Row Stochastic Matrix Research Articles

Distributed Privacy Preserving Algorithm for Economic Dispatch Over Time-Varying Communication

The economic dispatch problem (EDP) is the solid rock to guarantee the stable operation of power systems, which is essentially an optimization problem, i.e., dispatching generators to meet the total demands with the minimum generation cost under the constraint of generation capacity. With the increasing complexity of power systems, higher requirements for the security and robustness of EDP algorithm have been put forward. On the one hand, the explicit information communication between the generators may bring serious security risks; on the other hand, in practical applications, there may be communication delays and communication link failures due to network instability. However, many existing EDP algorithms can hardly meet the requirements of both security and robustness. Thus, this article develops a fully distributed privacy preserving algorithm for EDP with line losses over time-varying and directed communication and extends it to non-ideal communication environments. In particular, the proposed algorithm only relies on the row-stochastic weight matrix, where each generator can allocate the information weight of its neighbor nodes locally, making the approach easier to implement. Moreover, the conditional noise is added to the auxiliary variable to achieve the generator's privacy preservation. The rigorous mathematical analysis reveals that the proposed approach can effectively find the optimal dispatch while achieving privacy preservation of generators under the assumption that the communication delays are arbitrarily significant but bounded. Finally, several cases are presented to testify the convergence and robustness of the proposed approach.

Privacy preserving distributed online projected residual feedback optimization over unbalanced directed graphs

The distributed online optimization (DOO) problem with privacy guarantees over an unbalanced directed graph is considered in this paper, where the cost function is explicitly unknown. To solve this problem, a distributed online one-point residual feedback optimization algorithm based on differential privacy is designed. For the objective function explicitly unknown, this algorithm employs the residual feedback to estimate the true gradient information. In addition, only the row stochastic adjacency matrix utilized releases the requirement of double stochastic weighting matrices, making the algorithm easier to implement on directed graphs. Theoretical results show that the algorithm not only protects the privacy information of the nodes but also achieves the same sublinear rate regret as the DOO algorithm based on two-point feedback. Moreover, our regret bound depends on weaker assumptions than the traditional DOO algorithm based on one-point feedback. Finally, the simulation results verify the effectiveness of the algorithm.

Agents Attraction Competition in an Extended Friedkin-Johnsen Social Network

Unlike many complex social networks investigated in the existing literatures concerning about the communication, consensus, or cooperative behaviors, this paper is devoted to an agents attraction competition problem in an extended Friedkin-Johnsen network. In this problem, two strategic agents compete to attract the nonstrategic agents to adopt opinions as closer to theirs as possible, by connecting with these nonstrategic agents and allocating their strengths on these connections. Therefore a 2-person zero-sum competitive game is characterized. We develop mathematical tools to provide some useful properties of the game, including the partial derivatives and positive definiteness of the payoff functions, based on which the Nash equilibrium of the game is analyzed. We rigorously examine that each player will allocate their strengths according to the susceptibilities, the intrinsic opinions and the costs of the nonstrategic agents, and the weighted column-sums of the row-stochastic matrix which describes the interpersonal influences of the network, if the network satisfies a condition. Otherwise, a Nash equilibrium seeking algorithm is proposed with simplex constraints, by which each strategic agent can asymptotically learn its optimal strategy. Finally, numerical examples are presented to illustrate the validity of the obtained results.

Asymptotics of the quantization errors for some Markov-type measures with complete overlaps

Let G be a directed graph with 2N vertices 1,2,…,2N. Let T=(Ti,j)(i,j)∈G be a family of contractive similitudes on Rq. We assume that Ti˜,j˜=Ti,j for every (i˜,j˜) in {(i,j),(i,j+N),(i+N,j),(i+N,j+N)}∩G. We denote by K the Mauldin-Williams fractal determined by T. Let χ be a positive probability vector and P a 2N×2N row-stochastic matrix. We denote by ν the Markov-type measure associated with χ and P. Let μ be the image measure of ν under the natural projection, which is supported on K. We consider the following two cases: 1. G has two strongly connected components consisting of N vertices; 2. G is strongly connected. With some assumptions for G and T, for case 1, we determine the quantization dimension for μ in terms of the spectral radius of a related matrix; we prove that the lower quantization coefficient is always positive and establish a necessary and sufficient condition for the upper one to be finite. For case 2, we express the quantization dimension in terms of a pressure-like function and prove that the upper and lower quantization coefficient are always positive and finite.

