mechanics. In this article, we review some of the interesting
relations between statistical physics and networked systems and discuss
applications in the engineering of organic networked computing systems with
predictable, quantifiable and controllable self-* properties.Comment: 17 pages, 14 figures, preprint of submission to Informatik-Spektrum
published by Springe
The statistical mechanics of networks
We study the family of network models derived by requiring the expected
properties of a graph ensemble to match a given set of measurements of a
real-world network, while maximizing the entropy of the ensemble. Models of
this type play the same role in the study of networks as is played by the
Boltzmann distribution in classical statistical mechanics; they offer the best
prediction of network properties subject to the constraints imposed by a given
set of observations. We give exact solutions of models within this class that
incorporate arbitrary degree distributions and arbitrary but independent edge
probabilities. We also discuss some more complex examples with correlated edges
that can be solved approximately or exactly by adapting various familiar
methods, including mean-field theory, perturbation theory, and saddle-point
expansions.Comment: 15 pages, 4 figure
Nonextensive statistical mechanics and complex scale-free networks
One explanation for the impressive recent boom in network theory might be
that it provides a promising tool for an understanding of complex systems.
Network theory is mainly focusing on discrete large-scale topological
structures rather than on microscopic details of interactions of its elements.
This viewpoint allows to naturally treat collective phenomena which are often
an integral part of complex systems, such as biological or socio-economical
phenomena. Much of the attraction of network theory arises from the discovery
that many networks, natural or man-made, seem to exhibit some sort of
universality, meaning that most of them belong to one of three classes: {\it
random}, {\it scale-free} and {\it small-world} networks. Maybe most important
however for the physics community is, that due to its conceptually intuitive
nature, network theory seems to be within reach of a full and coherent
understanding from first principles ..
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