286 research outputs found
Expansion properties of a random regular graph after random vertex deletions
We investigate the following vertex percolation process. Starting with a
random regular graph of constant degree, delete each vertex independently with
probability p, where p=n^{-alpha} and alpha=alpha(n) is bounded away from 0. We
show that a.a.s. the resulting graph has a connected component of size n-o(n)
which is an expander, and all other components are trees of bounded size.
Sharper results are obtained with extra conditions on alpha. These results have
an application to the cost of repairing a certain peer-to-peer network after
random failures of nodes.Comment: 14 page
Sharp threshold for percolation on expanders
We study the appearance of the giant component in random subgraphs of a given
large finite graph G=(V,E) in which each edge is present independently with
probability p. We show that if G is an expander with vertices of bounded
degree, then for any c in ]0,1[, the property that the random subgraph contains
a giant component of size c|V| has a sharp threshold.Comment: Published in at http://dx.doi.org/10.1214/10-AOP610 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Broadcasting on Random Directed Acyclic Graphs
We study a generalization of the well-known model of broadcasting on trees.
Consider a directed acyclic graph (DAG) with a unique source vertex , and
suppose all other vertices have indegree . Let the vertices at
distance from be called layer . At layer , is given a random
bit. At layer , each vertex receives bits from its parents in
layer , which are transmitted along independent binary symmetric channel
edges, and combines them using a -ary Boolean processing function. The goal
is to reconstruct with probability of error bounded away from using
the values of all vertices at an arbitrarily deep layer. This question is
closely related to models of reliable computation and storage, and information
flow in biological networks.
In this paper, we analyze randomly constructed DAGs, for which we show that
broadcasting is only possible if the noise level is below a certain degree and
function dependent critical threshold. For , and random DAGs with
layer sizes and majority processing functions, we identify the
critical threshold. For , we establish a similar result for NAND
processing functions. We also prove a partial converse for odd
illustrating that the identified thresholds are impossible to improve by
selecting different processing functions if the decoder is restricted to using
a single vertex.
Finally, for any noise level, we construct explicit DAGs (using expander
graphs) with bounded degree and layer sizes admitting
reconstruction. In particular, we show that such DAGs can be generated in
deterministic quasi-polynomial time or randomized polylogarithmic time in the
depth. These results portray a doubly-exponential advantage for storing a bit
in DAGs compared to trees, where but layer sizes must grow exponentially
with depth in order to enable broadcasting.Comment: 33 pages, double column format. arXiv admin note: text overlap with
arXiv:1803.0752
Glauber dynamics on nonamenable graphs: Boundary conditions and mixing time
We study the stochastic Ising model on finite graphs with n vertices and
bounded degree and analyze the effect of boundary conditions on the mixing
time. We show that for all low enough temperatures, the spectral gap of the
dynamics with (+)-boundary condition on a class of nonamenable graphs, is
strictly positive uniformly in n. This implies that the mixing time grows at
most linearly in n. The class of graphs we consider includes hyperbolic graphs
with sufficiently high degree, where the best upper bound on the mixing time of
the free boundary dynamics is polynomial in n, with exponent growing with the
inverse temperature. In addition, we construct a graph in this class, for which
the mixing time in the free boundary case is exponentially large in n. This
provides a first example where the mixing time jumps from exponential to linear
in n while passing from free to (+)-boundary condition. These results extend
the analysis of Martinelli, Sinclair and Weitz to a wider class of nonamenable
graphs.Comment: 31 pages, 4 figures; added reference; corrected typo
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