License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.DISC.2020.9
URN: urn:nbn:de:0030-drops-130873
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/13087/
Giakkoupis, George ;
Saribekyan, Hayk ;
Sauerwald, Thomas
Spread of Information and Diseases via Random Walks in Sparse Graphs
Abstract
We consider a natural network diffusion process, modeling the spread of information or infectious diseases. Multiple mobile agents perform independent simple random walks on an n-vertex connected graph G. The number of agents is linear in n and the walks start from the stationary distribution. Initially, a single vertex has a piece of information (or a virus). An agent becomes informed (or infected) the first time it visits some vertex with the information (or virus); thereafter, the agent informs (infects) all vertices it visits. Giakkoupis et al. [George Giakkoupis et al., 2019] have shown that the spreading time, i.e., the time before all vertices are informed, is asymptotically and w.h.p. the same as in the well-studied randomized rumor spreading process, on any d-regular graph with d = Ω(log n). The case of sub-logarithmic degree was left open, and is the main focus of this paper.
First, we observe that the equivalence shown in [George Giakkoupis et al., 2019] does not hold for small d: We give an example of a 3-regular graph with logarithmic diameter for which the expected spreading time is Ω(log² n/log log n), whereas randomized rumor spreading is completed in time Θ(log n), w.h.p. Next, we show a general upper bound of Õ(d ⋅ diam(G) + log³ n /d), w.h.p., for the spreading time on any d-regular graph. We also provide a version of the bound based on the average degree, for non-regular graphs. Next, we give tight analyses for specific graph families. We show that the spreading time is O(log n), w.h.p., for constant-degree regular expanders. For the binary tree, we show an upper bound of O(log n⋅ log log n), w.h.p., and prove that this is tight, by giving a matching lower bound for the cover time of the tree by n random walks. Finally, we show a bound of O(diam(G)), w.h.p., for k-dimensional grids (k ≥ 1 is constant), by adapting a technique by Kesten and Sidoravicius [Kesten and Sidoravicius, 2003; Kesten and Sidoravicius, 2005].
BibTeX - Entry
@InProceedings{giakkoupis_et_al:LIPIcs:2020:13087,
author = {George Giakkoupis and Hayk Saribekyan and Thomas Sauerwald},
title = {{Spread of Information and Diseases via Random Walks in Sparse Graphs}},
booktitle = {34th International Symposium on Distributed Computing (DISC 2020)},
pages = {9:1--9:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-168-9},
ISSN = {1868-8969},
year = {2020},
volume = {179},
editor = {Hagit Attiya},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/13087},
URN = {urn:nbn:de:0030-drops-130873},
doi = {10.4230/LIPIcs.DISC.2020.9},
annote = {Keywords: parallel random walks, information dissemination, infectious diseases}
}
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Keywords: |
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parallel random walks, information dissemination, infectious diseases |
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Collection: |
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34th International Symposium on Distributed Computing (DISC 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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07.10.2020 |
