13,337 research outputs found
Absorption Time of the Moran Process
The Moran process models the spread of mutations in populations on graphs. We
investigate the absorption time of the process, which is the time taken for a
mutation introduced at a randomly chosen vertex to either spread to the whole
population, or to become extinct. It is known that the expected absorption time
for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which
allows the behaviour of the process on undirected graphs to be analysed using
the Markov chain Monte Carlo method. We show that this does not extend to
directed graphs by exhibiting an infinite family of directed graphs for which
the expected absorption time is exponential in the number of vertices. However,
for regular directed graphs, we show that the expected absorption time is
Omega(n log n) and O(n^2). We exhibit families of graphs matching these bounds
and give improved bounds for other families of graphs, based on isoperimetric
number. Our results are obtained via stochastic dominations which we
demonstrate by establishing a coupling in a related continuous-time model. The
coupling also implies several natural domination results regarding the fixation
probability of the original (discrete-time) process, resolving a conjecture of
Shakarian, Roos and Johnson.Comment: minor change
Limited packings of closed neighbourhoods in graphs
The k-limited packing number, , of a graph , introduced by
Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set
of vertices of such that every vertex of has at most elements of
in its closed neighbourhood. The main aim in this paper is to prove the
best-possible result that if is a cubic graph, then , improving the previous lower bound given by Gallant, \emph{et al.}
In addition, we construct an infinite family of graphs to show that lower
bounds given by Gagarin and Zverovich are asymptotically best-possible, up to a
constant factor, when is fixed and tends to infinity. For
tending to infinity and tending to infinity sufficiently
quickly, we give an asymptotically best-possible lower bound for ,
improving previous bounds
Zero forcing in iterated line digraphs
Zero forcing is a propagation process on a graph, or digraph, defined in
linear algebra to provide a bound for the minimum rank problem. Independently,
zero forcing was introduced in physics, computer science and network science,
areas where line digraphs are frequently used as models. Zero forcing is also
related to power domination, a propagation process that models the monitoring
of electrical power networks.
In this paper we study zero forcing in iterated line digraphs and provide a
relationship between zero forcing and power domination in line digraphs. In
particular, for regular iterated line digraphs we determine the minimum
rank/maximum nullity, zero forcing number and power domination number, and
provide constructions to attain them. We conclude that regular iterated line
digraphs present optimal minimum rank/maximum nullity, zero forcing number and
power domination number, and apply our results to determine those parameters on
some families of digraphs often used in applications
Explicit isoperimetric constants and phase transitions in the random-cluster model
The random-cluster model is a dependent percolation model that has
applications in the study of Ising and Potts models. In this paper, several new
results are obtained for the random-cluster model on nonamenable graphs with
cluster parameter . Among these, the main ones are the absence of
percolation for the free random-cluster measure at the critical value, and
examples of planar regular graphs with regular dual where \pc^\f (q) > \pu^\w
(q) for large enough. The latter follows from considerations of
isoperimetric constants, and we give the first nontrivial explicit calculations
of such constants. Such considerations are also used to prove non-robust phase
transition for the Potts model on nonamenable regular graphs
On Weak Odd Domination and Graph-based Quantum Secret Sharing
A weak odd dominated (WOD) set in a graph is a subset B of vertices for which
there exists a distinct set of vertices C such that every vertex in B has an
odd number of neighbors in C. We point out the connections of weak odd
domination with odd domination, [sigma,rho]-domination, and perfect codes. We
introduce bounds on \kappa(G), the maximum size of WOD sets of a graph G, and
on \kappa'(G), the minimum size of non WOD sets of G. Moreover, we prove that
the corresponding decision problems are NP-complete. The study of weak odd
domination is mainly motivated by the design of graph-based quantum secret
sharing protocols: a graph G of order n corresponds to a secret sharing
protocol which threshold is \kappa_Q(G) = max(\kappa(G), n-\kappa'(G)). These
graph-based protocols are very promising in terms of physical implementation,
however all such graph-based protocols studied in the literature have
quasi-unanimity thresholds (i.e. \kappa_Q(G)=n-o(n) where n is the order of the
graph G underlying the protocol). In this paper, we show using probabilistic
methods, the existence of graphs with smaller \kappa_Q (i.e. \kappa_Q(G)<
0.811n where n is the order of G). We also prove that deciding for a given
graph G whether \kappa_Q(G)< k is NP-complete, which means that one cannot
efficiently double check that a graph randomly generated has actually a
\kappa_Q smaller than 0.811n.Comment: Subsumes arXiv:1109.6181: Optimal accessing and non-accessing
structures for graph protocol
Random interlacements and amenability
We consider the model of random interlacements on transient graphs, which was
first introduced by Sznitman [Ann. of Math. (2) (2010) 171 2039-2087] for the
special case of (with ). In Sznitman [Ann. of Math.
(2) (2010) 171 2039-2087], it was shown that on : for any
intensity , the interlacement set is almost surely connected. The main
result of this paper says that for transient, transitive graphs, the above
property holds if and only if the graph is amenable. In particular, we show
that in nonamenable transitive graphs, for small values of the intensity u the
interlacement set has infinitely many infinite clusters. We also provide
examples of nonamenable transitive graphs, for which the interlacement set
becomes connected for large values of u. Finally, we establish the monotonicity
of the transition between the "disconnected" and the "connected" phases,
providing the uniqueness of the critical value where this transition
occurs.Comment: Published in at http://dx.doi.org/10.1214/12-AAP860 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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