4,112 research outputs found
Unavoidable doubly connected large graphs
A connected graph is doubly connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), defined on the set of integers exceeding three, such that every doubly connected graph on at least h(n) vertices must contain, as an induced subgraph, a doubly connected graph, which is either one of the following graphs or the complement of one of the following graphs: (1) Pn, a path on n vertices; (2) K1,ns, the graph obtained from K 1,n by subdividing an edge once; (3) K2,n\e, the graph obtained from K2,n by deleting an edge;(4) K2,n+, the graph obtained from K2,n by adding an edge between the two degree-n vertices x1 and x2, and a pendent edge at each xi. Two applications of this result are also discussed in the paper. © 2003 Elsevier B.V. All rights reserved
Unavoidable induced subgraphs in large graphs with no homogeneous sets
A homogeneous set of an -vertex graph is a set of vertices () such that every vertex not in is either complete or
anticomplete to . A graph is called prime if it has no homogeneous set. A
chain of length is a sequence of vertices such that for every vertex
in the sequence except the first one, its immediate predecessor is its unique
neighbor or its unique non-neighbor among all of its predecessors. We prove
that for all , there exists such that every prime graph with at least
vertices contains one of the following graphs or their complements as an
induced subgraph: (1) the graph obtained from by subdividing every
edge once, (2) the line graph of , (3) the line graph of the graph in
(1), (4) the half-graph of height , (5) a prime graph induced by a chain of
length , (6) two particular graphs obtained from the half-graph of height
by making one side a clique and adding one vertex.Comment: 13 pages, 3 figure
Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling
The goal of decentralized optimization over a network is to optimize a global
objective formed by a sum of local (possibly nonsmooth) convex functions using
only local computation and communication. It arises in various application
domains, including distributed tracking and localization, multi-agent
co-ordination, estimation in sensor networks, and large-scale optimization in
machine learning. We develop and analyze distributed algorithms based on dual
averaging of subgradients, and we provide sharp bounds on their convergence
rates as a function of the network size and topology. Our method of analysis
allows for a clear separation between the convergence of the optimization
algorithm itself and the effects of communication constraints arising from the
network structure. In particular, we show that the number of iterations
required by our algorithm scales inversely in the spectral gap of the network.
The sharpness of this prediction is confirmed both by theoretical lower bounds
and simulations for various networks. Our approach includes both the cases of
deterministic optimization and communication, as well as problems with
stochastic optimization and/or communication.Comment: 40 pages, 4 figure
Void Formation and Roughening in Slow Fracture
Slow crack propagation in ductile, and in certain brittle materials, appears
to take place via the nucleation of voids ahead of the crack tip due to plastic
yields, followed by the coalescence of these voids. Post mortem analysis of the
resulting fracture surfaces of ductile and brittle materials on the m-mm
and the nm scales respectively, reveals self-affine cracks with anomalous
scaling exponent in 3-dimensions and in
2-dimensions. In this paper we present an analytic theory based on the method
of iterated conformal maps aimed at modelling the void formation and the
fracture growth, culminating in estimates of the roughening exponents in
2-dimensions. In the simplest realization of the model we allow one void ahead
of the crack, and address the robustness of the roughening exponent. Next we
develop the theory further, to include two voids ahead of the crack. This
development necessitates generalizing the method of iterated conformal maps to
include doubly connected regions (maps from the annulus rather than the unit
circle). While mathematically and numerically feasible, we find that the
employment of the stress field as computed from elasticity theory becomes
questionable when more than one void is explicitly inserted into the material.
Thus further progress in this line of research calls for improved treatment of
the plastic dynamics.Comment: 15 pages, 20 figure
Directed Percolation with long-range interactions: modeling non-equilibrium wetting
It is argued that some phase--transitions observed in models of
non-equilibrium wetting phenomena are related to contact processes with
long-range interactions. This is investigated by introducing a model where the
activation rate of a site at the edge of an inactive island of length is
. Mean--field analysis and numerical simulations indicate
that for the transition is continuous and belongs to the
universality class of directed percolation, while for , the
transition becomes first order. This criterion is then applied to discuss
critical properties of various models of non--equilibrium wetting.Comment: 12 Figures. V Version resubmitted to Phys. Rev. E after referees
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