Set Of Stochastic Matrices Research Articles

Метод представления множеств стохастических матриц для многопараметрического анализа укрупненных и расширенных цепей Маркова

Метод представления множеств стохастических матриц для многопараметрического анализа укрупненных и расширенных цепей Маркова

Convergence of Infinite Products of Stochastic Matrices: A Graphical Decomposition Criterion

This technical note presents a convergence criterion for infinite products of stochastic matrices which is based on graphical decomposition of the associated graphs. We show that if the associated graphs of a set of stochastic matrices share a common graphical decomposition and the corresponding reduced graphs are rooted, then any infinite products of the given set of stochastic matrices is convergent. Specifically, we propose a numerical algorithm for finding the common graphical decomposition of the associated graphs, which has been proved to be polynomial-time fast. The proposed criterion can be applied directly to a series of classical results in distributed coordination algorithm.

A combinatorial necessary and sufficient condition for cluster consensus

In this letter, cluster consensus of discrete-time linear multi-agent systems is investigated. A set of stochastic matrices P is said to be a cluster consensus set if the system achieves cluster consensus for any initial state and any sequence of matrices taken from P. By introducing a cluster ergodicity coefficient, we present an equivalence relation between a range of characterization of cluster consensus set under some mild conditions including the widely adopted inter-cluster common influence. We obtain a combinatorial necessary and sufficient condition for a compact set P to be a cluster consensus set. This combinatorial condition is an extension of the avoiding set condition for global consensus, and can be easily checked by an elementary routine. As a byproduct, our result unveils that the cluster-spanning tree condition is not only sufficient but necessary in some sense for cluster consensus problems.

On pseudo-stochastic matrices and pseudo-positive maps

Stochastic matrices and positive maps in matrix algebras have proved to be very important tools for analysing classical and quantum systems. In particular they represent a natural set of transformations for classical and quantum states, respectively. Here we introduce the notion of pseudo-stochastic matrices and consider their semigroup property. Unlike stochastic matrices, pseudo-stochastic matrices are permitted to have matrix elements which are negative while respecting the requirement that the sum of the elements of each column is one. They also allow for convex combinations, and carry a Lie group structure which permits the introduction of Lie algebra generators. The quantum analog of a pseudo-stochastic matrix exists and is called a pseudo-positive map. They have the property of transforming a subset of quantum states (characterized by maximal purity or minimal von Neumann entropy requirements) into quantum states. Examples of qubit dynamics connected with ‘diamond’ sets of stochastic matrices and pseudo-positive maps are dealt with.

Load balancing for Markov chains with a specified directed graph

Given a strongly directed graph , let be the set of stochastic matrices whose directed graph is a spanning subgraph of . We consider the problem of finding the infimum of as ranges over the set of stationary distribution vectors of irreducible matrices in Using techniques from non-linear programming, combinatorial matrix theory and non-negative matrix theory, we find this infimum, which is given in terms the cardinality of a certain collection of vertex–disjoint cycles in . The situation that the infimum is attained as a minimum for the stationary distribution vector of some irreducible matrix in is also considered.

How to Decide Consensus? A Combinatorial Necessary and Sufficient Condition and a Proof that Consensus is Decidable but NP-Hard

A set of stochastic matrices ${\cal P}$ is a consensus set if for every sequence of matrices $P(1), P(2), \ldots$ whose elements belong to ${\cal P}$ and every initial state $x(0)$, the sequence of states defined by $x(t) = P(t) P(t-1) \cdots P(1) x(0)$ converges to a vector whose entries are all identical. In this paper, we introduce an “avoiding set condition” for compact sets of matrices and prove in our main theorem that this explicit combinatorial condition is both necessary and sufficient for consensus. We show that several of the conditions for consensus proposed in the literature can be directly derived from the avoiding set condition. The avoiding set condition is easy to check with an elementary algorithm, and so our result also establishes that consensus is algorithmically decidable. Direct verification of the avoiding set condition may require more than a polynomial time number of operations. This is, however, likely to be the case for any consensus checking algorithm since we also prove in th...

Dynamics on certain sets of stochastic matrices

We study iteration of polynomials on symmetric stochastic matrices. In particular, we focus on a certain one-parameter family of quadratic maps which exhibits chaotic behavior for a wide range of the parameters. The well-known dynamical behavior of the quadratic family on the interval, and its dependence on the parameter, is reproduced on the spectrum of the stochastic matrices. For certain subclasses of stochastic matrices the referred dynamical behavior is also obtained in the matrix entries. Since a stochastic matrix characterizes a Markov chain, we obtain a discrete dynamical system on the space of reversible Markov chains. Therefore, depending on the parameter, there are initial conditions for which the corresponding reversible Markov chains will lead under iteration to a fixed point, to a periodic point, or to an aperiodic point. Moreover, there are sensitivity to initial conditions and the coexistence of infinite repulsive periodic orbits, both features of chaos.

