Maximum-Entropy Representations in Convex Polytopes: Applications to Spatial Interaction

1989 ◽  
Vol 21 (11) ◽  
pp. 1541-1546 ◽  
Author(s):  
P B Slater
Keyword(s):  
Maximum Entropy ◽  
Convex Combination ◽  
Spatial Interaction ◽  
Stochastic Matrix ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Extreme Points ◽  
Doubly Stochastic ◽  
Interaction Modeling

Of all representations of a given point situated in a convex polytope, as a convex combination of extreme points, there exists one for which the probability or weighting distribution has maximum entropy. The determination of this multiplicative or exponential distribution can be accomplished by inverting a certain bijection—developed by Rothaus and by Bregman—of convex polytopes into themselves. An iterative algorithm is available for this procedure. The doubly stochastic matrix with a given set of transversals (generalized diagonal products) can be found by means of this method. Applications are discussed of the Rothaus -Bregman map to a proof of Birkhoff's theorem and to the calculation of trajectories of points leading to stationary or equilibrium values of the generalized permanent, in particular in spatial interaction modeling.

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

On the Extreme Rays of the Metric Cone

1980 ◽  
Vol 32 (1) ◽  
pp. 126-144 ◽  
Author(s):  
David Avis
Keyword(s):  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Convex Polyhedra ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Image Position ◽  
Polyhedral Convex Set ◽  
Extreme Rays

A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equationsBirkhoff [2] and Von Neuman have shown that the extreme points of this bounded polyhedron are just the n × n permutation matrices. The importance of this result for mathematical programming is that it tells us that the maximum of any linear form over P will occur for a permutation matrix X.

scholarly journals The computation of stationary distributions of Markov chains through perturbations

1991 ◽  
Vol 4 (1) ◽  
pp. 29-46 ◽  
Author(s):  
Jeffery J. Hunter
Keyword(s):  
Stochastic Matrix ◽  
Linear Equations ◽  
Probability Vector ◽  
Doubly Stochastic ◽  
Rank One ◽  
Systems Of Linear Equations ◽  
Stationary Probability Vector ◽  
Irreducible Markov Chain ◽  
Algorithmic Procedure

An algorithmic procedure for the determination of the stationary distribution of a finite, m-state, irreducible Markov chain, that does not require the use of methods for solving systems of linear equations, is presented. The technique is based upon a succession of m, rank one, perturbations of the trivial doubly stochastic matrix whose known steady state vector is updated at each stage to yield the required stationary probability vector.

An Elementary Proof of Johnson–Dulmage–Mendelsohn's Refinement of Birkhoff's Theorem on Doubly Stochastic Matrices

1979 ◽  
Vol 22 (1) ◽  
pp. 81-86 ◽  
Author(s):  
Akihiro Nishi
Keyword(s):  
Upper Bound ◽  
Elementary Proof ◽  
Convex Combination ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices ◽  
Birkhoff’S Theorem

SummaryA purely combinatorial and elementary proof of Johnson-Dulmage-Mendelsohn's theorem, which gives a quite sharp upper bound on the number of permutation matrices needed for representing a doubly stochastic matrix by their convex combination, is given.

scholarly journals A Nonlinear Convergence Consensus: Extreme Doubly Stochastic Quadratic Operators for Multi-Agent Systems

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 540 ◽  
Author(s):  
Rawad Abdulghafor ◽  
Sultan Almotairi ◽  
Hamad Almohamedh ◽  
Badr Almutairi ◽  
Abdullah Bajahzar ◽  
...  
Keyword(s):  
Stochastic Matrix ◽  
Extreme Points ◽  
Interaction Network ◽  
Low Complexity ◽  
Fast Time ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Common Value ◽  
Doubly Stochastic ◽  
Multi Agent

We investigate a novel nonlinear consensus from the extreme points of doubly stochastic quadratic operators (EDSQO), based on majorization theory and Markov chains for time-varying multi-agent distributed systems. We describe a dynamic system that has a local interaction network among agents. EDSQO has been applied for distributed agent systems, on a finite dimensional stochastic matrix. We prove that multi-agent systems converge at a center (common value) via the extreme waited value of doubly stochastic quadratic operators (DSQO), which are only 1 or 0 or 1/2 1 2 if the exchanges of each agent member has no selfish communication. Applying this rule means that the consensus is nonlinear and low-complexity computational for fast time convergence. The investigated nonlinear model of EDSQO follows the structure of the DeGroot linear (DGL) consensus model. However, EDSQO is nonlinear and faster convergent than the DGL model and is of lower complexity than DSQO and cubic stochastic quadratic operators (CSQO). The simulation result and theoretical proof are illustrated.

