5,816 research outputs found
Local Access to Random Walks
For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work ?(t). We desire local access algorithms supporting position_G(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/poly(n) close to those of a uniformly random walk in ?? distance.
We first give an algorithm for local access to random walks on a given undirected d-regular graph with O?(1/(1-?)?n) runtime per query, where ? is the second-largest eigenvalue of the random walk matrix of the graph in absolute value. Since random d-regular graphs G(n,d) are expanders with high probability, this gives an O?(?n) algorithm for a graph drawn from G(n,d) whp, which improves on the naive method for small numbers of queries.
We then prove that no algorithm with subconstant error given probe access to an input d-regular graph can have runtime better than ?(?n/log(n)) per query in expectation when the input graph is drawn from G(n,d), obtaining a nearly matching lower bound. We further show an ?(n^{1/4}) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance).
We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree n^? graphs for any ? ? (0,1]) and Cartesian product
On the spectral distribution of large weighted random regular graphs
McKay proved that the limiting spectral measures of the ensembles of
-regular graphs with vertices converge to Kesten's measure as
. In this paper we explore the case of weighted graphs. More
precisely, given a large -regular graph we assign random weights, drawn from
some distribution , to its edges. We study the relationship
between and the associated limiting spectral distribution
obtained by averaging over the weighted graphs. Among other results, we
establish the existence of a unique `eigendistribution', i.e., a weight
distribution such that the associated limiting spectral
distribution is a rescaling of . Initial investigations suggested
that the eigendistribution was the semi-circle distribution, which by Wigner's
Law is the limiting spectral measure for real symmetric matrices. We prove this
is not the case, though the deviation between the eigendistribution and the
semi-circular density is small (the first seven moments agree, and the
difference in each higher moment is ). Our analysis uses
combinatorial results about closed acyclic walks in large trees, which may be
of independent interest.Comment: Version 1.0, 19 page
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
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