3,444 research outputs found
Dynamics of Epidemics
This article examines how diseases on random networks spread in time. The
disease is described by a probability distribution function for the number of
infected and recovered individuals, and the probability distribution is
described by a generating function. The time development of the disease is
obtained by iterating the generating function. In cases where the disease can
expand to an epidemic, the probability distribution function is the sum of two
parts; one which is static at long times, and another whose mean grows
exponentially. The time development of the mean number of infected individuals
is obtained analytically. When epidemics occur, the probability distributions
are very broad, and the uncertainty in the number of infected individuals at
any given time is typically larger than the mean number of infected
individuals.Comment: 4 pages and 3 figure
Generalized percolation in random directed networks
We develop a general theory for percolation in directed random networks with
arbitrary two point correlations and bidirectional edges, that is, edges
pointing in both directions simultaneously. These two ingredients alter the
previously known scenario and open new views and perspectives on percolation
phenomena. Equations for the percolation threshold and the sizes of the giant
components are derived in the most general case. We also present simulation
results for a particular example of uncorrelated network with bidirectional
edges confirming the theoretical predictions
Evolution equation for a model of surface relaxation in complex networks
In this paper we derive analytically the evolution equation of the interface
for a model of surface growth with relaxation to the minimum (SRM) in complex
networks. We were inspired by the disagreement between the scaling results of
the steady state of the fluctuations between the discrete SRM model and the
Edward-Wilkinson process found in scale-free networks with degree distribution
for [Pastore y Piontti {\it et al.},
Phys. Rev. E {\bf 76}, 046117 (2007)]. Even though for Euclidean lattices the
evolution equation is linear, we find that in complex heterogeneous networks
non-linear terms appear due to the heterogeneity and the lack of symmetry of
the network; they produce a logarithmic divergency of the saturation roughness
with the system size as found by Pastore y Piontti {\it et al.} for .Comment: 9 pages, 2 figure
Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities
Many complex networks display a mesoscopic structure with groups of nodes
sharing many links with the other nodes in their group and comparatively few
with nodes of different groups. This feature is known as community structure
and encodes precious information about the organization and the function of the
nodes. Many algorithms have been proposed but it is not yet clear how they
should be tested. Recently we have proposed a general class of undirected and
unweighted benchmark graphs, with heterogenous distributions of node degree and
community size. An increasing attention has been recently devoted to develop
algorithms able to consider the direction and the weight of the links, which
require suitable benchmark graphs for testing. In this paper we extend the
basic ideas behind our previous benchmark to generate directed and weighted
networks with built-in community structure. We also consider the possibility
that nodes belong to more communities, a feature occurring in real systems,
like, e. g., social networks. As a practical application, we show how
modularity optimization performs on our new benchmark.Comment: 9 pages, 13 figures. Final version published in Physical Review E.
The code to create the benchmark graphs can be freely downloaded from
http://santo.fortunato.googlepages.com/inthepress
Percolation in invariant Poisson graphs with i.i.d. degrees
Let each point of a homogeneous Poisson process in R^d independently be
equipped with a random number of stubs (half-edges) according to a given
probability distribution mu on the positive integers. We consider
translation-invariant schemes for perfectly matching the stubs to obtain a
simple graph with degree distribution mu. Leaving aside degenerate cases, we
prove that for any mu there exist schemes that give only finite components as
well as schemes that give infinite components. For a particular matching scheme
that is a natural extension of Gale-Shapley stable marriage, we give sufficient
conditions on mu for the absence and presence of infinite components
Synchronization in Scale Free networks: The role of finite size effects
Synchronization problems in complex networks are very often studied by
researchers due to its many applications to various fields such as
neurobiology, e-commerce and completion of tasks. In particular, Scale Free
networks with degree distribution , are widely used in
research since they are ubiquitous in nature and other real systems. In this
paper we focus on the surface relaxation growth model in Scale Free networks
with , and study the scaling behavior of the fluctuations, in
the steady state, with the system size . We find a novel behavior of the
fluctuations characterized by a crossover between two regimes at a value of
that depends on : a logarithmic regime, found in previous
research, and a constant regime. We propose a function that describes this
crossover, which is in very good agreement with the simulations. We also find
that, for a system size above , the fluctuations decrease with
, which means that the synchronization of the system improves as
increases. We explain this crossover analyzing the role of the
network's heterogeneity produced by the system size and the exponent of the
degree distribution.Comment: 9 pages and 5 figures. Accepted in Europhysics Letter
Assortative mixing in networks
A network is said to show assortative mixing if the nodes in the network that
have many connections tend to be connected to other nodes with many
connections. We define a measure of assortative mixing for networks and use it
to show that social networks are often assortatively mixed, but that
technological and biological networks tend to be disassortative. We propose a
model of an assortative network, which we study both analytically and
numerically. Within the framework of this model we find that assortative
networks tend to percolate more easily than their disassortative counterparts
and that they are also more robust to vertex removal.Comment: 5 pages, 1 table, 1 figur
Non-equilibrium mean-field theories on scale-free networks
Many non-equilibrium processes on scale-free networks present anomalous
critical behavior that is not explained by standard mean-field theories. We
propose a systematic method to derive stochastic equations for mean-field order
parameters that implicitly account for the degree heterogeneity. The method is
used to correctly predict the dynamical critical behavior of some binary spin
models and reaction-diffusion processes. The validity of our non-equilibrium
theory is furtherly supported by showing its relation with the generalized
Landau theory of equilibrium critical phenomena on networks.Comment: 4 pages, no figures, major changes in the structure of the pape
Dynamical Scaling Behavior of Percolation Clusters in Scale-free Networks
In this work we investigate the spectra of Laplacian matrices that determine
many dynamic properties of scale-free networks below and at the percolation
threshold. We use a replica formalism to develop analytically, based on an
integral equation, a systematic way to determine the ensemble averaged
eigenvalue spectrum for a general type of tree-like networks. Close to the
percolation threshold we find characteristic scaling functions for the density
of states rho(lambda) of scale-free networks. rho(lambda) shows characteristic
power laws rho(lambda) ~ lambda^alpha_1 or rho(lambda) ~ lambda^alpha_2 for
small lambda, where alpha_1 holds below and alpha_2 at the percolation
threshold. In the range where the spectra are accessible from a numerical
diagonalization procedure the two methods lead to very similar results.Comment: 9 pages, 6 figure
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