[Paper Review] Fast Distributed Computation in Dynamic Networks via Random Walks

Atish Das Sarma, Anisur Rahaman Molla+1 more|arXiv (Cornell University)|May 24, 2012

Distributed systems and fault tolerance28 references4 citations

TL;DR

This paper presents the first fast, fully distributed algorithms for performing random walks in dynamic networks where topology changes arbitrarily over time. It introduces a novel framework for defining stationary distributions in evolving graphs and achieves a round complexity of Õ(√(τΦ)) for sampling from the stationary distribution, significantly improving upon naive O(ℓ) methods for random walks of length ℓ.

ABSTRACT

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 $ ilde{O}(\sqrt{τΦ})$ rounds (with high probability), where $τ$ is the dynamic mixing time and $Φ$ 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 $ ilde{O}(\min\{n^{1/3}k^{2/3}(τΦ)^{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.

Motivation & Objective

• To design efficient distributed algorithms for random walk sampling in dynamic networks where topology changes arbitrarily from round to round.
• To establish a rigorous framework for defining and computing the stationary distribution in evolving graphs under an oblivious adversary model.
• To develop fast distributed algorithms for key problems like information dissemination (gossip) and spectral property estimation in dynamic networks.
• To achieve sublinear round complexity for random walk sampling, improving over the naive O(ℓ) time for walks of length ℓ.
• To provide provable guarantees for convergence to the stationary distribution and estimation of dynamic mixing time and spectral properties.

Proposed method

• Proposes a formal framework for defining the stationary distribution in dynamic networks under an oblivious adversary that controls topology changes.
• Uses multiple independent random walks of increasing length ℓ (doubling in each iteration) to estimate mixing time via statistical hypothesis testing.
• Employs a distributed simulation of the algorithm from Batu et al. [4] to compare empirical distribution of walk samples with uniform (stationary) distribution.
• Leverages the CONGEST model with small message sizes and assumes only local knowledge and no global topology awareness.
• Applies the random walk primitive as a subroutine to solve gossip (k-token dissemination) and spectral property estimation problems.
• Uses binary search over powers of two to efficiently locate the mixing time threshold with high probability.

Experimental results

Research questions

• RQ1 Can random walks be used to efficiently sample from the stationary distribution in dynamic networks with arbitrary topology changes?
• RQ2 What is the optimal round complexity for performing a single random walk in a dynamic network under the CONGEST model?
• RQ3 Can random walks be used to design fast distributed algorithms for k-token dissemination (gossip) in dynamic networks?
• RQ4 How can the dynamic mixing time and related spectral properties (e.g., spectral gap, conductance) be estimated efficiently in a distributed manner?
• RQ5 Are the proposed random walk-based algorithms optimal in terms of round complexity for dynamic network computation?

Key findings

• The proposed random walk algorithm runs in Õ(√(τΦ)) rounds with high probability, where τ is the dynamic mixing time and Φ is the dynamic diameter.
• The algorithm successfully samples from a distribution close to the stationary distribution, even in networks with arbitrary topology changes.
• A fast distributed gossip algorithm is designed that runs in Õ(min{n^{1/3}k^{2/3}(τΦ)^{1/3}, nk}) rounds, improving over the previous O(nk) bound.
• The dynamic mixing time τ^x_mix for a source node x can be estimated in Õ(n^{1/4}√(Φτ^x(ε))) rounds, with τ^x_mix ≤ τ^x(ε) for ε = 1/(6912e√n log n).
• The estimated mixing time allows approximation of spectral gap and conductance via known inequalities, enabling distributed spectral analysis.
• The framework and algorithms are fully decentralized, require no global knowledge, and are robust under an oblivious adversary model.