2026-07-26T16:15:06Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:13087 2024-03-06T10:51:32Z ddc:004 open_access

Spread of Information and Diseases via Random Walks in Sparse Graphs Giakkoupis, George Saribekyan, Hayk Sauerwald, Thomas parallel random walks information dissemination infectious diseases 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]. Schloss Dagstuhl – Leibniz-Zentrum für Informatik George Giakkoupis and Hayk Saribekyan and Thomas Sauerwald 2020 Is Part Of LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.DISC.2020.9 urn:nbn:de:0030-drops-130873 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.9 eng https://creativecommons.org/licenses/by/3.0/legalcode