200,321 research outputs found
Bipartite quantum states and random complex networks
We introduce a mapping between graphs and pure quantum bipartite states and
show that the associated entanglement entropy conveys non-trivial information
about the structure of the graph. Our primary goal is to investigate the family
of random graphs known as complex networks. In the case of classical random
graphs we derive an analytic expression for the averaged entanglement entropy
while for general complex networks we rely on numerics. For large
number of nodes we find a scaling where both
the prefactor and the sub-leading O(1) term are a characteristic of
the different classes of complex networks. In particular, encodes
topological features of the graphs and is named network topological entropy.
Our results suggest that quantum entanglement may provide a powerful tool in
the analysis of large complex networks with non-trivial topological properties.Comment: 4 pages, 3 figure
Markov Chain Methods For Analyzing Complex Transport Networks
We have developed a steady state theory of complex transport networks used to
model the flow of commodity, information, viruses, opinions, or traffic. Our
approach is based on the use of the Markov chains defined on the graph
representations of transport networks allowing for the effective network
design, network performance evaluation, embedding, partitioning, and network
fault tolerance analysis. Random walks embed graphs into Euclidean space in
which distances and angles acquire a clear statistical interpretation. Being
defined on the dual graph representations of transport networks random walks
describe the equilibrium configurations of not random commodity flows on
primary graphs. This theory unifies many network concepts into one framework
and can also be elegantly extended to describe networks represented by directed
graphs and multiple interacting networks.Comment: 26 pages, 4 figure
Clustering and the hyperbolic geometry of complex networks
Clustering is a fundamental property of complex networks and it is the
mathematical expression of a ubiquitous phenomenon that arises in various types
of self-organized networks such as biological networks, computer networks or
social networks. In this paper, we consider what is called the global
clustering coefficient of random graphs on the hyperbolic plane. This model of
random graphs was proposed recently by Krioukov et al. as a mathematical model
of complex networks, under the fundamental assumption that hyperbolic geometry
underlies the structure of these networks. We give a rigorous analysis of
clustering and characterize the global clustering coefficient in terms of the
parameters of the model. We show how the global clustering coefficient can be
tuned by these parameters and we give an explicit formula for this function.Comment: 51 pages, 1 figur
Modularity from Fluctuations in Random Graphs and Complex Networks
The mechanisms by which modularity emerges in complex networks are not well
understood but recent reports have suggested that modularity may arise from
evolutionary selection. We show that finding the modularity of a network is
analogous to finding the ground-state energy of a spin system. Moreover, we
demonstrate that, due to fluctuations, stochastic network models give rise to
modular networks. Specifically, we show both numerically and analytically that
random graphs and scale-free networks have modularity. We argue that this fact
must be taken into consideration to define statistically-significant modularity
in complex networks.Comment: 4 page
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.
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
Random Networks Tossing Biased Coins
In statistical mechanical investigations on complex networks, it is useful to
employ random graphs ensembles as null models, to compare with experimental
realizations. Motivated by transcription networks, we present here a simple way
to generate an ensemble of random directed graphs with, asymptotically,
scale-free outdegree and compact indegree. Entries in each row of the adjacency
matrix are set to be zero or one according to the toss of a biased coin, with a
chosen probability distribution for the biases. This defines a quick and simple
algorithm, which yields good results already for graphs of size n ~ 100.
Perhaps more importantly, many of the relevant observables are accessible
analytically, improving upon previous estimates for similar graphs
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