### Abstract

In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph. Coalescing random walks can be used to achieve consensus in distributed networks, and is the basis of the self-stabilizing mutual exclusion algorithm of Israeli and Jalfon [14]. Let G=(V,E), be an undirected, connected n vertex graph. Let C(n) be the expected time for all particles to coalesce, when initially one particle is located at each vertex. We study the problem of bounding the coalescence time C(n) for general classes of graphs. Our general result is, that C(n)= O(n/(v(1 - λ _{2}))), where v = Σ _{u∈V}d ^{2}(u)/(d ^{2} n), d(u) is the degree of vertex u, d is the average vertex degree, and λ ^{2} is the second eigenvalue of the transition matrix of the random walk. The parameter v is an indicator of the variability of vertex degrees: 1 ≤ v = O(n), with v = 1 for regular graphs. Our general bound on C(n) holds provided the maximum vertex degree is O( ^{m}1-ε), where m is the number of edges in the graph. This result implies, for example, that C(n)=O(n/(1-λ _{2})) for d-regular graphs with expansion parameterized by the eigenvalue gap 1-λ _{2}. The O(n/(v(1-λ _{2}))) bound is sublinear for some classes of graphs with skewed degree distributions. A system of coalescing particles where initially one particle is located at each vertex, corresponds to the following voter model. Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbour. The voting process can be used for leader election in a distributed context. Let E(C _{v}) be the expected time for voting to complete, that is, for a unique opinion to emerge. It is known that E(C _{v})=C(n ), so our results imply that E(C _{v}) = O(n/(v(1-λ _{2}))). We also investigate how the voting time improves when a vertex elicits more than one opinion at each step. In a model which we call min-voting, each vertex initially holds a distinct opinion drawn from a linearly ordered domain. At each step each vertex takes the opinions of two random neighbours and keeps the smaller. We show that for regular graphs with very good expansion properties, voting is completed in O(log n) time with high probability. This result can be viewed as an example of the "power of two choices" in distributed voting.

Original language | English |
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Title of host publication | PODC'12 - Proceedings of the 2012 ACM Symposium on Principles of Distributed Computing |

Pages | 47-55 |

Number of pages | 9 |

DOIs | |

Publication status | Published - Aug 20 2012 |

Event | 2012 ACM Symposium on Principles of Distributed Computing, PODC'12 - Madeira, Portugal Duration: Jul 16 2012 → Jul 18 2012 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Principles of Distributed Computing |
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### Other

Other | 2012 ACM Symposium on Principles of Distributed Computing, PODC'12 |
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Country | Portugal |

City | Madeira |

Period | 7/16/12 → 7/18/12 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software
- Hardware and Architecture
- Computer Networks and Communications

### Cite this

*PODC'12 - Proceedings of the 2012 ACM Symposium on Principles of Distributed Computing*(pp. 47-55). (Proceedings of the Annual ACM Symposium on Principles of Distributed Computing). https://doi.org/10.1145/2332432.2332440

**Coalescing random walks and voting on graphs.** / Cooper, Colin; Elsässer, Robert; Ono, Hirotaka; Radzik, Tomasz.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*PODC'12 - Proceedings of the 2012 ACM Symposium on Principles of Distributed Computing.*Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, pp. 47-55, 2012 ACM Symposium on Principles of Distributed Computing, PODC'12, Madeira, Portugal, 7/16/12. https://doi.org/10.1145/2332432.2332440

