1,956 research outputs found
On Vertex Bisection Width of Random -Regular Graphs
Vertex bisection is a graph partitioning problem in which the aim is to find
a partition into two equal parts that minimizes the number of vertices in one
partition set that have a neighbor in the other set. We are interested in
giving upper bounds on the vertex bisection width of random -regular graphs
for constant values of . Our approach is based on analyzing a greedy
algorithm by using the Differential Equations Method. In this way, we obtain
the first known upper bounds for the vertex bisection width in random regular
graphs. The results are compared with experimental ones and with lower bounds
obtained by Kolesnik and Wormald, (Lower Bounds for the Isoperimetric Numbers
of Random Regular Graphs, SIAM J. on Disc. Math. 28(1), 553-575, 2014).Comment: 31 pages, 2 figure
The isoperimetric constant of the random graph process
The isoperimetric constant of a graph on vertices, , is the
minimum of , taken over all nonempty subsets
of size at most , where denotes the set of
edges with precisely one end in . A random graph process on vertices,
, is a sequence of graphs, where
is the edgeless graph on vertices, and
is the result of adding an edge to ,
uniformly distributed over all the missing edges. We show that in almost every
graph process equals the minimal degree of
as long as the minimal degree is . Furthermore,
we show that this result is essentially best possible, by demonstrating that
along the period in which the minimum degree is typically , the
ratio between the isoperimetric constant and the minimum degree falls from 1 to
1/2, its final value
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
Quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights
We consider a stationary and ergodic random field
that is parameterized by the edge set of the Euclidean lattice ,
. The random variable , taking values in and
satisfying certain moment bounds, is thought of as the conductance of the edge
. Assuming that the set of edges with positive conductances give rise to a
unique infinite cluster , we prove a quenched
invariance principle for the continuous-time random walk among random
conductances under relatively mild conditions on the structure of the infinite
cluster. An essential ingredient of our proof is a new anchored relative
isoperimetric inequality.Comment: 22 page
Percolation and local isoperimetric inequalities
In this paper we establish some relations between percolation on a given
graph G and its geometry. Our main result shows that, if G has polynomial
growth and satisfies what we call the local isoperimetric inequality of
dimension d > 1, then p_c(G) < 1. This gives a partial answer to a question of
Benjamini and Schramm. As a consequence of this result we derive, under the
additional condition of bounded degree, that these graphs also undergo a
non-trivial phase transition for the Ising-Model, the Widom-Rowlinson model and
the beach model. Our techniques are also applied to dependent percolation
processes with long range correlations. We provide results on the uniqueness of
the infinite percolation cluster and quantitative estimates on the size of
finite components. Finally we leave some remarks and questions that arise
naturally from this work.Comment: 21 pages, 2 figure
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
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