Distributed computation in dynamic networks via random walks

Enabling Robust and Efficient Distributed Computation in Dynamic Peer-to-Peer Networks

“…is regular (i.e., all nodes have ∆ degree with no missing edges) and there is no churn in this process. Regularity is obtained by introducing the so called ghost edges (which are simply virtual edges that take the place of missing edges in G i -they are determined in lines 10-14 of Algorithm IV.2) which allow us to apply techniques from [57]. Tokens that travel through ghost edges are lost (and ghost edges are adversarially determined, since missing edges are) and some bias is also introduced.…”

“…Algorithm III.1) on G. Our goal is to leverage the result of [57] in which it is shown that random walks mix in dynamic networks without churn. Towards this goal, for a given G, we construct another sequence of graphsḠ = (Ḡ 1 ,Ḡ 2 , .…”

“…Algorithm IV.2). For the purpose of analysis, we simulate the random walks based sampling algorithm onḠ so that we can apply the result from [57] (cf. Lemma 5, sinceḠ i is an expander it has a second largest eigenvalue bounded away from 1), and show how to translate the result back to G in a suitable manner (cf.…”

“…Tokens that never passed along ghost edges are called real tokens. Therefore, we can apply the results from [57] to get the following lemma.…”

“…[57]). For the dynamic graphḠ = (Ḡ i ) i 1 described above, there is a τ ∈ Θ(log n) that we call its mixing time such that, if a node received a random walk token t in round i that originated in round i = i − τ , then, the probability that t originated from any particular node inV i is within [1/n − 1/n c , 1/n + 1/n c ] for any fixed c that depends on the constant hidden in the asymptotic description of τ .…”