Distributed primal-dual method on unbalanced digraphs with row stochasticity

This study investigates distributed convex optimisation that accounts for the inequality constraints over unbalanced directed graphs. In distributed optimisation, agents exchange information on a network to obtain an optimal solution when they only know their own cost function. We propose a distributed primal-dual subgradient method based on a row stochastic weight matrix that is associated with a communication network. In the proposed method, the normalised left eigenvector of the weight matrix is estimated by a consensus algorithm. Then, the subgradient of the Lagrange function is scaled by the estimated values of the left eigenvector to compensate for the imbalance of the information flow in the network. We show that the pair of estimations for the primal and dual problems converge to an optimal primal-dual solution. We also show the relation between the convergence rate of the proposed algorithm and the step-size rule. A numerical example confirms the validity of the proposed method.

Distributed Data Collection Algorithm With Unknown Objective Function in Adaptive Duty-Cycle WSNs

With the rapid development of wireless sensor networks (WSNs), higher requirements are raised for the flexibility and applicability of data collection algorithms. Therefore, a novel distributed algorithm for data collection in adaptive duty-cycle WSNs is proposed in this article. On the one hand, the priority in data transmission is considered, in which the queues are used to realize the data storage and data transmission with different priorities and then a scheme for selecting the optimal transmission path for the sensor nodes is developed to ensure that high-priority data can be transmitted more efficiently. Next, a method that can adaptively adjust the duty cycle of the sensor nodes is designed to reduce the transmission delay, where each sensor can update its own duty cycle according to the amount of data with different priorities. On the other hand, the relation between the objective function and the variables in the data collection problem may be difficult to establish accurately or even unknown due to the complexity of WSNs, while the most existing data collection algorithms require the objective function to be explicitly expressed, making them inapplicable to the above scenarios. Hence, the data collection problem with unknown objective function in adaptive duty-cycle WSNs is also investigated in this article. Since the objective function cannot be established, a distributed optimization algorithm with randomized gradient-free is developed to solve that problem. In particular, the proposed algorithm solely relies on the row-stochastic weight matrix, where each sensor needs to communicate only with its neighbors to make decision and locally allocates the information weight of its neighbors, making the algorithm easier to implement. Under some reasonable assumptions, the explicit analysis for the convergence of the proposed algorithm is provided. Finally, the convergence and effectiveness of the proposed algorithm are substantiated through numerical simulations.

Distributed MPC algorithm with row‐stochastic weight matrix over non‐ideal time‐varying directed communication

Distributed model predictive control (DMPC) approaches have achieved remarkable results in complex multiple subsystems network applications, such as unmanned aerial vehicle and sensor control networks. However, most of the existing DMPC algorithms require that the communication network of subsystems is time-invariant or undirected with local constraints, by ignoring the cooperation of multiple subsystems with global constraint in the non-ideal communication network, which greatly limits the applicability of the algorithms. To this end, the authors develop a fully DMPC algorithm of linear system with global constraint over time-varying unbalanced directed communication. Considering the uncertainty of communication network, this algorithm can handle the non-ideal communication network (e.g. communication noise, communication delay). Specifically, the row-stochastic weight matrix is adopted to improve the independent controllability of subsystems. Under reasonable assumptions, it is proved that the algorithm can converge to the optimal solution while guaranteeing the recursive feasibility and exponential stability of closed-loop system. Finally, the simulation experiments are shown to substantiate the convergence and robustness of the proposed algorithm.

Privacy Preservation of Optimization Algorithm Over Unbalanced Directed Graph

This paper proposes an algorithm with the property of privacy preservation to solve the multi-agent optimization problem, where the communication topology is a fixed and strongly connected directed graph with a row stochastic weight matrix. Here, we consider the external eavesdropper attack model and take the gradient information of the objective function as the agent's privacy information. In this algorithm, each agent adds an additional state variable that interacts with it with a time-varying weight. And this new variable performs gradient iterative calculation. The original state variable interacts with the new variable and the original neighbors. It is proved that the algorithm converges to the optimal solution of the problem and at the same time achieves the purpose of privacy preservation. In addition, the algorithm neither requires additional hidden signals, nor does it increase a large amount of calculation. Finally, a simulation example is given to verify the performance of the algorithm.

