2,752 research outputs found
Complementary algorithms for graphs and percolation
A pair of complementary algorithms are presented. One of the pair is a fast
method for connecting graphs with an edge. The other is a fast method for
removing edges from a graph. Both algorithms employ the same tree based graph
representation and so, in concert, can arbitrarily modify any graph. Since the
clusters of a percolation model may be described as simple connected graphs, an
efficient Monte Carlo scheme can be constructed that uses the algorithms to
sweep the occupation probability back and forth between two turning points.
This approach concentrates computational sampling time within a region of
interest. A high precision value of pc = 0.59274603(9) was thus obtained, by
Mersenne twister, for the two dimensional square site percolation threshold.Comment: 5 pages, 3 figures, poster version presented at statphys23 (2007
Continuum Percolation in the Relative Neighborhood Graph
In the present study, we establish the existence of nontrivial site
percolation threshold in the Relative Neighborhood Graph (RNG) for Poisson
stationary point process with unit intensity in the plane
Statistical mechanics of the vertex-cover problem
We review recent progress in the study of the vertex-cover problem (VC). VC
belongs to the class of NP-complete graph theoretical problems, which plays a
central role in theoretical computer science. On ensembles of random graphs, VC
exhibits an coverable-uncoverable phase transition. Very close to this
transition, depending on the solution algorithm, easy-hard transitions in the
typical running time of the algorithms occur.
We explain a statistical mechanics approach, which works by mapping VC to a
hard-core lattice gas, and then applying techniques like the replica trick or
the cavity approach. Using these methods, the phase diagram of VC could be
obtained exactly for connectivities , where VC is replica symmetric.
Recently, this result could be confirmed using traditional mathematical
techniques. For , the solution of VC exhibits full replica symmetry
breaking.
The statistical mechanics approach can also be used to study analytically the
typical running time of simple complete and incomplete algorithms for VC.
Finally, we describe recent results for VC when studied on other ensembles of
finite- and infinite-dimensional graphs.Comment: review article, 26 pages, 9 figures, to appear in J. Phys. A: Math.
Ge
On percolation and the bunkbed conjecture
We study a problem on edge percolation on product graphs . Here
is any finite graph and consists of two vertices connected
by an edge. Every edge in is present with probability
independent of other edges. The Bunkbed conjecture states that for all and
the probability that is in the same component as is greater
than or equal to the probability that is in the same component as
for every pair of vertices .
We generalize this conjecture and formulate and prove similar statements for
randomly directed graphs. The methods lead to a proof of the original
conjecture for special classes of graphs , in particular outerplanar graphs.Comment: 13 pages, improved exposition thanks to anonymous referee. To appear
in CP
A trust model for spreading gossip in social networks
We introduce here a multi-type bootstrap percolation model, which we call
T-Bootstrap Percolation (T-BP), and apply it to study information propagation
in social networks. In this model, a social network is represented by a graph G
whose vertices have different labels corresponding to the type of role the
person plays in the network (e.g. a student, an educator, etc.). Once an
initial set of vertices of G is randomly selected to be carrying a gossip (e.g.
to be infected), the gossip propagates to a new vertex provided it is
transmitted by a minimum threshold of vertices with different labels. By
considering random graphs, which have been shown to closely represent social
networks, we study different properties of the T-BP model through numerical
simulations, and describe its implications when applied to rumour spread, fake
news, and marketing strategies.Comment: 9 pages, 9 figure
Bernoulli and self-destructive percolation on non-amenable graphs
In this note we study some properties of infinite percolation clusters on
non-amenable graphs. In particular, we study the percolative properties of the
complement of infinite percolation clusters. An approach based on
mass-transport is adapted to show that for a large class of non-amenable
graphs, the graph obtained by removing each site contained in an infinite
percolation cluster has critical percolation threshold which can be arbitrarily
close to the critical threshold for the original graph, almost surely, as p
approaches p_c. Closely related is the self-destructive percolation process,
introduced by J. van den Berg and R. Brouwer, for which we prove that an
infinite cluster emerges for any small reinforcement.Comment: 7 page
Space-Time Hierarchical-Graph Based Cooperative Localization in Wireless Sensor Networks
It has been shown that cooperative localization is capable of improving both
the positioning accuracy and coverage in scenarios where the global positioning
system (GPS) has a poor performance. However, due to its potentially excessive
computational complexity, at the time of writing the application of cooperative
localization remains limited in practice. In this paper, we address the
efficient cooperative positioning problem in wireless sensor networks. A
space-time hierarchical-graph based scheme exhibiting fast convergence is
proposed for localizing the agent nodes. In contrast to conventional methods,
agent nodes are divided into different layers with the aid of the space-time
hierarchical-model and their positions are estimated gradually. In particular,
an information propagation rule is conceived upon considering the quality of
positional information. According to the rule, the information always
propagates from the upper layers to a certain lower layer and the message
passing process is further optimized at each layer. Hence, the potential error
propagation can be mitigated. Additionally, both position estimation and
position broadcasting are carried out by the sensor nodes. Furthermore, a
sensor activation mechanism is conceived, which is capable of significantly
reducing both the energy consumption and the network traffic overhead incurred
by the localization process. The analytical and numerical results provided
demonstrate the superiority of our space-time hierarchical-graph based
cooperative localization scheme over the benchmarking schemes considered.Comment: 14 pages, 15 figures, 4 tables, accepted to appear on IEEE
Transactions on Signal Processing, Sept. 201
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