3,759 research outputs found
Matrices of forests, analysis of networks, and ranking problems
The matrices of spanning rooted forests are studied as a tool for analysing
the structure of networks and measuring their properties. The problems of
revealing the basic bicomponents, measuring vertex proximity, and ranking from
preference relations / sports competitions are considered. It is shown that the
vertex accessibility measure based on spanning forests has a number of
desirable properties. An interpretation for the stochastic matrix of
out-forests in terms of information dissemination is given.Comment: 8 pages. This article draws heavily from arXiv:math/0508171.
Published in Proceedings of the First International Conference on Information
Technology and Quantitative Management (ITQM 2013). This version contains
some corrections and addition
Boundary Partitions in Trees and Dimers
Given a finite planar graph, a grove is a spanning forest in which every
component tree contains one or more of a specified set of vertices (called
nodes) on the outer face. For the uniform measure on groves, we compute the
probabilities of the different possible node connections in a grove. These
probabilities only depend on boundary measurements of the graph and not on the
actual graph structure, i.e., the probabilities can be expressed as functions
of the pairwise electrical resistances between the nodes, or equivalently, as
functions of the Dirichlet-to-Neumann operator (or response matrix) on the
nodes. These formulae can be likened to generalizations (for spanning forests)
of Cardy's percolation crossing probabilities, and generalize Kirchhoff's
formula for the electrical resistance. Remarkably, when appropriately
normalized, the connection probabilities are in fact integer-coefficient
polynomials in the matrix entries, where the coefficients have a natural
algebraic interpretation and can be computed combinatorially. A similar
phenomenon holds in the so-called double-dimer model: connection probabilities
of boundary nodes are polynomial functions of certain boundary measurements,
and as formal polynomials, they are specializations of the grove polynomials.
Upon taking scaling limits, we show that the double-dimer connection
probabilities coincide with those of the contour lines in the Gaussian free
field with certain natural boundary conditions. These results have direct
application to connection probabilities for multiple-strand SLE_2, SLE_8, and
SLE_4.Comment: 46 pages, 12 figures. v4 has additional diagrams and other minor
change
Exponents and bounds for uniform spanning trees in d dimensions
Uniform spanning trees are a statistical model obtained by taking the set of
all spanning trees on a given graph (such as a portion of a cubic lattice in d
dimensions), with equal probability for each distinct tree. Some properties of
such trees can be obtained in terms of the Laplacian matrix on the graph, by
using Grassmann integrals. We use this to obtain exact exponents that bound
those for the power-law decay of the probability that k distinct branches of
the tree pass close to each of two distinct points, as the size of the lattice
tends to infinity.Comment: 5 pages. v2: references added. v3: closed form results can be
extended slightly (thanks to C. Tanguy). v4: revisions, and a figure adde
The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs
One major open conjecture in the area of critical random graphs, formulated
by statistical physicists, and supported by a large amount of numerical
evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array
of random graph models with degree exponent , distances between
typical points both within maximal components in the critical regime as well as
on the minimal spanning tree on the giant component in the supercritical regime
scale like .
In this paper we study the metric space structure of maximal components of
the multiplicative coalescent, in the regime where the sizes converge to
excursions of L\'evy processes "without replacement" [10], yielding a
completely new class of limiting random metric spaces. A by-product of the
analysis yields the continuum scaling limit of one fundamental class of random
graph models with degree exponent where edges are rescaled by
yielding the first rigorous proof of the above
conjecture. The limits in this case are compact "tree-like" random fractals
with finite fractal dimensions and with a dense collection of hubs (infinite
degree vertices) a finite number of which are identified with leaves to form
shortcuts. In a special case, we show that the Minkowski dimension of the
limiting spaces equal a.s., in stark contrast to the
Erd\H{o}s-R\'{e}nyi scaling limit whose Minkowski dimension is 2 a.s. It is
generally believed that dynamic versions of a number of fundamental random
graph models, as one moves from the barely subcritical to the critical regime
can be approximated by the multiplicative coalescent. In work in progress, the
general theory developed in this paper is used to prove analogous limit results
for other random graph models with degree exponent .Comment: 71 pages, 5 figures, To appear in Probability Theory and Related
Field
Dimers, Tilings and Trees
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others
we describe a natural equivalence between three planar objects: weighted
bipartite planar graphs; planar Markov chains; and tilings with convex
polygons. This equivalence provides a measure-preserving bijection between
dimer coverings of a weighted bipartite planar graph and spanning trees on the
corresponding Markov chain. The tilings correspond to harmonic functions on the
Markov chain and to ``discrete analytic functions'' on the bipartite graph.
The equivalence is extended to infinite periodic graphs, and we classify the
resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure
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