When infinite flow is sufficient for ergodicity

Behrouz Touri, Angelia Nedić

Research output: Chapter in Book/Report/Conference proceedingConference contribution

17 Scopus citations


We consider the consensus and ergodicity for a random linear discrete-time system driven by stochastic matrices. We focus on independent models with certain properties, and we study the ergodicity and consensus of such random models through a novel property, termed infinite flow property. Our key result is the establishment that for a class of independent random models, this property is a necessary and sufficient condition for ergodicity. Using this result, we show that the ergodicity of these models and the ergodicity of their expected models are the same. The result provides us with new tools for studying various aspects of dynamic networks and beyond. We demonstrate the potential use of our key result through several different applications. In particular, we apply it to provide a generalization of the randomized gossip algorithm and to study a consensus over a dynamic network with link failures. Also, we use the result to investigate necessary and sufficient conditions for the ergodicity of an equal-neighbor average algorithm on Erdös-Rényi random graphs. Finally, we demonstrate that our result can be employed to provide an alternative proof of the second Borel-Cantelli lemma.

Original languageEnglish (US)
Title of host publication2010 49th IEEE Conference on Decision and Control, CDC 2010
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages8
ISBN (Print)9781424477456
StatePublished - 2010
Externally publishedYes
Event49th IEEE Conference on Decision and Control, CDC 2010 - Atlanta, United States
Duration: Dec 15 2010Dec 17 2010

Publication series

NameProceedings of the IEEE Conference on Decision and Control
ISSN (Print)0743-1546
ISSN (Electronic)2576-2370


Conference49th IEEE Conference on Decision and Control, CDC 2010
Country/TerritoryUnited States


  • Borel-Cantelli lemma
  • Ergodicity, product of random stochastic matrices
  • Gossip algorithm
  • Linear random model
  • Link failure

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization


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