22,190 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
Resiliently evolving supply-demand networks
Peer reviewedPublisher PD
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
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
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
Clustering and the hyperbolic geometry of complex networks
Clustering is a fundamental property of complex networks and it is the
mathematical expression of a ubiquitous phenomenon that arises in various types
of self-organized networks such as biological networks, computer networks or
social networks. In this paper, we consider what is called the global
clustering coefficient of random graphs on the hyperbolic plane. This model of
random graphs was proposed recently by Krioukov et al. as a mathematical model
of complex networks, under the fundamental assumption that hyperbolic geometry
underlies the structure of these networks. We give a rigorous analysis of
clustering and characterize the global clustering coefficient in terms of the
parameters of the model. We show how the global clustering coefficient can be
tuned by these parameters and we give an explicit formula for this function.Comment: 51 pages, 1 figur
The effect of network structure on phase transitions in queuing networks
Recently, De Martino et al have presented a general framework for the study
of transportation phenomena on complex networks. One of their most significant
achievements was a deeper understanding of the phase transition from the
uncongested to the congested phase at a critical traffic load. In this paper,
we also study phase transition in transportation networks using a discrete time
random walk model. Our aim is to establish a direct connection between the
structure of the graph and the value of the critical traffic load. Applying
spectral graph theory, we show that the original results of De Martino et al
showing that the critical loading depends only on the degree sequence of the
graph -- suggesting that different graphs with the same degree sequence have
the same critical loading if all other circumstances are fixed -- is valid only
if the graph is dense enough. For sparse graphs, higher order corrections,
related to the local structure of the network, appear.Comment: 12 pages, 7 figure
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