}

TY - GEN

T1 - Coalescing random walks and voting on graphs

AU - Cooper, Colin

AU - Elsässer, Robert

AU - Ono, Hirotaka

AU - Radzik, Tomasz

PY - 2012/8/20

Y1 - 2012/8/20

N2 - In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph. Coalescing random walks can be used to achieve consensus in distributed networks, and is the basis of the self-stabilizing mutual exclusion algorithm of Israeli and Jalfon [14]. Let G=(V,E), be an undirected, connected n vertex graph. Let C(n) be the expected time for all particles to coalesce, when initially one particle is located at each vertex. We study the problem of bounding the coalescence time C(n) for general classes of graphs. Our general result is, that C(n)= O(n/(v(1 - λ 2))), where v = Σ u∈Vd 2(u)/(d 2 n), d(u) is the degree of vertex u, d is the average vertex degree, and λ 2 is the second eigenvalue of the transition matrix of the random walk. The parameter v is an indicator of the variability of vertex degrees: 1 ≤ v = O(n), with v = 1 for regular graphs. Our general bound on C(n) holds provided the maximum vertex degree is O( m1-ε), where m is the number of edges in the graph. This result implies, for example, that C(n)=O(n/(1-λ 2)) for d-regular graphs with expansion parameterized by the eigenvalue gap 1-λ 2. The O(n/(v(1-λ 2))) bound is sublinear for some classes of graphs with skewed degree distributions. A system of coalescing particles where initially one particle is located at each vertex, corresponds to the following voter model. Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbour. The voting process can be used for leader election in a distributed context. Let E(C v) be the expected time for voting to complete, that is, for a unique opinion to emerge. It is known that E(C v)=C(n ), so our results imply that E(C v) = O(n/(v(1-λ 2))). We also investigate how the voting time improves when a vertex elicits more than one opinion at each step. In a model which we call min-voting, each vertex initially holds a distinct opinion drawn from a linearly ordered domain. At each step each vertex takes the opinions of two random neighbours and keeps the smaller. We show that for regular graphs with very good expansion properties, voting is completed in O(log n) time with high probability. This result can be viewed as an example of the "power of two choices" in distributed voting.

AB - In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph. Coalescing random walks can be used to achieve consensus in distributed networks, and is the basis of the self-stabilizing mutual exclusion algorithm of Israeli and Jalfon [14]. Let G=(V,E), be an undirected, connected n vertex graph. Let C(n) be the expected time for all particles to coalesce, when initially one particle is located at each vertex. We study the problem of bounding the coalescence time C(n) for general classes of graphs. Our general result is, that C(n)= O(n/(v(1 - λ 2))), where v = Σ u∈Vd 2(u)/(d 2 n), d(u) is the degree of vertex u, d is the average vertex degree, and λ 2 is the second eigenvalue of the transition matrix of the random walk. The parameter v is an indicator of the variability of vertex degrees: 1 ≤ v = O(n), with v = 1 for regular graphs. Our general bound on C(n) holds provided the maximum vertex degree is O( m1-ε), where m is the number of edges in the graph. This result implies, for example, that C(n)=O(n/(1-λ 2)) for d-regular graphs with expansion parameterized by the eigenvalue gap 1-λ 2. The O(n/(v(1-λ 2))) bound is sublinear for some classes of graphs with skewed degree distributions. A system of coalescing particles where initially one particle is located at each vertex, corresponds to the following voter model. Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbour. The voting process can be used for leader election in a distributed context. Let E(C v) be the expected time for voting to complete, that is, for a unique opinion to emerge. It is known that E(C v)=C(n ), so our results imply that E(C v) = O(n/(v(1-λ 2))). We also investigate how the voting time improves when a vertex elicits more than one opinion at each step. In a model which we call min-voting, each vertex initially holds a distinct opinion drawn from a linearly ordered domain. At each step each vertex takes the opinions of two random neighbours and keeps the smaller. We show that for regular graphs with very good expansion properties, voting is completed in O(log n) time with high probability. This result can be viewed as an example of the "power of two choices" in distributed voting.

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U2 - 10.1145/2332432.2332440

DO - 10.1145/2332432.2332440

M3 - Conference contribution

AN - SCOPUS:84864979372

SN - 9781450314503

T3 - Proceedings of the Annual ACM Symposium on Principles of Distributed Computing

SP - 47

EP - 55

BT - PODC'12 - Proceedings of the 2012 ACM Symposium on Principles of Distributed Computing

ER -