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.identically distributed (i.i.d.) random models. We then investigate the random models with a feedback property and the models with a common steady state for the expected matrices. Both of these properties have been used in the analysis of consensus models, but a deeper understanding of their roles has not been observed. We classify feedback property in three basic types from weak to strong and show some relations for them. Then, we study the models with a common steady state in expectation. By putting all the pieces together, we show that the ergodicity of the model is equivalent to the infinite flow property for a class of independent random models with feedback property and a common steady state in expectation, as given in infinite flow theorem (Theorem 7). The infinite flow theorem also establishes the equivalence between the infinite flow properties of the model and the expected model. Furthermore, the theorem also shows the equivalence between the ergodicity of the model and the ergodicity of the expected model. As such, the theorem provides a novel deterministic characterization of the ergodicity, thus rendering another tool for studying the consensus over random networks and convergence of random consensus algorithms.The main contributions of this paper include: 1) the equivalence of the ergodicity of the model and the expected model for a class of independent random models with a feedback property and a common steady state in expectation; 2) the new insights and understanding of the ergodicity and consensus events over random networks brought to light through a new phenomena of infinite flow event, which to the best of our knowledge has not been known prior to this work; 3) novel comprehensive study of the fundamental properties of the consensus and ergodicity events for general class of independent random models; 4) new insights into the role of feedback property and the role of a common steady state in expectation for the ergodicity and consensus.The study of the random product of stochastic matrices dates back to the earl...
We study distributed non-convex optimization on a time-varying multi-agent network. Each node has access to its own smooth local cost function, and the collective goal is to minimize the sum of these functions. The perturbed push-sum algorithm was previously used for convex distributed optimization. We generalize the result obtained for the convex case to the case of non-convex functions. Under some additional technical assumptions on the gradients we prove the convergence of the distributed push-sum algorithm to some critical point of the objective function. By utilizing perturbations on the update process, we show the almost sure convergence of the perturbed dynamics to a local minimum of the global objective function, if the objective function has no saddle points. Our analysis shows that this perturbed procedure converges at a rate of O(1/t).
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 introduce balanced chains and using some geometric properties of these chains, we characterize the stability of a subclass of 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 lim k→∞ A k 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.
This thesis is mainly concerned with the study of product of random stochastic matrices and random weighted averaging dynamics. It will be shown that a generalization of a fundamental result in the theory of ergodic Markov chains not only holds for inhomogeneous chains of stochastic matrices, but also remains true for random stochastic matrices. To do this, the concept of infinite flow property will be introduced for a deterministic chain of stochastic matrices and it will be proven that it is necessary for ergodicity of any stochastic chain. This result will further be extended to ergodic classes, through the development of the concept of the infinite flow graph and ℓ 1 -approximation technique.For the converse implications, the product of stochastic matrices will be studied in the more general setting of random adapted stochastic chains. Using a result of A. Kolmogorov, it will be shown that any averaging dynamics admits infinitely many comparison functions including a quadratic one. By identifying the decrease of the quadratic comparison function along the trajectories of the dynamics, it will be proven that under general assumptions on a random chain, the chain is infinite flow stable, i.e. the product of random stochastic matrices is convergent almost surely and, also, the limiting matrices admit certain structures that can be deduced from the infinite flow graph of the chain. It will be shown that a general class of stochastic chains, the balanced chains with feedback property, satisfy the conditions of this result.Some implications of the developed results for products of independent random stochastic matrices will be provided. Furthermore, it will be proven that under general conditions, an independent random chain and its expected chain exhibit the same ergodic behavior. It will be proven that an extension of a well-known result in the theory of homogeneous Markov chains holds for a sequence of inhomogeneous stochastic matrices. Then, link-failure models for averaging dynamics will be introduced and it will be shown that under general conditions, link failure does not affect the limiting behavior of averaging dynamics. Then, the application of the developed methods to the study of Hegselmann-Krause model will be considered. Using the developed results, an upper bound O(m 4 ) will be established for the Hegselmann-Krause dynamics, which is an improvement to the previously known bound ii O(m 5 ). As a final application for the developed tools, an alternative proof for the second Borel-Cantelli lemma will be provided. Motivated by the infinite flow property, a stronger one, the absolute infinite flow property will be introduced. It will be shown that this stronger property is in fact necessary for ergodicity of any stochastic chain. Moreover, the equivalency of the absolute infinite flow property with ergodicity of doubly stochastic chains will be proven. These results will be driven by introduction and study of the rotational transformation of a stochastic chain.Finally, motivated by the study of ...
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.
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