Maximum Entropy Convex Decompositions of Doubly Stochastic and Nonnegative Matrices

1987 ◽  
Vol 19 (3) ◽  
pp. 403-407 ◽  
Author(s):  
P B Slater
Keyword(s):  
Maximum Entropy ◽  
Extreme Points ◽  
Social Classes ◽  
Nonnegative Matrices ◽  
Iterative Proportional Fitting ◽  
Mathematical Structures ◽  
Stochastic Version ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Fitting Procedure

Two maximum entropy convex decompositions are computed with the use of the iterative proportional fitting procedure. First, a doubly stochastic version of a 5 × 5 British social mobility table is represented as the sum of 120 5 × 5 permutation matrices. The most heavily weighted permutations display a bandwidth form, indicative of relatively strong movements within social classes and between neighboring classes. Then the mobility table itself is expressed as the sum of 6 985 5 × 5 transportation matrices—possessing the same row and column sums as the mobility table. A particular block-diagonal structure is evident in the matrices assigned the greatest weight. The methodology can be applied as well to the representation of other nonnegative matrices in terms of their extreme points, and should be extendable to higher-order mathematical structures—for example, operators and functions.

scholarly journals Diagonal Sums of Doubly Substochastic Matrices

2019 ◽  
Vol 35 ◽  
pp. 42-52
Author(s):  
Lei Cao ◽  
Zhi Chen ◽  
Xuefeng Duan ◽  
Selcuk Koyuncu ◽  
Huilan Li
Keyword(s):  
Symmetric Group ◽  
Stochastic Matrix ◽  
Convex Polytope ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices

Let $\Omega_n$ denote the convex polytope of all $n\times n$ doubly stochastic matrices, and $\omega_{n}$ denote the convex polytope of all $n\times n$ doubly substochastic matrices. For a matrix $A\in\omega_n$, define the sub-defect of $A$ to be the smallest integer $k$ such that there exists an $(n+k)\times(n+k)$ doubly stochastic matrix containing $A$ as a submatrix. Let $\omega_{n,k}$ denote the subset of $\omega_n$ which contains all doubly substochastic matrices with sub-defect $k$. For $\pi$ a permutation of symmetric group of degree $n$, the sequence of elements $a_{1\pi(1)},a_{2\pi(2)}, \ldots, a_{n\pi(n)}$ is called the diagonal of $A$ corresponding to $\pi$. Let $h(A)$ and $l(A)$ denote the maximum and minimum diagonal sums of $A\in \omega_{n,k}$, respectively. In this paper, existing results of $h$ and $l$ functions are extended from $\Omega_n$ to $\omega_{n,k}.$ In addition, an analogue of Sylvesters law of the $h$ function on $\omega_{n,k}$ is proved.

A short proof of the Farahat-Mirsky refinement of Birkhoff's theorem on doubly-stochastic matrices

1961 ◽  
Vol 57 (3) ◽  
pp. 681-681 ◽  
Author(s):  
J. M. Hammersley
Keyword(s):  
Short Proof ◽  
Convex Combination ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

A doubly-stochastic matrix is an n × n matrix with non-negative elements such that each row and each column sums to 1. A permutation matrix is the result of permuting the rows of the unit n × n matrix. Birkhoff's theorem states that the doubly-stochastic matrices constitute the convex hull of the permutation matrices. Using Birkhoff's theorem, Farahat and Mirsky (1) showed that each doubly-stochastic matrix could be expressed as a convex combination of n2 − 2n + 2 permutation matrices, though not in general of fewer. Given Birkhoff's theorem, the Farahat-Mirsky refinement can also be proved quite shortly as follows.

scholarly journals The semigroup of doubly-stochastic matrices

1966 ◽  
Vol 7 (4) ◽  
pp. 178-183 ◽  
Author(s):  
H. K. Farahat
Keyword(s):  
Stochastic Matrix ◽  
Matrix Multiplication ◽  
Symmetric Groups ◽  
Maximal Subgroups ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Yield Information ◽  
Doubly Stochastic Matrices

The set Dn of all n × n doubly-stochastic matrices is a semigroup with respect to ordinary matrix multiplication. This note is concerned with the determination of the maximal subgroups of Dn. It is shown that the number of subgroups is finite, that each subgroup is finite and is in fact isomorphic to a direct product of symmetric groups. These results are applied in § 3 to yield information about the least number of permutation matrices whose convex hull contains a given doubly-stochastic matrix.

scholarly journals Extreme Points of Certain Transportation Polytopes with Fixed Total Sums

2021 ◽  
Vol 37 ◽  
pp. 256-271
Author(s):  
Zhi Chen ◽  
Zelin Zhu ◽  
Jiawei Li ◽  
Lizhen Yang ◽  
Lei Cao
Keyword(s):  
Convex Polytopes ◽  
Convex Polytope ◽  
Extreme Points ◽  
Nonnegative Matrices ◽  
New Class ◽  
Term Rank ◽  
Transportation Polytope ◽  
Transportation Polytopes ◽  
The Given ◽  
Sum Vector

Transportation matrices are $m\times n$ nonnegative matrices with given row sum vector $R$ and column sum vector $S$. All such matrices form the convex polytope $\mathcal{U}(R,S)$ which is called a transportation polytope and its extreme points have been classified. In this article, we consider a new class of convex polytopes $\Delta(\bar{R},\bar{S},\sigma)$ consisting of certain transportation polytopes satisfying that the sum of all elements is $\sigma$, and the row and column sum vectors are dominated componentwise by the given positive vectors $\bar{R}$ and $\bar{S}$, respectively. We characterize the extreme points of $\Delta(\bar{R},\bar{S},\sigma)$. Moreover, we give the minimal term rank and maximal permanent of $\Delta(\bar{R},\bar{S},\sigma)$.

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