Behrouz Touri

On Ergodicity, Infinite Flow, and Consensus in Random Models

We consider the ergodicity and consensus problem for a discrete-time linear dynamic model driven by random stochastic matrices, which is equivalent to studying these concepts for the product of such matrices. Our focus is on the model where the random matrices have independent but time-variant distribution. We introduce a new phenomenon, the infinite flow, and we study its fundamental properties and relations with the ergodicity and consensus. The central result is the infinite flow theorem establishing the equivalence between the infinite flow and the ergodicity for a class of independent random models, where the matrices in the model have a common steady state in expectation and a feedback property. For such models, this result demonstrates that the expected infinite flow is both necessary and sufficient for the ergodicity. The result is providing a deterministic characterization of the ergodicity, which can be used for studying the consensus and average consensus over random graphs.

Product of Random Stochastic Matrices

The paper deals with the convergence properties of the products of random (row-)stochastic matrices. The limiting behavior of such products is studied from a dynamical system point of view. In particular, by appropriately defining a dynamic associated with a given sequence of random (row-)stochastic matrices, we prove that the dynamics admits a class of time-varying Lyapunov functions, including a quadratic one. Then, we discuss a special class of stochastic matrices, a class P*, which plays a central role in this work. We then study cut-balanced chains and using some geometric properties of these chains, we characterize the stability of a subclass of cut-balanced chains. As a special consequence of this stability result, we obtain an extension of a central result in the non-negative matrix theory stating that, for any aperiodic and irreducible row-stochastic matrix A, the limit limk→∞ Ak exists and it is a rank one stochastic matrix. We show that a generalization of this result holds not only for sequences of stochastic matrices but also for independent random sequences of such matrices.

Product of random stochastic matrices and distributed averaging

Introduction.- Products of Stochastic Matrices and Averaging Dynamics.- Ergodicity of Random Chains.- Infinite Flow Stability.- Implications.- Absolute Infinite Flow Property.- Averaging Dynamics in General State Spaces.- Conclusion and Suggestions for Future Works.- Appendices.

On Approximations and Ergodicity Classes in Random Chains

We study the limiting behavior of a random dynamic system driven by a stochastic chain. Our interest is in the chains that are not necessarily ergodic but are decomposable into ergodic classes. To investigate the conditions under which the ergodic classes of a model can be identified, we introduce and study an l1 -approximation and infinite flow graph of the model. We show that the l1-approximations of random chains preserve certain limiting behavior. Using the l1-approximations, we show how the connectivity of the infinite flow graph is related to the structure of the ergodic groups of the model. Our main result of this paper provides conditions under which the ergodicity groups of the model can be identified by considering the connected components in the infinite flow graph. We provide two applications of our main result to random networks, namely broadcast over time-varying networks and networks with random link failure.

Multi-dimensional Hegselmann-Krause dynamics

We consider multi-dimensional Hegselmann-Krause model for opinion dynamics in discrete-time for a set of homogeneous agents. Using dynamic system point of view, we investigate stability properties of the dynamics and show its finite time convergence. The novelty of this work lies in the use of dynamic system approach and the development of Lyapunov-type tools for the analysis of the Hegselmann-Krause model. Furthermore, some new insights and results are provided. The results are valid for any norm that is used to define the neighbor sets.

On backward product of stochastic matrices

We study the ergodicity of backward product of stochastic and doubly stochastic matrices by introducing the concept of absolute infinite flow property. We show that this property is necessary for ergodicity of any chain of stochastic matrices, by defining and exploring the properties of a rotational transformation for a stochastic chain. Then, we establish that the absolute infinite flow property is equivalent to ergodicity for doubly stochastic chains. Furthermore, we develop a rate of convergence result for ergodic doubly stochastic chains. We also investigate the limiting behavior of a doubly stochastic chain and show that the product of doubly stochastic matrices is convergent up to a permutation sequence. Finally, we apply the results to provide a necessary and sufficient condition for the absolute asymptotic stability of a discrete linear inclusion driven by doubly stochastic matrices.

Consensus in the presence of an adversary

Abstract In this work, we consider two types of adversarial attacks on a network of nodes seeking to reach consensus. The first type involves an adversary that is capable of breaking a specific number of links at each time instant. In the second attack, the adversary is capable of corrupting the values of the nodes by adding a noise signal. In this latter case, we assume that the adversary is constrained by a power budget. We consider the optimization problem of the adversary and fully characterize its optimum strategy for each scenario.

Termination time of multidimensional Hegselmann-Krause opinion dynamics

We consider the Hegselmann-Krause model for opinion dynamics in higher dimensions. Our goal is to investigate the termination time of these dynamics, which has been investigated for a scalar case, but remained an open question for dimensions higher than one. We provide a polynomial upper bound for the termination time of the dynamics when the connectivity among the agents maintains a certain structure. Our approach is based on the use of an adjoint dynamics for the Hegselmann-Krause model and a Lyapunov comparison function that is defined in terms of the adjoint dynamics.

On existence of a quadratic comparison function for random weighted averaging dynamics and its implications

In this paper we study the stability and limiting behavior of discrete-time deterministic and random weighted averaging dynamics. We show that any such dynamics admits infinitely many comparison functions including a quadratic one. Using a quadratic comparison function, we establish the stability and characterize the set of equilibrium points of a broad class of random and deterministic averaging dynamics. This class includes a set of balanced chains, which itself contains many of the previously studied chains. Finally, we provide some implications of the developed results for products of independent random stochastic matrices.

On convergence rate of scalar Hegselmann-Krause dynamics

In this work, we derive a new upper bound on the termination time of the Hegselmann-Krause model for opinion dynamics. Using a novel method, we show that the process terminates in no more than O(n3) iterations, which improves the best known upper bound of O(n4) by a factor of n.