142 research outputs found
Efficient algorithm to study interconnected networks
Interconnected networks have been shown to be much more vulnerable to random
and targeted failures than isolated ones, raising several interesting questions
regarding the identification and mitigation of their risk. The paradigm to
address these questions is the percolation model, where the resilience of the
system is quantified by the dependence of the size of the largest cluster on
the number of failures. Numerically, the major challenge is the identification
of this cluster and the calculation of its size. Here, we propose an efficient
algorithm to tackle this problem. We show that the algorithm scales as O(N log
N), where N is the number of nodes in the network, a significant improvement
compared to O(N^2) for a greedy algorithm, what permits studying much larger
networks. Our new strategy can be applied to any network topology and
distribution of interdependencies, as well as any sequence of failures.Comment: 5 pages, 6 figure
Controlling percolation with limited resources
Connectivity - or the lack thereof - is crucial for the function of many
man-made systems, from financial and economic networks over epidemic spreading
in social networks to technical infrastructure. Often, connections are
deliberately established or removed to induce, maintain, or destroy global
connectivity. Thus, there has been a great interest in understanding how to
control percolation, the transition to large-scale connectivity. Previous work,
however, studied control strategies assuming unlimited resources. Here, we
depart from this unrealistic assumption and consider the effect of limited
resources on the effectiveness of control. We show that, even for scarce
resources, percolation can be controlled with an efficient intervention
strategy. We derive this strategy and study its implications, revealing a
discontinuous transition as an unintended side-effect of optimal control.Comment: 5 pages, 4 figures, additional supplemental material (19 pages
Shock waves on complex networks
Power grids, road maps, and river streams are examples of infrastructural
networks which are highly vulnerable to external perturbations. An abrupt local
change of load (voltage, traffic density, or water level) might propagate in a
cascading way and affect a significant fraction of the network. Almost
discontinuous perturbations can be modeled by shock waves which can eventually
interfere constructively and endanger the normal functionality of the
infrastructure. We study their dynamics by solving the Burgers equation under
random perturbations on several real and artificial directed graphs. Even for
graphs with a narrow distribution of node properties (e.g., degree or
betweenness), a steady state is reached exhibiting a heterogeneous load
distribution, having a difference of one order of magnitude between the highest
and average loads. Unexpectedly we find for the European power grid and for
finite Watts-Strogatz networks a broad pronounced bimodal distribution for the
loads. To identify the most vulnerable nodes, we introduce the concept of
node-basin size, a purely topological property which we show to be strongly
correlated to the average load of a node
Inheritances, social classes, and wealth distribution
We consider a simple theoretical model to investigate the impact of
inheritances on the wealth distribution. Wealth is described as a finite
resource, which remains constant over different generations and is divided
equally among offspring. All other sources of wealth are neglected. We consider
different societies characterized by a different offspring probability
distribution. We find that, if the population remains constant, the society
reaches a stationary wealth distribution. We show that inequality emerges every
time the number of children per family is not always the same. For realistic
offspring distributions from developed countries, the model predicts a Gini
coefficient of . If we divide the society into wealth classes and
set the probability of getting married to depend on the distance between
classes, the stationary wealth distribution crosses over from an exponential to
a power-law regime as the number of wealth classes and the level of class
distinction increase.Comment: 7 pages, 7 figure
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