Distributed continuous-time algorithm for resource allocation over unbalanced directed networks

The problem we aim to solve in this study is the distributed resource allocation problem (DRAP) over unbalanced directed networks, in which the local inequality constraint and global equality constraint are considered. On the basis of the fixed-time projection method, a distributed gradient algorithm is developed. The corresponding eigenvalue is obtained within fixed-time by tapping the row-stochastic or column-stochastic Laplacian matrix to overcome the unbalance of directed weights. Therefore, the optimal solution can be obtained asymptotically. Moreover, the proposed algorithm is initialization-free and its control parameters are constant. Finally, the effectiveness is illustrated by several case studies.

English

An m × n matrix R with nonnegative entries is called row stochastic if the sum of entries on every row of R is 1. Let Mm,n be the set of all m × n real matrices. For A, B ∈ Mm,n, we say that A is row Hadamard majorized by B (denoted by A ≺ RHB) if there exists an m × n row stochastic matrix R such that A = R ο B, where X ο Y is the Hadamard product (entrywise product) of matrices X, Y ∈ Mm,n. In this paper, we consider the concept of row Hadamard majorization as a relation on Mm,n and characterize the structure of all linear operators T: Mm,n → Mm,n preserving (or strongly preserving) row Hadamard majorization. Also, we find a theoretic graph connection with linear preservers (or strong linear preservers) of row Hadamard majorization, and we give some equivalent conditions for these linear operators on Mn.

Privacy-Preserving Distributed Online Optimization Over Unbalanced Digraphs via Subgradient Rescaling

In this article, we investigate a distributed online constrained optimization problem with differential privacy where the network is modeled by an unbalanced digraph with a row-stochastic adjacency matrix. To address such a problem, a distributed differentially private algorithm without introducing a trusted third-party is proposed to preserve the privacy of the participating nodes. Under mild conditions, we show that the proposed algorithm attains an O(log T) expected regret bound for strongly convex local cost functions, where T is the time horizon. Moreover, we remove the need for knowing the time horizon T in advance by adopting doubling trick scheme, and derive an O(√T) expected regret bound for general convex local cost functions. Our results coincide with the best theoretical regrets that can be achieved in the state-of-the-art algorithms. Finally, simulation results are conducted to validate the efficiency of our proposed algorithm.

Distributed constrained optimization for multi-agent systems over a directed graph with piecewise stepsize

In this paper, the distributed constrained optimization problem over a directed graph is considered. We assume that the digraph has a row-stochastic adjacency matrix, which corresponds to the case when agents assign weights to the received information individually. We present an algorithm that converges to the optimal solution even with a time-varying stepsize which is not attenuated to zero. The choice of stepsizes is relatively easy. The usable type of stepsizes is added. The diminishing or constant stepsizes can be used in our algorithm. Equality constraints and set constraints are also considered in this paper. Convergence analysis of the proposed algorithm relies on a conversion theorem between column-stochastic and row-stochastic matrices. Finally, the results of the numerical simulation are provided to verify the effectiveness of the proposed algorithm.

Joint measurability of POVM measurement assemblages and applications

The POVM measurement assemblage consisting of a set of positive operator-value measurements (POVMs) is an important tool for describing and studying the non-locality, quantum steering and quantum correlation of composite quantum systems. In this paper, we first introduce the definition of joint measurability of the POVM measurement assemblage $\mathcal{M}_S$ consisting of a finite number of POVMs and then prove that the joint measurability of $\mathcal{M}_S$ is equivalent to its compatibility. We give a general form of a POVM measurement which is jointly measurable with the one-rank projection measurement. It is proved that the POVM measurement assemblage $\mathcal{M}_S$ is jointly measurable if and only if the corresponding “row stochastic operator matrix" $\textbf{M}_S$ composed of $\mathcal{M}_S$ can be expressed as an “operator convex combination" of $\{0,I_S\}$—the row stochastic operator matrices. In particular, it is proved that the POVM measurement assemblage $\mathcal{M}(\textbf{n}_{0~},\textbf{n}_{1})$ induced by two three-dimensional real unit vectors $\textbf{n}_{i}(i=0,1)$ is jointly measurable if and only if $\textbf{n}_{0}=\pm\textbf{n}_{1}$. As an application, we obtain a necessary and sufficient condition under which a two-qubit quantum state is unsteerable with $\mathcal{M}(\textbf{n}_{0},\textbf{n}_{1})$.

