15 research outputs found
A decomposition based proof for fast mixing of a Markov chain over balanced realizations of a joint degree matrix
A joint degree matrix (JDM) specifies the number of connections between nodes
of given degrees in a graph, for all degree pairs and uniquely determines the
degree sequence of the graph. We consider the space of all balanced
realizations of an arbitrary JDM, realizations in which the links between any
two degree groups are placed as uniformly as possible. We prove that a swap
Markov Chain Monte Carlo (MCMC) algorithm in the space of all balanced
realizations of an {\em arbitrary} graphical JDM mixes rapidly, i.e., the
relaxation time of the chain is bounded from above by a polynomial in the
number of nodes . To prove fast mixing, we first prove a general
factorization theorem similar to the Martin-Randall method for disjoint
decompositions (partitions). This theorem can be used to bound from below the
spectral gap with the help of fast mixing subchains within every partition and
a bound on an auxiliary Markov chain between the partitions. Our proof of the
general factorization theorem is direct and uses conductance based methods
(Cheeger inequality).Comment: submitted, 18 pages, 4 figure
Sampling for conditional inference on contingency tables, multigraphs, and high dimensional tables
We propose new sequential importance sampling methods for sampling contingency tables with fixed margins, loopless, undirected multigraphs, and high-dimensional tables. In each case, the proposals for the method are constructed by leveraging approximations to the total number of structures (tables, multigraphs, or high-dimensional tables), based on results in the literature. The methods generate structures that are very close to the target uniform distribution. Along with their importance weights, the data structures are used to approximate the null distribution of test statistics. In the case of contingency tables, we apply the methods to a number of applications and demonstrate an improvement over competing methods. For loopless, undirected multigraphs, we apply the method to ecological and security problems, and demonstrate excellent performance. In the case of high-dimensional tables, we apply the sequential importance sampling method to the analysis of multimarker linkage disequilibrium data and also demonstrate excellent performance
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
Barvinok's Rational Functions: Algorithms and Applications to Optimization, Statistics, and Algebra
The main theme of this dissertation is the study of the lattice points in a
rational convex polyhedron and their encoding in terms of Barvinok's short
rational functions. The first part of this thesis looks into theoretical
applications of these rational functions to Optimization, Statistics, and
Computational Algebra. The main theorem on Chapter 2 concerns the computation
of the \emph{toric ideal} of an integral matrix . We
encode the binomials belonging to the toric ideal associated with
using Barvinok's rational functions. If we fix and , this representation
allows us to compute a universal Gr\"obner basis and the reduced Gr\"obner
basis of the ideal , with respect to any term order, in polynomial time.
We derive a polynomial time algorithm for normal form computations which
replaces in this new encoding the usual reductions of the division algorithm.
Chapter 3 presents three ways to use Barvinok's rational functions to solve
Integer Programs.
The second part of the thesis is experimental and consists mainly of the
software package {\tt LattE}, the first implementation of Barvinok's algorithm.
We report on experiments with families of well-known rational polytopes:
multiway contingency tables, knapsack type problems, and rational polygons. We
also developed a new algorithm, {\em the homogenized Barvinok's algorithm} to
compute the generating function for a rational polytope. We showed that it runs
in polynomial time in fixed dimension. With the homogenized Barvinok's
algorithm, we obtained new combinatorial formulas: the generating function for
the number of magic squares and the generating function for the
number of magic cubes as rational functions.Comment: Thesi
Enumerating contingency tables via random permanents
Given m positive integers R=(r_i), n positive integers C=(c_j) such that sum
r_i = sum c_j =N, and mn non-negative weights W=(w_{ij}), we consider the total
weight T=T(R, C; W) of non-negative integer matrices (contingency tables)
D=(d_{ij}) with the row sums r_i, column sums c_j, and the weight of D equal to
prod w_{ij}^{d_{ij}}. We present a randomized algorithm of a polynomial in N
complexity which computes a number T'=T'(R,C; W) such that T' < T < alpha(R, C)
T' where alpha(R,C) = min{prod r_i! r_i^{-r_i}, prod c_j! c_j^{-c_j}} N^N/N!.
In many cases, ln T' provides an asymptotically accurate estimate of ln T. The
idea of the algorithm is to express T as the expectation of the permanent of an
N x N random matrix with exponentially distributed entries and approximate the
expectation by the integral T' of an efficiently computable log-concave
function on R^{mn}. Applications to counting integer flows in graphs are also
discussed.Comment: 19 pages, bounds are sharpened, references are adde
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A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees
Random graphs with a given degree sequence are a useful model capturing several features absent in the classical Erd˝os-R´enyi model, such as dependent edges and non-binomial degrees. In this paper, we
use a characterization due to Erd˝os and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows surprisingly
efficient sequential importance sampling. Applications are given, including simulating a biological network and estimating the number of graphs with a given degree sequence.Statistic
New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling
In network modelling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in the form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a Joint Degree Matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence, and conjectured that it mixes rapidly (in polynomial time) for arbitrary degree sequences. Although the conjecture is still open, it has been proved for special degree sequences, in particular for those of undirected and directed regular simple graphs, half-regular bipartite graphs, and graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite ‘splitted’ degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs. Copyright © Cambridge University Press 201
Sampling Hypergraphs with Given Degrees
There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm for sampling bipartite graphs with given degrees, uniformly or nearly uniformly, in (expected) polynomial time. The expected runtime of the hypergraph sampling algorithm depends on the (expected) runtime of the bipartite graph sampling algorithm , and the probability that a uniformly random bipartite graph with given degrees corresponds to a simple hypergraph. We give some conditions on the hypergraph degree sequence which guarantee that this probability is bounded below by a constant