12,708 research outputs found
Setting Parameters by Example
We introduce a class of "inverse parametric optimization" problems, in which
one is given both a parametric optimization problem and a desired optimal
solution; the task is to determine parameter values that lead to the given
solution. We describe algorithms for solving such problems for minimum spanning
trees, shortest paths, and other "optimal subgraph" problems, and discuss
applications in multicast routing, vehicle path planning, resource allocation,
and board game programming.Comment: 13 pages, 3 figures. To be presented at 40th IEEE Symp. Foundations
of Computer Science (FOCS '99
Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs
We show how to compute the probabilities of various connection topologies for
uniformly random spanning trees on graphs embedded in surfaces. As an
application, we show how to compute the "intensity" of the loop-erased random
walk in , that is, the probability that the walk from (0,0) to
infinity passes through a given vertex or edge. For example, the probability
that it passes through (1,0) is 5/16; this confirms a conjecture from 1994
about the stationary sandpile density on . We do the analogous
computation for the triangular lattice, honeycomb lattice and , for which the probabilities are 5/18, 13/36, and
respectively.Comment: 45 pages, many figures. v2 has an expanded introduction, a revised
section on the LERW intensity, and an expanded appendix on the annular matri
Maximum Performance at Minimum Cost in Network Synchronization
We consider two optimization problems on synchronization of oscillator
networks: maximization of synchronizability and minimization of synchronization
cost. We first develop an extension of the well-known master stability
framework to the case of non-diagonalizable Laplacian matrices. We then show
that the solution sets of the two optimization problems coincide and are
simultaneously characterized by a simple condition on the Laplacian
eigenvalues. Among the optimal networks, we identify a subclass of hierarchical
networks, characterized by the absence of feedback loops and the normalization
of inputs. We show that most optimal networks are directed and
non-diagonalizable, necessitating the extension of the framework. We also show
how oriented spanning trees can be used to explicitly and systematically
construct optimal networks under network topological constraints. Our results
may provide insights into the evolutionary origin of structures in complex
networks for which synchronization plays a significant role.Comment: 29 pages, 9 figures, accepted for publication in Physica D, minor
correction
The Tensor Track, III
We provide an informal up-to-date review of the tensor track approach to
quantum gravity. In a long introduction we describe in simple terms the
motivations for this approach. Then the many recent advances are summarized,
with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion,
Osterwalder-Schrader positivity...) which, while important for the tensor track
program, are not detailed in the usual quantum gravity literature. We list open
questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure
Spanning forests and the vector bundle Laplacian
The classical matrix-tree theorem relates the determinant of the
combinatorial Laplacian on a graph to the number of spanning trees. We
generalize this result to Laplacians on one- and two-dimensional vector
bundles, giving a combinatorial interpretation of their determinants in terms
of so-called cycle rooted spanning forests (CRSFs). We construct natural
measures on CRSFs for which the edges form a determinantal process. This theory
gives a natural generalization of the spanning tree process adapted to graphs
embedded on surfaces. We give a number of other applications, for example, we
compute the probability that a loop-erased random walk on a planar graph
between two vertices on the outer boundary passes left of two given faces. This
probability cannot be computed using the standard Laplacian alone.Comment: Published in at http://dx.doi.org/10.1214/10-AOP596 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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