Linear Preservers of Ultraweak, Row, and Weakly Hadamard Majorization on $$M_{mn}$$

Let $$X,Y\in M_{mn}$$ . We say that X is ultraweak Hadamard majorized by Y, denoted by $$X\prec _H^{uw} Y$$ , if there exists a matrix $$D=[d_{ij}]\in M_{mn}$$ , where $$0\le d_{ij}\le 1$$ , such that $$X=D\circ Y$$ . Also, we say that X is row Hadamard majorized (resp. weakly Hadamard majorized) by Y, denoted by $$X \prec ^{r}_{H} Y$$ (resp. $$X \prec ^{w}_{H} Y$$ ), if there exists a row stochastic matrix R (resp. doubly substochastic matrix D), such that $$X=R\circ Y$$ (resp. $$X=D\circ Y$$ ). In this paper, some properties of $$ \prec _H^{uw}, \prec ^{r}_{H}$$ and $$\prec ^{w}_{H}$$ on $$M_{mn}$$ are first obtained, and then, the (strong) linear preservers of $$ \prec _H^{uw}, \prec ^{r}_{H}$$ and $$\prec ^{w}_{H}$$ on $$M_{mn}$$ are characterized.

Linear operators preserving majorization of matrix tuples

In this paper, we consider weak, directional and strong matrix majorizations. Namely, for square matrices A and B of the same size we say that A is weakly majorized by B if there is a row stochastic matrix X such that A = XB. Further, A is strongly majorized by B if there is a doubly stochastic matrix X such that A = XB. Finally, A is directionally majorized by B if Ax is majorized by Bx for any vector x where the usual vector majorization is used. We introduce the notion of majorization of matrix tuples which is defined as a natural generalization of matrix majorizations: for a chosen type of majorization we say that one tuple of matrices is majorized by another tuple of the same size if every matrix of the “smaller” tuple is majorized by a matrix in the same position in the “bigger” tuple. We say that a linear operator preserves majorization if it maps ordered pairs to ordered pairs and the image of the smaller element does not exceed the image of the bigger one. This paper contains a full characterization of linear operators that preserve weak, strong or directional majorization of tuples of matrices and linear operators that map tuples that are ordered with respect to strong majorization to tuples that are ordered with respect to directional majorization. We have shown that every such operator preserves respective majorization of each component. For all types of majorization we provide counterexamples that demonstrate that the inverse statement does not hold, that is if majorization of each component is preserved, majorization of tuples may not.

Semi-global state synchronization for discrete-time multi-agent systems subject to actuator saturation and unknown nonuniform input delay

This paper studies semi-global state synchronization of discrete-time homogeneous networks with diffusive full-state coupling or partial-state coupling subject to actuator saturation and unknown nonuniform input delay. We assume that agents are at most critical unstable, that is the agents have all its eigenvalues in the closed unit disc. The communication network is associated with an undirected and weighted graph, which is represented by a row stochastic matrix. In this paper, we derive an upper bound for the input delay tolerance, which explicitly depends only on the agent dynamics. Moreover, for any unknown delay less than the upper bound, we propose a linear static protocol for MAS with full-state coupling and a linear dynamic protocol for MAS with partial-state coupling based on a low-gain methodology such that state synchronization is achieved among agents for any initial conditions in a priori given compact set.

