47,406 research outputs found
Component sizes in networks with arbitrary degree distributions
We give an exact solution for the complete distribution of component sizes in
random networks with arbitrary degree distributions. The solution tells us the
probability that a randomly chosen node belongs to a component of size s, for
any s. We apply our results to networks with the three most commonly studied
degree distributions -- Poisson, exponential, and power-law -- as well as to
the calculation of cluster sizes for bond percolation on networks, which
correspond to the sizes of outbreaks of SIR epidemic processes on the same
networks. For the particular case of the power-law degree distribution, we show
that the component size distribution itself follows a power law everywhere
below the phase transition at which a giant component forms, but takes an
exponential form when a giant component is present.Comment: 5 pages, 1 figur
Optimization in Gradient Networks
Gradient networks can be used to model the dominant structure of complex
networks. Previous works have focused on random gradient networks. Here we
study gradient networks that minimize jamming on substrate networks with
scale-free and Erd\H{o}s-R\'enyi structure. We introduce structural
correlations and strongly reduce congestion occurring on the network by using a
Monte Carlo optimization scheme. This optimization alters the degree
distribution and other structural properties of the resulting gradient
networks. These results are expected to be relevant for transport and other
dynamical processes in real network systems.Comment: 5 pages, 4 figure
Community detection and graph partitioning
Many methods have been proposed for community detection in networks. Some of
the most promising are methods based on statistical inference, which rest on
solid mathematical foundations and return excellent results in practice. In
this paper we show that two of the most widely used inference methods can be
mapped directly onto versions of the standard minimum-cut graph partitioning
problem, which allows us to apply any of the many well-understood partitioning
algorithms to the solution of community detection problems. We illustrate the
approach by adapting the Laplacian spectral partitioning method to perform
community inference, testing the resulting algorithm on a range of examples,
including computer-generated and real-world networks. Both the quality of the
results and the running time rival the best previous methods.Comment: 5 pages, 2 figure
Complex Systems: A Survey
A complex system is a system composed of many interacting parts, often called
agents, which displays collective behavior that does not follow trivially from
the behaviors of the individual parts. Examples include condensed matter
systems, ecosystems, stock markets and economies, biological evolution, and
indeed the whole of human society. Substantial progress has been made in the
quantitative understanding of complex systems, particularly since the 1980s,
using a combination of basic theory, much of it derived from physics, and
computer simulation. The subject is a broad one, drawing on techniques and
ideas from a wide range of areas. Here I give a survey of the main themes and
methods of complex systems science and an annotated bibliography of resources,
ranging from classic papers to recent books and reviews.Comment: 10 page
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