The number of graphs and a random graph with a given degree sequence

We consider the set of all graphs on n labeled vertices with prescribed degrees D=(d_1, ..., d_n). For a wide class of tame degree sequences D we prove a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0-1 matrices with prescribed row and column sums.Comment: 52 pages, minor improvement

Convex Geometry and its Applications

The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of other algorithms in computer science. High-dimensional geometry is an extremely active area of research: the participation of a considerable number of talented young mathematicians at this meeting is testament to the fact that the field is flourishing

Deterministic Approximation Algorithms for Volumes of Spectrahedra

We give a method for computing asymptotic formulas and approximations for the volumes of spectrahedra, based on the maximum-entropy principle from statistical physics. The method gives an approximate volume formula based on a single convex optimization problem of minimizing $-\log \det P$ over the spectrahedron. Spectrahedra can be described as affine slices of the convex cone of positive semi-definite (PSD) matrices, and the method yields efficient deterministic approximation algorithms and asymptotic formulas whenever the number of affine constraints is sufficiently dominated by the dimension of the PSD cone. Our approach is inspired by the work of Barvinok and Hartigan who used an analogous framework for approximately computing volumes of polytopes. Spectrahedra, however, possess a remarkable feature not shared by polytopes, a new fact that we also prove: central sections of the set of density matrices (the quantum version of the simplex) all have asymptotically the same volume. This allows for very general approximation algorithms, which apply to large classes of naturally occurring spectrahedra. We give two main applications of this method. First, we apply this method to what we call the "multi-way Birkhoff spectrahedron" and obtain an explicit asymptotic formula for its volume. This spectrahedron is the set of quantum states with maximal entanglement (i.e., the quantum states having univariant quantum marginals equal to the identity matrix) and is the quantum analog of the multi-way Birkhoff polytope. Second, we apply this method to explicitly compute the asymptotic volume of central sections of the set of density matrices

Volumes and Integer Points of Multi-Index Transportation Polytopes.

Counting the integer points of transportation polytopes has important applications in statistics for tests of statistical significance, as well as in several applications in combinatorics. In this dissertation, we derive asymptotic formulas for the number of integer and binary integer points in a wide class of multi-index transportation polytopes. A simple closed form approximation is given as the size of the corresponding arrays goes to infinity. A formula for the volume of $4$-index transportation polytopes is also given. We follow the approach of Barvinok and Hartigan to estimate the quantities through a type of local Central Limit Theorem. By carefully estimating eigenvalues and eigenvectors of certain quadratic forms, we are able to prove strong concentration results for the corresponding multivariate Gaussian random variables. We also estimate correlations between linear functions of Gaussian random variables to produce concentration results for the distribution of certain exponential functions. Combined, these techniques allow us to give a complete accounting of the integrals of several functions that are key to estimating the number of integer or binary integer points in multi-index transportation polytopes. As an additional result, we transform a standard concentration of measure on the sphere argument to a concentration result for Gaussian measures whose quadratic forms contain several small eigenvalues, allowing a similar calculation for the volume of multi-index transportation polytopes.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111456/1/dputnins_1.pd

Improved estimates for the number of non-negative integer matrices with given row and column sums

The number of non-negative integer matrices with given row and column sums appears in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations of various kinds. Here we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We also use this new estimate as the starting point for an improved numerical method for either counting or sampling matrices using sequential importance sampling. Code implementing our methods is provided.Comment: 25 pages, 6 figure