Containment control of fractional-order multi-agent systems with time-varying delays

In this paper, containment control problems of networked fractional-order multi-agent systems with time-varying delays are studied. The normalized directed graphs are employed to characterize the communication topologies. Two sampled-data based containment control protocols are proposed, which can overcome the time-varying delays and switching topologies. It is interestingly found that the decays of the closed-loop systems correspond to the Mittag-Leffler function and its approximation, which are the extensions of the exponential function and its approximation, respectively. Based on the algebraic graph theory, the properties of row-stochastic matrix, and the relation between the topologies and the matrices, some conditions for containment control are established. For the fixed topology, a necessary and sufficient condition is obtained; and for the switching topology, a sufficient condition is provided. Finally, the theoretical results are illustrated by several numerical simulations.

Convergence Analysis of a Distributed Optimization Algorithm with a General Unbalanced Directed Communication Network

In this paper, we discuss a class of distributed constrained optimization problems in power systems where the target is to optimize the sum of all agents’ local convex objective functions over a general unbalanced directed communication network. Each local convex objective function is known exclusively to a single agent, and the agents’ variables are constrained to global coupling linear constraint and individual box constraints. To collaboratively solve the optimization problems, existing distributed methods mostly require the communication network to be balanced or have the knowledge of in-neighbors’ out-degree for all agents, which are quite restrictive and hardly inevitable in practical applications. In contrast, we investigate a novel distributed primal-dual augmented (sub)gradient algorithm which utilizes a row-stochastic matrix (does not need each agent to know its in-neighbors out-degree) and employs uncoordinated step-sizes, and yet exactly converges to the optimal solution over a general unbalanced directed communication network. Under the assumptions of the strong convexity and smoothness on the aggregate objective functions, it is proved that the algorithm geometrically converges to the optimal solution if the uncoordinated step-sizes do not exceed the upper bound. An explicit analysis for the convergence rate of the proposed algorithm is also characterized. To manifest effectiveness and applicability of the proposed algorithm, three case studies are presented to solve two practical problems in power systems.

Relative tempo of distributed averaging on networks

Distributed averaging has applications in a wide range of distributed algorithms. This paper introduces relative tempo for distributed averaging on networks, revealing a novel facet of the consensus problem. Intuitively, the relative tempo characterizes the relative rate of change in state of a pair of subgroups of agents in a network adopting distributed averaging. We show that there exists a quantitative link between the relative tempo and the eigenvector corresponding to the second largest eigenvalue of the row-stochastic matrix related to the underlying network. We further show that the dominant mode of a distributed averaging process can be efficiently estimated by utilizing the relative tempo. Distributed neighbor selection protocols are further proposed for the reduction of network topology, with potential applications in the context of fast consensus and eliminating the influence of network intruders. The correlation between relative tempo and network symmetry is also examined. Numerical simulation is provided to demonstrate the theoretical results.

Distributed optimization over directed graphs with row stochasticity and constraint regularity

This paper deals with an optimization problem over a network of agents, where the cost function is the sum of the individual (possibly nonsmooth) objectives of the agents and the constraint set is the intersection of local constraints. Most existing methods employing subgradient and consensus steps for solving this problem require the weight matrix associated with the network to be column stochastic or even doubly stochastic, conditions that can be hard to arrange in directed networks. Moreover, known convergence analyses for distributed subgradient methods vary depending on whether the problem is unconstrained or constrained, and whether the local constraint sets are identical or nonidentical and compact. The main goals of this paper are: (i) removing the common column stochasticity requirement; (ii) relaxing the compactness assumption, and (iii) providing a unified convergence analysis. Specifically, assuming the communication graph to be fixed and strongly connected and the weight matrix to (only) be row stochastic, a distributed projected subgradient algorithm and a variation of this algorithm are presented to solve the problem for cost functions that are convex and Lipschitz continuous. The key component of the algorithms is to adjust the subgradient of each agent by an estimate of its corresponding entry in the normalized left Perron eigenvector of the weight matrix. These estimates are obtained locally from an augmented consensus iteration using the same row stochastic weight matrix and requiring very limited global information about the network. Moreover, based on a regularity assumption on the local constraint sets, a unified analysis is given that can be applied to both unconstrained and constrained problems and without assuming compactness of the constraint sets or an interior point in their intersection. Further, we also establish an upper bound on the absolute objective error evaluated at each agent’s available local estimate under a nonincreasing step size sequence. This bound allows us to analyze the convergence rate of both algorithms.