102 research outputs found
Cycles in Random Bipartite Graphs
In this paper we study cycles in random bipartite graph . We prove
that if , then a.a.s. satisfies the following. Every
subgraph with more than edges contains a
cycle of length for all even . Our theorem
complements a previous result on bipancyclicity, and is closely related to a
recent work of Lee and Samotij.Comment: 8 pages, 2 figure
Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (II)
Following the derivation of the trace formulae in the first paper in this
series, we establish here a connection between the spectral statistics of
random regular graphs and the predictions of Random Matrix Theory (RMT). This
follows from the known Poisson distribution of cycle counts in regular graphs,
in the limit that the cycle periods are kept constant and the number of
vertices increases indefinitely. The result is analogous to the so called
"diagonal approximation" in Quantum Chaos. We also show that by assuming that
the spectral correlations are given by RMT to all orders, we can compute the
leading deviations from the Poisson distribution for cycle counts. We provide
numerical evidence which supports this conjecture.Comment: 15 pages, 5 figure
Exchangeable pairs, switchings, and random regular graphs
We consider the distribution of cycle counts in a random regular graph, which
is closely linked to the graph's spectral properties. We broaden the asymptotic
regime in which the cycle counts are known to be approximately Poisson, and we
give an explicit bound in total variation distance for the approximation. Using
this result, we calculate limiting distributions of linear eigenvalue
functionals for random regular graphs.
Previous results on the distribution of cycle counts by McKay, Wormald, and
Wysocka (2004) used the method of switchings, a combinatorial technique for
asymptotic enumeration. Our proof uses Stein's method of exchangeable pairs and
demonstrates an interesting connection between the two techniques.Comment: Very minor changes; 23 page
Strong Secrecy for Erasure Wiretap Channels
We show that duals of certain low-density parity-check (LDPC) codes, when
used in a standard coset coding scheme, provide strong secrecy over the binary
erasure wiretap channel (BEWC). This result hinges on a stopping set analysis
of ensembles of LDPC codes with block length and girth , for some
. We show that if the minimum left degree of the ensemble is
, the expected probability of block error is
\calO(\frac{1}{n^{\lceil l_\mathrm{min} k /2 \rceil - k}}) when the erasure
probability , where
depends on the degree distribution of the ensemble. As long as and , the dual of this LDPC code provides strong secrecy over a
BEWC of erasure probability greater than .Comment: Submitted to the Information Theory Workship (ITW) 2010, Dubli
Approximating max-min linear programs with local algorithms
A local algorithm is a distributed algorithm where each node must operate
solely based on the information that was available at system startup within a
constant-size neighbourhood of the node. We study the applicability of local
algorithms to max-min LPs where the objective is to maximise subject to for each and
for each . Here , , and the support sets , ,
and have bounded size. In the distributed setting,
each agent is responsible for choosing the value of , and the
communication network is a hypergraph where the sets and
constitute the hyperedges. We present inapproximability results for a
wide range of structural assumptions; for example, even if and
are bounded by some constants larger than 2, there is no local approximation
scheme. To contrast the negative results, we present a local approximation
algorithm which achieves good approximation ratios if we can bound the relative
growth of the vertex neighbourhoods in .Comment: 16 pages, 2 figure
- β¦