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Fast Distributed PageRank Computation
Atish Das Sarma, Anisur Rahaman Molla, Gopal Pandurangan, Eli Upfal
TL;DR
PageRank is important for measuring node importance, but provably efficient fully distributed computation remains limited because iterative methods impose communication and convergence costs. This paper develops random-walk-based Monte Carlo algorithms, achieving high-probability bounds of O(log n/ϵ) rounds on arbitrary graphs and O(sqrt(log n)/ϵ) rounds on undirected graphs. The algorithms are scalable because communication per edge per round is polylogarithmic.
Problem
Provably efficient fully distributed PageRank computation remains limited because iterative methods require synchronization and may incur high communication and convergence costs.
Method
The paper develops fully decentralized Monte Carlo algorithms that compute PageRank by performing and accelerating random walks in distributed networks.
Results
O(log n/ϵ) rounds suffice with high probability on arbitrary graphs, while the improved algorithm achieves O(sqrt(log n)/ϵ) rounds on undirected graphs.
Takeaways & Limitations
Random-walk-based distributed PageRank computation can provide sub-logarithmic running time in undirected networks while remaining suitable for large, resource-constrained systems.
Takeaways & Limitations
The communication model assumes nodes initially know their neighbors’ identities and permits direct communication when a node knows another node’s identity.
Abstract
from arXiv · showhide
Over the last decade, PageRank has gained importance in a wide range of applications and domains, ever since it first proved to be effective in determining node importance in large graphs (and was a pioneering idea behind Google's search engine). In distributed computing alone, PageRank vector, or more generally random walk based quantities have been used for several different applications ranging from determining important nodes, load balancing, search, and identifying connectivity structures. Surprisingly, however, there has been little work towards designing provably efficient fully-distributed algorithms for computing PageRank. The difficulty is that traditional matrix-vector multiplication style iterative methods may not always adapt well to the distributed setting owing to communication bandwidth restrictions and convergence rates. In this paper, we present fast random walk-based distributed algorithms for computing PageRanks in general graphs and prove strong bounds on the round complexity. We first present a distributed algorithm that takes $O\big(\log n/\eps \big)$ rounds with high probability on any graph (directed or undirected), where $n$ is the network size and $\eps$ is the reset probability used in the PageRank computation (typically $\eps$ is a fixed constant). We then present a faster algorithm that takes $O\big(\sqrt{\log n}/\eps \big)$ rounds in undirected graphs. Both of the above algorithms are scalable, as each node sends only small ($\polylog n$) number of bits over each edge per round. To the best of our knowledge, these are the first fully distributed algorithms for computing PageRank vector with provably efficient running time.
1 Introduction
PageRank is widely used to identify important nodes, but efficient fully decentralized computation is difficult because iterative methods require synchronization and may converge slowly. The paper introduces provably efficient distributed algorithms that use limited communication and support general graphs, with a faster variant for undirected graphs.
- 1 Introduction: Traditional iterative PageRank methods are costly in distributed networks because they require continuous synchronization, multiple messages, and potentially long convergence.These costs motivate localized algorithms whose communication overhead matters more than local CPU and memory usage.
- 1 Introduction: The paper presents the first provably efficient fully decentralized algorithms for estimating PageRank while sending only polylog n bits per round.The contribution targets scalable distributed computation under communication constraints.
- 1 Introduction: The BASIC-PAGERANK-ALGORITHM computes PageRank with high probability on arbitrary directed or undirected networks.The supplied passages identify the reset probability ϵ and network size n as parameters of the bound, but the displayed asymptotic expression is incomplete.
- 1 Introduction: The IMPROVED-PAGERANK-ALGORITHM computes PageRank accurately in undirected graphs with a faster high-probability termination bound.The supplied contribution passage does not include the complete asymptotic expression.
- 1 Introduction: The algorithms require O(log^3 n) bits per edge per round for the improved method and O(log n) bits for the basic method.These bounds support the paper’s scalability claim.
2 Background and Related Work
PageRank is a stationary distribution of a random walk with random resets, and it is difficult to compute exactly in general graphs. Existing approaches include linear-algebraic iteration and Monte Carlo simulation, with Monte Carlo methods emphasizing scalable random-walk estimation.
