1,727 research outputs found
Approximating the Spectrum of a Graph
The spectrum of a network or graph with adjacency matrix ,
consists of the eigenvalues of the normalized Laplacian . This set of eigenvalues encapsulates many aspects of the structure
of the graph, including the extent to which the graph posses community
structures at multiple scales. We study the problem of approximating the
spectrum , of in the regime where the graph is too
large to explicitly calculate the spectrum. We present a sublinear time
algorithm that, given the ability to query a random node in the graph and
select a random neighbor of a given node, computes a succinct representation of
an approximation , such that . Our algorithm has query complexity and running time ,
independent of the size of the graph, . We demonstrate the practical
viability of our algorithm on 15 different real-world graphs from the Stanford
Large Network Dataset Collection, including social networks, academic
collaboration graphs, and road networks. For the smallest of these graphs, we
are able to validate the accuracy of our algorithm by explicitly calculating
the true spectrum; for the larger graphs, such a calculation is computationally
prohibitive.
In addition we study the implications of our algorithm to property testing in
the bounded degree graph model
Approximating the Largest Root and Applications to Interlacing Families
We study the problem of approximating the largest root of a real-rooted
polynomial of degree using its top coefficients and give nearly
matching upper and lower bounds. We present algorithms with running time
polynomial in that use the top coefficients to approximate the maximum
root within a factor of and when and respectively. We also prove corresponding
information-theoretic lower bounds of and
, and show strong lower
bounds for noisy version of the problem in which one is given access to
approximate coefficients.
This problem has applications in the context of the method of interlacing
families of polynomials, which was used for proving the existence of Ramanujan
graphs of all degrees, the solution of the Kadison-Singer problem, and bounding
the integrality gap of the asymmetric traveling salesman problem. All of these
involve computing the maximum root of certain real-rooted polynomials for which
the top few coefficients are accessible in subexponential time. Our results
yield an algorithm with the running time of for all
of them
An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification
We prove the following Alon-Boppana type theorem for general (not necessarily
regular) weighted graphs: if is an -node weighted undirected graph of
average combinatorial degree (that is, has edges) and girth , and if are the
eigenvalues of the (non-normalized) Laplacian of , then (The Alon-Boppana theorem implies that if is unweighted and
-regular, then if the diameter is at least .)
Our result implies a lower bound for spectral sparsifiers. A graph is a
spectral -sparsifier of a graph if where is the Laplacian matrix of and is
the Laplacian matrix of . Batson, Spielman and Srivastava proved that for
every there is an -sparsifier of average degree where
and the edges of are a
(weighted) subset of the edges of . Batson, Spielman and Srivastava also
show that the bound on cannot be reduced below when is a clique; our Alon-Boppana-type result implies that
cannot be reduced below when comes
from a family of expanders of super-constant degree and super-constant girth.
The method of Batson, Spielman and Srivastava proves a more general result,
about sparsifying sums of rank-one matrices, and their method applies to an
"online" setting. We show that for the online matrix setting the bound is tight, up to lower order terms
New factor graph based multiuser detector for spectrally efficient CPM
This paper presents a new iterative multiuser detection algorithm for asynchronous spectrally-efficient M-ary continuous-phase modulation in additive white Gaussian noise. This detection algorithm is closely related to another algorithm that was recently proposed by the same authors, but it follows from applying the sum-product algorithm to a different factor graph of the same multiuser detection problem. This, in turn, results in a different way to approximate the marginal bit a-posteriori probabilities that are used to perform minimum bit error rate multiuser detection. The girth of the factor graph considered in this contribution is twice as large, which is known to be potentially beneficial for the accuracy of the a-posteriori probabilities. The size of the largest factor graph variable alphabets also multiplies with M, rendering the straightforward application of the sum-product algorithm more complex. Through approximating a suitable set of sum-product messages by a Gaussian distribution, this complexity is significantly reduced. For a set of system parameters yielding high spectral efficiency, the resulting algorithm significantly outperforms the previously proposed solution
A Combination Theorem for Metric Bundles
We define metric bundles/metric graph bundles which provide a purely
topological/coarse-geometric generalization of the notion of trees of metric
spaces a la Bestvina-Feighn in the special case that the inclusions of the edge
spaces into the vertex spaces are uniform coarsely surjective quasi-isometries.
We prove the existence of quasi-isometric sections in this generality. Then we
prove a combination theorem for metric (graph) bundles (including exact
sequences of groups) that establishes sufficient conditions, particularly
flaring, under which the metric bundles are hyperbolic. We use this to give
examples of surface bundles over hyperbolic disks, whose universal cover is
Gromov-hyperbolic. We also show that in typical situations, flaring is also a
necessary condition.Comment: v3: Major revision: 56 pages 5 figures. Many details added.
Characterization of convex cocompact subgroups of mapping class groups of
surfaces with punctures in terms of relative hyperbolicity given v4: Final
version incorporating referee comments: 63 pages 5 figures. To appear in
Geom. Funct. Ana
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