31 research outputs found
Asymptotic normality of the size of the giant component via a random walk
In this paper we give a simple new proof of a result of Pittel and Wormald
concerning the asymptotic value and (suitably rescaled) limiting distribution
of the number of vertices in the giant component of above the scaling
window of the phase transition. Nachmias and Peres used martingale arguments to
study Karp's exploration process, obtaining a simple proof of a weak form of
this result. We use slightly different martingale arguments to obtain a much
sharper result with little extra work.Comment: 11 pages; slightly expanded, reference adde
Between 2- and 3-colorability
We consider the question of the existence of homomorphisms between
and odd cycles when . We show that for any positive integer
, there exists such that if then
w.h.p. has a homomorphism from to so long as
its odd-girth is at least . On the other hand, we show that if
then w.h.p. there is no homomorphism from to . Note that in our
range of interest, w.h.p., implying that there is a
homomorphism from to
Essential edges in Poisson random hypergraphs
Consider a random hypergraph on a set of N vertices in which, for k between 1
and N, a Poisson(N beta_k) number of hyperedges is scattered randomly over all
subsets of size k. We collapse the hypergraph by running the following
algorithm to exhaustion: pick a vertex having a 1-edge and remove it; collapse
the hyperedges over that vertex onto their remaining vertices; repeat until
there are no 1-edges left. We call the vertices removed in this process
"identifiable". Also any hyperedge all of whose vertices are removed is called
"identifiable". We say that a hyperedge is "essential" if its removal prior to
collapse would have reduced the number of identifiable vertices. The limiting
proportions, as N tends to infinity, of identifiable vertices and hyperedges
were obtained by Darling and Norris. In this paper, we establish the limiting
proportion of essential hyperedges. We also discuss, in the case of a random
graph, the relation of essential edges to the 2-core of the graph, the maximal
sub-graph with minimal vertex degree 2.Comment: 12 pages, 3 figures. Revised version with minor
corrections/clarifications and slightly expanded introductio
Monotonicity, asymptotic normality and vertex degrees in random graphs
We exploit a result by Nerman which shows that conditional limit theorems
hold when a certain monotonicity condition is satisfied. Our main result is an
application to vertex degrees in random graphs, where we obtain asymptotic
normality for the number of vertices with a given degree in the random graph
with a fixed number of edges from the corresponding result for the
random graph with independent edges. We also give some simple
applications to random allocations and to spacings. Finally, inspired by these
results, but logically independent of them, we investigate whether a one-sided
version of the Cram\'{e}r--Wold theorem holds. We show that such a version
holds under a weak supplementary condition, but not without it.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6103 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
The Peculiar Phase Structure of Random Graph Bisection
The mincut graph bisection problem involves partitioning the n vertices of a
graph into disjoint subsets, each containing exactly n/2 vertices, while
minimizing the number of "cut" edges with an endpoint in each subset. When
considered over sparse random graphs, the phase structure of the graph
bisection problem displays certain familiar properties, but also some
surprises. It is known that when the mean degree is below the critical value of
2 log 2, the cutsize is zero with high probability. We study how the minimum
cutsize increases with mean degree above this critical threshold, finding a new
analytical upper bound that improves considerably upon previous bounds.
Combined with recent results on expander graphs, our bound suggests the unusual
scenario that random graph bisection is replica symmetric up to and beyond the
critical threshold, with a replica symmetry breaking transition possibly taking
place above the threshold. An intriguing algorithmic consequence is that
although the problem is NP-hard, we can find near-optimal cutsizes (whose ratio
to the optimal value approaches 1 asymptotically) in polynomial time for
typical instances near the phase transition.Comment: substantially revised section 2, changed figures 3, 4 and 6, made
minor stylistic changes and added reference
On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees
We consider a Bernoulli bond percolation on a random recursive tree of size
, with supercritical parameter for some fixed. It
is known that with high probability, there exists then a unique giant cluster
of size G_n\sim \e^{-c}, and it follows from a recent result of Schweinsberg
\cite{Sch} that has non-gaussian fluctuations. We provide an explanation
of this by analyzing the effect of percolation on different phases of the
growth of recursive trees. This alternative approach may be useful for studying
percolation on other classes of trees, such as for instance regular trees
Finding paths in sparse random graphs requires many queries
We discuss a new algorithmic type of problem in random graphs studying the
minimum number of queries one has to ask about adjacency between pairs of
vertices of a random graph in order to find a
subgraph which possesses some target property with high probability. In this
paper we focus on finding long paths in when
for some fixed constant . This
random graph is known to have typically linearly long paths.
To have edges with high probability in one
clearly needs to query at least pairs of
vertices. Can we find a path of length economically, i.e., by querying
roughly that many pairs? We argue that this is not possible and one needs to
query significantly more pairs. We prove that any randomised algorithm which
finds a path of length
with at least constant probability in with
must query at least
pairs of vertices. This is
tight up to the factor.Comment: 14 page