Stochastic matrices generated by entangled states of qubit-qutrit systems

Considering the spin-tomogram of entangled qubit-qutrit states, three sets of stochastic matrices are generated. These matrices realize representations of three semigroups, which are mutually related. The representations are characterized by different Bell numbers — one of them is the Cirelson bound \(2\sqrt 2 \), and the other one corresponds (up to accuracy 10−4) to \(\sqrt 5 \). The correlations corresponding to the latter number are stronger than the classical ones and weaker than purely quantum ones. The nonpositive Hermitian matrix obtained by the positive partial transpose density matrix of an entangled qubit-qutrit state is shown to create under local unitary transformations the semigroup of stochastic 4×4 matrices having the same bounds \(\sqrt 5 \) and \(2\sqrt 2 \), which are the characteristics of the semigroup obtained from the nonnegative density matrix of the entangled state.

Digraph-based conditioning for Markov chains

For an irreducible stochastic matrix T, we consider a certain condition number c( T), which measures the stability of the corresponding stationary distribution when T is perturbed. We characterize the strongly connected directed graphs D such that c( T) is bounded as T ranges over S D , the set of stochastic matrices whose directed graph is contained in D. For those digraphs D for which c( T) is bounded, we find the maximum value of c( T) as T ranges over S D .

On probabilistic automata in deterministic environment

We establish the basic notion of automata in environments and solve some of the basic problems in the general theory of automata in deterministic environment (ADE). ADE are, in general, stronger than probabilistic automata. In an effort to find when an ADE and a PA have the same capability, we introduce the concept of simulation of an ADE by a PA. Every ADE with a finite environment set can be simulated by a PA (Theorem 1). There are sets which cannot be defined by PA but can be defined by ADE with finite environment sets and, hence, can be simulated by a PA. ADE for which the environment can be reduced to a finite set can be simulated by a PA (Theorem 2). The set of stochastic matrices is a semi-group. We investigate the case where the environment set is also a semi-group and the transition function is a homomorphism between the semi-groups. If the semi-group environment set can be generated by a finite set, then the ADE can be simulated by a PA (Theorem 3). The continuous time PA of Knast [2] is seen to be a special case of the ADE.

Quasifiniteness of a set of stochastic matrices

Quasifiniteness of a set of stochastic matrices

Some Notes on Stability Problems of Probabilistic Automata

Abstract The behavior of a probabilistic automaton is essentially characterized by products of matrices selected from a given finite set of stochastic matrices. It is of interest to know under what conditions these matrix products are stable against perturbations of the entries in the matrices.

The Decision Problems of Definite Stochastic Automata

A k-definite stochastic automaton is defined as a finite stochastic automaton in which the internal state probability distribution depends only on the last k input symbols for some definite integer k. Whether or not a stochastic automaton is definite depends on its state transition matrices. A set of stochastic matrices is called definite of order k by Paz if, for a fixed integer k, any product of k or more of the matrices is a matrix with all rows equal. A study is made of conditions in which linearly independent row vectors in the component matrices will result in identical row vectors in the product matrix. It is established that a definite stochastic automaton with n internal states and with highest rank of its transition matrices $n - m$ is at least ${{(n - m)} / m}$-definite. A time saving decision procedure for the definiteness of stochastic automata is presented and illustrated.

Some algebraic properties of sets of stochastic matrices

This paper presents some algebraic properties of some certain sets of stochastic matrices of probabilistic automata from the standpoint of their characteristic values. At first, a definite set and a quasidefinite set introduced by Paz (1965) , are algebraically investigated. Then we introduce new concepts, a periodic set and a quasiperiodic set, which are defined among stochastic matrices with a period t . The results obtained from a periodic set and a quasiperiodic set have a remarkable correspondence to those from a definite set and a quasidefinite set, respectively, and the formers in the case of the period t = 1 coincide with the latters. It is known that a definite set and a quasidefinite set are the special case and the extended concept of actual automata, respectively, and a definite set can be associated with a definite automaton. On the other hand, it is shown that a periodic set can be associated with a bounded transient automation which is constructed from autonomous and definite machines.