1,724 research outputs found
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
Giant strongly connected component of directed networks
We describe how to calculate the sizes of all giant connected components of a
directed graph, including the {\em strongly} connected one. Just to the class
of directed networks, in particular, belongs the World Wide Web. The results
are obtained for graphs with statistically uncorrelated vertices and an
arbitrary joint in,out-degree distribution . We show that if
does not factorize, the relative size of the giant strongly
connected component deviates from the product of the relative sizes of the
giant in- and out-components. The calculations of the relative sizes of all the
giant components are demonstrated using the simplest examples. We explain that
the giant strongly connected component may be less resilient to random damage
than the giant weakly connected one.Comment: 4 pages revtex, 4 figure
Discrete surface growth process as a synchronization mechanism for scale free complex networks
We consider the discrete surface growth process with relaxation to the
minimum [F. Family, J. Phys. A {\bf 19} L441, (1986).] as a possible
synchronization mechanism on scale-free networks, characterized by a degree
distribution , where is the degree of a node and
his broadness, and compare it with the usually applied
Edward-Wilkinson process [S. F. Edwards and D. R. Wilkinson, Proc. R. Soc.
London Ser. A {\bf 381},17 (1982) ]. In spite of both processes belong to the
same universality class for Euclidean lattices, in this work we demonstrate
that for scale-free networks with exponents this is not true.
Moreover, we show that for these ubiquitous cases the Edward-Wilkinson process
enhances spontaneously the synchronization when the system size is increased,
which is a non-physical result. Contrarily, the discrete surface growth process
do not present this flaw and is applicable for every .Comment: 8 pages, 4 figure
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
Network synchronization: Optimal and Pessimal Scale-Free Topologies
By employing a recently introduced optimization algorithm we explicitely
design optimally synchronizable (unweighted) networks for any given scale-free
degree distribution. We explore how the optimization process affects
degree-degree correlations and observe a generic tendency towards
disassortativity. Still, we show that there is not a one-to-one correspondence
between synchronizability and disassortativity. On the other hand, we study the
nature of optimally un-synchronizable networks, that is, networks whose
topology minimizes the range of stability of the synchronous state. The
resulting ``pessimal networks'' turn out to have a highly assortative
string-like structure. We also derive a rigorous lower bound for the Laplacian
eigenvalue ratio controlling synchronizability, which helps understanding the
impact of degree correlations on network synchronizability.Comment: 11 pages, 4 figs, submitted to J. Phys. A (proceedings of Complex
Networks 2007
Solution of the 2-star model of a network
The p-star model or exponential random graph is among the oldest and
best-known of network models. Here we give an analytic solution for the
particular case of the 2-star model, which is one of the most fundamental of
exponential random graphs. We derive expressions for a number of quantities of
interest in the model and show that the degenerate region of the parameter
space observed in computer simulations is a spontaneously symmetry broken phase
separated from the normal phase of the model by a conventional continuous phase
transition.Comment: 5 pages, 3 figure
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
Graph Partitioning Induced Phase Transitions
We study the percolation properties of graph partitioning on random regular
graphs with N vertices of degree . Optimal graph partitioning is directly
related to optimal attack and immunization of complex networks. We find that
for any partitioning process (even if non-optimal) that partitions the graph
into equal sized connected components (clusters), the system undergoes a
percolation phase transition at where is the fraction of
edges removed to partition the graph. For optimal partitioning, at the
percolation threshold, we find where is the size of the
clusters and where is their diameter. Additionally,
we find that undergoes multiple non-percolation transitions for
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