25,930 research outputs found
Adjacency labeling schemes and induced-universal graphs
We describe a way of assigning labels to the vertices of any undirected graph
on up to vertices, each composed of bits, such that given the
labels of two vertices, and no other information regarding the graph, it is
possible to decide whether or not the vertices are adjacent in the graph. This
is optimal, up to an additive constant, and constitutes the first improvement
in almost 50 years of an bound of Moon. As a consequence, we
obtain an induced-universal graph for -vertex graphs containing only
vertices, which is optimal up to a multiplicative constant,
solving an open problem of Vizing from 1968. We obtain similar tight results
for directed graphs, tournaments and bipartite graphs
Angular Normal Modes of a Circular Coulomb Cluster
We investigate the angular normal modes for small oscillations about an
equilibrium of a single-component coulomb cluster confined by a radially
symmetric external potential to a circle. The dynamical matrix for this system
is a Laplacian symmetrically circulant matrix and this result leads to an
analytic solution for the eigenfrequencies of the angular normal modes. We also
show the limiting dependence of the largest eigenfrequency for large numbers of
particles
Bidirectional PageRank Estimation: From Average-Case to Worst-Case
We present a new algorithm for estimating the Personalized PageRank (PPR)
between a source and target node on undirected graphs, with sublinear
running-time guarantees over the worst-case choice of source and target nodes.
Our work builds on a recent line of work on bidirectional estimators for PPR,
which obtained sublinear running-time guarantees but in an average-case sense,
for a uniformly random choice of target node. Crucially, we show how the
reversibility of random walks on undirected networks can be exploited to
convert average-case to worst-case guarantees. While past bidirectional methods
combine forward random walks with reverse local pushes, our algorithm combines
forward local pushes with reverse random walks. We also discuss how to modify
our methods to estimate random-walk probabilities for any length distribution,
thereby obtaining fast algorithms for estimating general graph diffusions,
including the heat kernel, on undirected networks.Comment: Workshop on Algorithms and Models for the Web-Graph (WAW) 201
Resiliently evolving supply-demand networks
Peer reviewedPublisher PD
Non-perturbative corrections to mean-field behavior: spherical model on spider-web graph
We consider the spherical model on a spider-web graph. This graph is
effectively infinite-dimensional, similar to the Bethe lattice, but has loops.
We show that these lead to non-trivial corrections to the simple mean-field
behavior. We first determine all normal modes of the coupled springs problem on
this graph, using its large symmetry group. In the thermodynamic limit, the
spectrum is a set of -functions, and all the modes are localized. The
fractional number of modes with frequency less than varies as for tending to zero, where is a constant. For an
unbiased random walk on the vertices of this graph, this implies that the
probability of return to the origin at time varies as ,
for large , where is a constant. For the spherical model, we show that
while the critical exponents take the values expected from the mean-field
theory, the free-energy per site at temperature , near and above the
critical temperature , also has an essential singularity of the type
.Comment: substantially revised, a section adde
Random Vibrational Networks and Renormalization Group
We consider the properties of vibrational dynamics on random networks, with
random masses and spring constants. The localization properties of the
eigenstates contrast greatly with the Laplacian case on these networks. We
introduce several real-space renormalization techniques which can be used to
describe this dynamics on general networks, drawing on strong disorder
techniques developed for regular lattices. The renormalization group is capable
of elucidating the localization properties, and provides, even for specific
network instances, a fast approximation technique for determining the spectra
which compares well with exact results.Comment: 4 pages, 3 figure
Induced Lorentz- and CPT-violating Chern-Simons term in QED: Fock-Schwinger proper time method
Using the Fock-Schwinger proper time method, we calculate the induced
Chern-Simons term arising from the Lorentz- and CPT-violating sector of quantum
electrodynamics with a term. Our
result to all orders in coincides with a recent linear-in- calculation
by Chaichian et al. [hep-th/0010129 v2]. The coincidence was pointed out by
Chung [Phys. Lett. {\bf B461} (1999) 138] and P\'{e}rez-Victoria [Phys. Rev.
Lett. {\bf 83} (1999) 2518] in the standard Feynman diagram calculation with
the nonperturbative-in- propagator.Comment: 11 pages, no figur
Active Semi-Supervised Learning Using Sampling Theory for Graph Signals
We consider the problem of offline, pool-based active semi-supervised
learning on graphs. This problem is important when the labeled data is scarce
and expensive whereas unlabeled data is easily available. The data points are
represented by the vertices of an undirected graph with the similarity between
them captured by the edge weights. Given a target number of nodes to label, the
goal is to choose those nodes that are most informative and then predict the
unknown labels. We propose a novel framework for this problem based on our
recent results on sampling theory for graph signals. A graph signal is a
real-valued function defined on each node of the graph. A notion of frequency
for such signals can be defined using the spectrum of the graph Laplacian
matrix. The sampling theory for graph signals aims to extend the traditional
Nyquist-Shannon sampling theory by allowing us to identify the class of graph
signals that can be reconstructed from their values on a subset of vertices.
This approach allows us to define a criterion for active learning based on
sampling set selection which aims at maximizing the frequency of the signals
that can be reconstructed from their samples on the set. Experiments show the
effectiveness of our method.Comment: 10 pages, 6 figures, To appear in KDD'1
Calculation of a Class of Three-Loop Vacuum Diagrams with Two Different Mass Values
We calculate analytically a class of three-loop vacuum diagrams with two
different mass values, one of which is one-third as large as the other, using
the method of Chetyrkin, Misiak, and M\"{u}nz in the dimensional regularization
scheme. All pole terms in \epsilon=4-D (D being the space-time dimensions in a
dimensional regularization scheme) plus finite terms containing the logarithm
of mass are kept in our calculation of each diagram. It is shown that
three-loop effective potential calculated using three-loop integrals obtained
in this paper agrees, in the large-N limit, with the overlap part of
leading-order (in the large-N limit) calculation of Coleman, Jackiw, and
Politzer [Phys. Rev. D {\bf 10}, 2491 (1974)].Comment: RevTex, 15 pages, 4 postscript figures, minor corrections in K(c),
Appendix B removed, typos corrected, acknowledgements change
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