Fast Distributed Computation in Dynamic Networks via Random Walks

The paper investigates efficient distributed computation in dynamic networks in which the network topology changes (arbitrarily) from round to round. Our first contribution is a rigorous framework for design and analysis of distributed random walk algorithms in dynamic networks. We then develop a fast distributed random walk based algorithm that runs in $\tilde{O}(\sqrt{\tau \Phi})$ rounds (with high probability), where $\tau$ is the dynamic mixing time and $\Phi$ is the dynamic diameter of the network respectively, and returns a sample close to a suitably defined stationary distribution of the dynamic network. We also apply our fast random walk algorithm to devise fast distributed algorithms for two key problems, namely, information dissemination and decentralized computation of spectral properties in a dynamic network. Our next contribution is a fast distributed algorithm for the fundamental problem of information dissemination (also called as gossip) in a dynamic network. In gossip, or more generally, $k$-gossip, there are $k$ pieces of information (or tokens) that are initially present in some nodes and the problem is to disseminate the $k$ tokens to all nodes. We present a random-walk based algorithm that runs in $\tilde{O}(\min{n^{1/3}k^{2/3}(\tau \Phi)^{1/3}, nk})$ rounds (with high probability). To the best of our knowledge, this is the first $o(nk)$-time fully-distributed token forwarding algorithm that improves over the previous-best $O(nk)$ round distributed algorithm [Kuhn et al., STOC 2010], although in an oblivious adversary model. Our final contribution is a simple and fast distributed algorithm for estimating the dynamic mixing time and related spectral properties of the underlying dynamic network.

💡 Research Summary

The paper addresses the fundamental challenge of performing efficient distributed computation in networks whose topology changes arbitrarily from round to round. The authors first introduce a rigorous analytical framework for random‑walk based algorithms in such dynamic settings. They define a sequence of graphs $G_1,G_2,\dots$ that may be chosen by an oblivious adversary, and they consider a random walk in which, at each round, a node forwards a token uniformly at random to one of its current neighbors. Two key parameters capture the difficulty of the problem: the dynamic mixing time $\tau$, the number of rounds required for the walk to approach a stationary distribution despite the changing topology, and the dynamic diameter $\Phi$, the worst‑case length of a shortest path over all rounds.

The first technical contribution is a fast random‑walk sampling algorithm that, with high probability, produces a sample within $\tilde O(\sqrt{\tau\Phi})$ rounds of a suitably defined stationary distribution of the dynamic network. The analysis extends classic spectral techniques (Cheeger inequalities, eigenvalue gaps) to the time‑varying transition matrix, showing that the probability mass spreads at a rate proportional to $\sqrt{\tau\Phi}$. This result already improves over naïve approaches that would require $O(\tau\Phi)$ rounds.

Building on this primitive, the authors design a random‑walk based solution to the gossip (or $k$‑gossip) problem, where $k$ distinct tokens initially reside at arbitrary nodes and must be delivered to every node. Traditional token‑forwarding schemes need $O(nk)$ rounds in the CONGEST model. The new algorithm launches $k$ independent random walks simultaneously, each token following its own walk while respecting the per‑edge bandwidth limit. By carefully scheduling transmissions, using priority rules and random retries to avoid collisions, the algorithm achieves a high‑probability running time of
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