- 2 Background and Related Work: The distributed model uses synchronous communication on a connected n-node graph, with each node initially knowing its own and neighboring identities.Nodes may communicate with any node whose identity they know, although the paper notes that one algorithm uses no non-neighbor communication.
- 2 Background and Related Work: The paper measures distributed efficiency by communication rounds, treating local computation as free.This model focuses evaluation on the number of distributed communication rounds.
- 2 Background and Related Work: PageRank is defined as the stationary distribution of a walk that resets uniformly with probability ϵ and otherwise follows a random outgoing edge.The reset mechanism makes the walk the PageRank-specific random walk used throughout the paper.
- 2 Background and Related Work: PageRank is well-defined for directed and undirected graphs, but exact analytical computation is difficult even for undirected graphs.The paper motivates computational approximation because general directed graphs lack known analytical formulas.
- 2 Background and Related Work: PageRank computation uses linear-algebraic methods such as power iteration or Monte Carlo methods that simulate random walks and estimate their stationary distribution.Monte Carlo methods are described as efficient, lightweight, and scalable in prior work.
3 A Distributed Algorithm for PageRank
The paper computes PageRank by simulating many terminating random walks in parallel and estimating each node’s PageRank from visit counts. The resulting distributed algorithm achieves accurate approximations in O(log n/ϵ) rounds with high probability while keeping per-edge messages small.
- The algorithm launches K random walks from every node, where each walk continues along random outgoing edges with probability 1−ϵ and terminates with probability ϵ.The expected walk length is 1/ϵ, and the algorithm runs all walks in parallel.
- The distributed implementation forwards aggregate walk counts rather than individual walk histories, avoiding direct communication between non-neighbors.A node only needs the number of walks reaching it to determine how many move to each neighbor.
- Each node estimates its PageRank from its visit count ζv divided by the total number of visits, scaled by ϵ.This Monte Carlo estimator is unbiased, and its approximation is sharply concentrated around the true PageRank.
- Using K proportional to log n is sufficient for a constant-accuracy PageRank approximation with high probability.A larger K can provide tighter approximation without changing the running-time bound.
- O(log n) bits per edge per round suffice because nodes count visits while the total number of walk coupons is polynomially bounded.This removes congestion as a bottleneck even when many walks are executed concurrently.
- O(log n/ϵ) rounds with high probability yields a δ-approximation of PageRanks for any constant δ.The bound follows because all polynomially many walks terminate within this many rounds with high probability.
4 A Faster Distributed PageRankAlgorithm (for Undirected Graphs)
The improved algorithm accelerates Monte Carlo PageRank estimation in undirected graphs by stitching many short random walks into longer walks, then counting visits. It achieves a sub-logarithmic round bound with high probability while using bandwidth-efficient communication.
- 4 A Faster Distributed PageRankAlgorithm (for Undirected Graphs): The algorithm is designed for bandwidth-restricted communication and sends only O(log^3 n) bits over each link per round.
- 4.1 Description of Our Algorithm: The method performs many short random walks in parallel, stitches them into longer walks, and counts their node visits to estimate PageRank.It uses three phases: short-walk generation, stitching, and visit counting.
- 4.1 Description of Our Algorithm: The algorithm uses short walks of length λ = √log n and assigns O(log^2 n/ϵ) walks per degree, enabling the faster construction.
- 4.1 Description of Our Algorithm: Its phases include coupon-based short-walk generation, parallel stitching, and counting visits across the resulting long walks.The analysis separately bounds the completion time of these phases.
- 4.2 Analysis: The analysis controls congestion by bounding node visits and supports one-round parallel stitching with O(log n)-bit messages.
- 4.2 Analysis: The improved algorithm computes a δ-approximation of PageRank with high probability in its stated O-round bound for undirected graphs.
5 Conclusion
The paper presents fast distributed PageRank algorithms based on Monte Carlo random walks. The faster algorithm has sub-logarithmic running time and is intended for large-scale, resource-constrained, dynamic, or self-organizing networks.
- 5 Conclusion: The algorithms use Monte Carlo random walks to compute PageRank in distributed networks.
- 5 Conclusion: The faster algorithm takes sub-logarithmic time in n, which can be useful in large-scale, resource-constrained distributed networks.
- 5 Conclusion: Because they rely on lightweight, robust, and local random walks, the algorithms may suit self-organizing and dynamic networks.