86,906 research outputs found
Exact calculation of network robustness
Finding the most critical nodes regarding network connectivity has attracted the attention of many researchers in infrastructure networks, power grids, transportation networks and physics in complex networks. Static robustness of networks under intentional attacks analyses the ability of a system to maintain its connectivity after the disconnection or deletion of a series of targeted nodes. In this context, connectivity is typically measured by the size of the remaining largest connected component. When targeting these nodes, previous literature has mostly used adaptive strategies that sequentially remove central nodes, or created heuristics in order to improve the results of the adaptive strategies. The proposed methodology based on mathematical programming allows to identify, for every fraction of disconnected or removed nodes, the set that minimizes the size of the largest connected component of a network, i.e. it allows to calculate the exact (most critical) robustness of a network.Peer ReviewedPostprint (author's final draft
Network robustness and fragility: Percolation on random graphs
Recent work on the internet, social networks, and the power grid has
addressed the resilience of these networks to either random or targeted
deletion of network nodes. Such deletions include, for example, the failure of
internet routers or power transmission lines. Percolation models on random
graphs provide a simple representation of this process, but have typically been
limited to graphs with Poisson degree distribution at their vertices. Such
graphs are quite unlike real world networks, which often possess power-law or
other highly skewed degree distributions. In this paper we study percolation on
graphs with completely general degree distribution, giving exact solutions for
a variety of cases, including site percolation, bond percolation, and models in
which occupation probabilities depend on vertex degree. We discuss the
application of our theory to the understanding of network resilience.Comment: 4 pages, 2 figure
The Influence of Canalization on the Robustness of Boolean Networks
Time- and state-discrete dynamical systems are frequently used to model
molecular networks. This paper provides a collection of mathematical and
computational tools for the study of robustness in Boolean network models. The
focus is on networks governed by -canalizing functions, a recently
introduced class of Boolean functions that contains the well-studied class of
nested canalizing functions. The activities and sensitivity of a function
quantify the impact of input changes on the function output. This paper
generalizes the latter concept to -sensitivity and provides formulas for the
activities and -sensitivity of general -canalizing functions as well as
canalizing functions with more precisely defined structure. A popular measure
for the robustness of a network, the Derrida value, can be expressed as a
weighted sum of the -sensitivities of the governing canalizing functions,
and can also be calculated for a stochastic extension of Boolean networks.
These findings provide a computationally efficient way to obtain Derrida values
of Boolean networks, deterministic or stochastic, that does not involve
simulation.Comment: 16 pages, 2 figures, 3 table
Self-similarity, small-world, scale-free scaling, disassortativity, and robustness in hierarchical lattices
In this paper, firstly, we study analytically the topological features of a
family of hierarchical lattices (HLs) from the view point of complex networks.
We derive some basic properties of HLs controlled by a parameter . Our
results show that scale-free networks are not always small-world, and support
the conjecture that self-similar scale-free networks are not assortative.
Secondly, we define a deterministic family of graphs called small-world
hierarchical lattices (SWHLs). Our construction preserves the structure of
hierarchical lattices, while the small-world phenomenon arises. Finally, the
dynamical processes of intentional attacks and collective synchronization are
studied and the comparisons between HLs and Barab{\'asi}-Albert (BA) networks
as well as SWHLs are shown. We show that degree distribution of scale-free
networks does not suffice to characterize their synchronizability, and that
networks with smaller average path length are not always easier to synchronize.Comment: 26 pages, 8 figure
Genetic networks with canalyzing Boolean rules are always stable
We determine stability and attractor properties of random Boolean genetic
network models with canalyzing rules for a variety of architectures. For all
power law, exponential, and flat in-degree distributions, we find that the
networks are dynamically stable. Furthermore, for architectures with few inputs
per node, the dynamics of the networks is close to critical. In addition, the
fraction of genes that are active decreases with the number of inputs per node.
These results are based upon investigating ensembles of networks using
analytical methods. Also, for different in-degree distributions, the numbers of
fixed points and cycles are calculated, with results intuitively consistent
with stability analysis; fewer inputs per node implies more cycles, and vice
versa. There are hints that genetic networks acquire broader degree
distributions with evolution, and hence our results indicate that for single
cells, the dynamics should become more stable with evolution. However, such an
effect is very likely compensated for by multicellular dynamics, because one
expects less stability when interactions among cells are included. We verify
this by simulations of a simple model for interactions among cells.Comment: Final version available through PNAS open access at
http://www.pnas.org/cgi/content/abstract/0407783101v
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