21,985 research outputs found
Building Damage-Resilient Dominating Sets in Complex Networks against Random and Targeted Attacks
We study the vulnerability of dominating sets against random and targeted
node removals in complex networks. While small, cost-efficient dominating sets
play a significant role in controllability and observability of these networks,
a fixed and intact network structure is always implicitly assumed. We find that
cost-efficiency of dominating sets optimized for small size alone comes at a
price of being vulnerable to damage; domination in the remaining network can be
severely disrupted, even if a small fraction of dominator nodes are lost. We
develop two new methods for finding flexible dominating sets, allowing either
adjustable overall resilience, or dominating set size, while maximizing the
dominated fraction of the remaining network after the attack. We analyze the
efficiency of each method on synthetic scale-free networks, as well as real
complex networks
The Distribution of the Domination Number of a Family of Random Interval Catch Digraphs
We study a new kind of proximity graphs called proportional-edge proximity
catch digraphs (PCDs)in a randomized setting. PCDs are a special kind of random
catch digraphs that have been developed recently and have applications in
statistical pattern classification and spatial point pattern analysis. PCDs are
also a special type of intersection digraphs; and for one-dimensional data, the
proportional-edge PCD family is also a family of random interval catch
digraphs. We present the exact (and asymptotic) distribution of the domination
number of this PCD family for uniform (and non-uniform) data in one dimension.
We also provide several extensions of this random catch digraph by relaxing the
expansion and centrality parameters, thereby determine the parameters for which
the asymptotic distribution is non-degenerate. We observe sudden jumps (from
degeneracy to non-degeneracy or from a non-degenerate distribution to another)
in the asymptotic distribution of the domination number at certain parameter
values.Comment: 29 pages, 3 figure
Localization for Linearly Edge Reinforced Random Walks
We prove that the linearly edge reinforced random walk (LRRW) on any graph
with bounded degrees is recurrent for sufficiently small initial weights. In
contrast, we show that for non-amenable graphs the LRRW is transient for
sufficiently large initial weights, thereby establishing a phase transition for
the LRRW on non-amenable graphs. While we rely on the description of the LRRW
as a mixture of Markov chains, the proof does not use the magic formula. We
also derive analogous results for the vertex reinforced jump process.Comment: 30 page
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