### Abstract

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.

Original language | English (US) |
---|---|

Article number | 5624571 |

Pages (from-to) | 1593-1605 |

Number of pages | 13 |

Journal | IEEE Transactions on Automatic Control |

Volume | 56 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2011 |

Externally published | Yes |

### Fingerprint

### Keywords

- Ergodicity
- infinite flow
- linear random model
- product of random matrices
- random consensus

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Control and Systems Engineering
- Computer Science Applications

### Cite this

*IEEE Transactions on Automatic Control*,

*56*(7), 1593-1605. [5624571]. https://doi.org/10.1109/TAC.2010.2091174

**On ergodicity, infinite flow, and consensus in random models.** / Touri, Behrouz; Nedich, Angelia.

Research output: Contribution to journal › Article

*IEEE Transactions on Automatic Control*, vol. 56, no. 7, 5624571, pp. 1593-1605. https://doi.org/10.1109/TAC.2010.2091174

}

TY - JOUR

T1 - On ergodicity, infinite flow, and consensus in random models

AU - Touri, Behrouz

AU - Nedich, Angelia

PY - 2011/7

Y1 - 2011/7

N2 - 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.

AB - 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.

KW - Ergodicity

KW - infinite flow

KW - linear random model

KW - product of random matrices

KW - random consensus

UR - http://www.scopus.com/inward/record.url?scp=79960122105&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960122105&partnerID=8YFLogxK

U2 - 10.1109/TAC.2010.2091174

DO - 10.1109/TAC.2010.2091174

M3 - Article

AN - SCOPUS:79960122105

VL - 56

SP - 1593

EP - 1605

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 7

M1 - 5624571

ER -