588 research outputs found
Optimization of Robustness of Complex Networks
Networks with a given degree distribution may be very resilient to one type
of failure or attack but not to another. The goal of this work is to determine
network design guidelines which maximize the robustness of networks to both
random failure and intentional attack while keeping the cost of the network
(which we take to be the average number of links per node) constant. We find
optimal parameters for: (i) scale free networks having degree distributions
with a single power-law regime, (ii) networks having degree distributions with
two power-law regimes, and (iii) networks described by degree distributions
containing two peaks. Of these various kinds of distributions we find that the
optimal network design is one in which all but one of the nodes have the same
degree, (close to the average number of links per node), and one node is
of very large degree, , where is the number of nodes in
the network.Comment: Accepted for publication in European Physical Journal
Statistics of leaders and lead changes in growing networks
We investigate various aspects of the statistics of leaders in growing
network models defined by stochastic attachment rules. The leader is the node
with highest degree at a given time (or the node which reached that degree
first if there are co-leaders). This comprehensive study includes the full
distribution of the degree of the leader, its identity, the number of
co-leaders, as well as several observables characterizing the whole history of
lead changes: number of lead changes, number of distinct leaders, lead
persistence probability. We successively consider the following network models:
uniform attachment, linear attachment (the Barabasi-Albert model), and
generalized preferential attachment with initial attractiveness.Comment: 28 pages, 14 figures, 1 tabl
Ising spin glass models versus Ising models: an effective mapping at high temperature II. Applications to graphs and networks
By applying a recently proposed mapping, we derive exactly the upper phase
boundary of several Ising spin glass models defined over static graphs and
random graphs, generalizing some known results and providing new ones.Comment: 11 pages, 1 Postscript figur
Why Do Cascade Sizes Follow a Power-Law?
We introduce random directed acyclic graph and use it to model the
information diffusion network. Subsequently, we analyze the cascade generation
model (CGM) introduced by Leskovec et al. [19]. Until now only empirical
studies of this model were done. In this paper, we present the first
theoretical proof that the sizes of cascades generated by the CGM follow the
power-law distribution, which is consistent with multiple empirical analysis of
the large social networks. We compared the assumptions of our model with the
Twitter social network and tested the goodness of approximation.Comment: 8 pages, 7 figures, accepted to WWW 201
Robustness of interdependent networks under targeted attack
When an initial failure of nodes occurs in interdependent networks, a cascade
of failure between the networks occurs. Earlier studies focused on random
initial failures. Here we study the robustness of interdependent networks under
targeted attack on high or low degree nodes. We introduce a general technique
and show that the {\it targeted-attack} problem in interdependent networks can
be mapped to the {\it random-attack} problem in a transformed pair of
interdependent networks. We find that when the highly connected nodes are
protected and have lower probability to fail, in contrast to single scale free
(SF) networks where the percolation threshold , coupled SF networks are
significantly more vulnerable with significantly larger than zero. The
result implies that interdependent networks are difficult to defend by
strategies such as protecting the high degree nodes that have been found useful
to significantly improve robustness of single networks.Comment: 11 pages, 2 figure
Dynamical signatures of the vulcanization transition
Dynamical properties of vulcanized polymer networks are addressed via a
Rouse-type model that incorporates the effect of permanent random crosslinks.
The incoherent intermediate scattering function is computed in the sol and gel
phases, and at the vulcanization transition between them. At any nonzero
crosslink density within the sol phase Kohlrausch relaxation is found. The
critical point is signalled by divergence of the longest time-scale, and at
this point the scattering function decays algebraically, whereas within the gel
phase it acquires a time-persistent part identified with the gel fraction.Comment: 4 page
Analytical Solution of a Stochastic Content Based Network Model
We define and completely solve a content-based directed network whose nodes
consist of random words and an adjacency rule involving perfect or approximate
matches, for an alphabet with an arbitrary number of letters. The analytic
expression for the out-degree distribution shows a crossover from a leading
power law behavior to a log-periodic regime bounded by a different power law
decay. The leading exponents in the two regions have a weak dependence on the
mean word length, and an even weaker dependence on the alphabet size. The
in-degree distribution, on the other hand, is much narrower and does not show
scaling behavior. The results might be of interest for understanding the
emergence of genomic interaction networks, which rely, to a large extent, on
mechanisms based on sequence matching, and exhibit similar global features to
those found here.Comment: 13 pages, 5 figures. Rewrote conclusions regarding the relevance to
gene regulation networks, fixed minor errors and replaced fig. 4. Main body
of paper (model and calculations) remains unchanged. Submitted for
publicatio
CONTEST : a Controllable Test Matrix Toolbox for MATLAB
Large, sparse networks that describe complex interactions are a common feature across a number of disciplines, giving rise to many challenging matrix computational tasks. Several random graph models have been proposed that capture key properties of real-life networks. These models provide realistic, parametrized matrices for testing linear system and eigenvalue solvers. CONTEST (CONtrollable TEST matrices) is a random network toolbox for MATLAB that implements nine models. The models produce unweighted directed or undirected graphs; that is, symmetric or unsymmetric matrices with elements equal to zero or one. They have one or more parameters that affect features such as sparsity and characteristic pathlength and all can be of arbitrary dimension. Utility functions are supplied for rewiring, adding extra shortcuts and subsampling in order to create further classes of networks. Other utilities convert the adjacency matrices into real-valued coefficient matrices for naturally arising computational tasks that reduce to sparse linear system and eigenvalue problems
A nullstellensatz for sequences over F_p
Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in
F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1
x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually
characterizes A up to a nonzero multiplicative constant, which is no longer
true for l < p. The critical case l=p is of particular interest. In this
context, we prove that whenever l=p and A is nonconstant, the above equation
has at least p-1 minimal 0-1 solutions, thus refining a theorem of Olson. The
subcritical case l=p-1 is studied in detail also. Our approach is algebraic in
nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper
type theorem.Comment: 23 page
Percolation of partially interdependent networks under targeted attack
The study of interdependent networks, and in particular the robustness on
networks, has attracted considerable attention. Recent studies mainly assume
that the dependence is fully interdependent. However, targeted attack for
partially interdependent networks simultaneously has the characteristics of
generality in real world. In this letter, the comprehensive percolation of
generalized framework of partially interdependent networks under targeted
attack is analyzed. As and , the percolation law is
presented. Especially, when , , , the
first and second lines of phase transition coincide with each other. The
corresponding phase transition diagram and the critical line between the first
and the second phase transition are found. We find that the tendency of
critical line is monotone decreasing with parameter . However, for
different , the tendency of critical line is monotone increasing with
. In a larger sense, our findings have potential application for
designing networks with strong robustness and can regulate the robustness of
some current networks.Comment: 6 pages, 9 figure
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