research-article

Coalescing-branching random walks on graphs

Authors:

Gopal Pandurangan,

Rajmohan Rajaraman,

Scott RocheAuthors Info & Claims

SPAA '13: Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures

Pages 176 - 185

https://doi.org/10.1145/2486159.2486197

Published: 23 July 2013 Publication History

Abstract

We study a distributed randomized information propagation mechanism in networks we call the coalescing-branching random walk (cobra walk, for short). A cobra walk is a generalization of the well-studied "standard" random walk, and is useful in modeling and understanding the Susceptible-Infected Susceptible (SIS)-type of epidemic processes in networks. It can also be helpful in performing light-weight information dissemination in resource-constrained networks. A cobra walk is parameterized by a branching factor k. The process starts from an arbitrary node, which is labeled active for step 1. (For instance, this could be a node that has a piece of data, rumor, or a virus.) In each step of a cobra walk, each active node chooses k random neighbors to become active for the next step ("branching"). A node is active for step t + 1 only if it is chosen by an active node in step t ("coalescing"). This results in a stochastic process in the underlying network with properties that are quite different from both the standard random walk (which is equivalent to the cobra walk with branching factor 1) as well as other gossip-based rumor spreading mechanisms.

We focus on the cover time of the cobra walk, which is the number of steps for the walk to reach all the nodes, and derive almost-tight bounds for various graph classes. Our main technical result is an O(log2 n) high probability bound for the cover time of cobra walks on expanders, if either the expansion factor or the branching factor is sufficiently large; we also obtain an O(log n) high probability bound for the partial cover time, which is the number of steps needed for the walk to reach at least a constant fraction of the nodes. We show that the cobra walk takes O(n log n) steps on any n-node tree for k ≥ 2, and Õ(n^1/d) steps on a d-dimensional grid for k ≥ 2, with high probability.

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Cited By

Cooper CRadzik TRivera N(2019)New Cover Time Bounds for the Coalescing-Branching Random Walk on GraphsACM Transactions on Parallel Computing10.1145/33642066:3(1-24)Online publication date: 2-Nov-2019
https://dl.acm.org/doi/10.1145/3364206
Berenbrink PGiakkoupis GKling PCzumaj A(2018)Tight bounds for coalescing-branching random walks on regular graphsProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175417(1715-1733)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175417
Cooper CRadzik TRivera NScheideler CHajiaghayi M(2017)Improved Cover Time Bounds for the Coalescing-Branching Random Walk on GraphsProceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3087556.3087564(305-312)Online publication date: 24-Jul-2017
https://dl.acm.org/doi/10.1145/3087556.3087564
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Index Terms

Coalescing-branching random walks on graphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
  2. Probability and statistics

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cover image ACM Conferences

SPAA '13: Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures

July 2013

348 pages

ISBN:9781450315722

DOI:10.1145/2486159

General Chair:
Guy Blelloch
Carnegie Mellon University, USA
,
Program Chair:
Berthold Vöcking
RWTH Aachen University, Germany

Copyright © 2013 ACM.

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Published: 23 July 2013

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Cited By

Cooper CRadzik TRivera N(2019)New Cover Time Bounds for the Coalescing-Branching Random Walk on GraphsACM Transactions on Parallel Computing10.1145/33642066:3(1-24)Online publication date: 2-Nov-2019
https://dl.acm.org/doi/10.1145/3364206
Berenbrink PGiakkoupis GKling PCzumaj A(2018)Tight bounds for coalescing-branching random walks on regular graphsProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175417(1715-1733)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175417
Cooper CRadzik TRivera NScheideler CHajiaghayi M(2017)Improved Cover Time Bounds for the Coalescing-Branching Random Walk on GraphsProceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3087556.3087564(305-312)Online publication date: 24-Jul-2017
https://dl.acm.org/doi/10.1145/3087556.3087564
Cooper CRadzik TRivera NGiakkoupis G(2016)The Coalescing-Branching Random Walk on Expanders and the Dual Epidemic ProcessProceedings of the 2016 ACM Symposium on Principles of Distributed Computing10.1145/2933057.2933119(461-467)Online publication date: 25-Jul-2016
https://dl.acm.org/doi/10.1145/2933057.2933119
Dediu AM. Matos JMartín-Vide C(2016)Theory and Practice of Discrete Interacting Agents ModelsEmergent Computation10.1007/978-3-319-46376-6_19(419-433)Online publication date: 5-Nov-2016
https://doi.org/10.1007/978-3-319-46376-6_19

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