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Distributed Randomized Algorithms for the PageRank Computation
Hideaki Ishii, Roberto Tempo
TL;DR
PageRank must rank relevant search results, and the reported scale of the web makes centralized computation costly. The paper develops randomized local updates for distributed PageRank computation. It reports mean-square convergence to the true values using time averages and relates the approach to consensus problems.
Problem
PageRank computation concerns ranking pages by link-based importance, while the reported web scale and week-long centralized power-method computation motivate more efficient methods.
Source: §1 · pp. 1–2Method
The paper proposes randomized distributed updates in which pages exchange values with linked neighbors and initiate communication at independent random times.
Source: §1 · p. 2Results
The proposed algorithm converges in the mean-square sense to true PageRank values by computing time averages at each page.
Source: §1 · p. 2Takeaways & Limitations
The approach has a clear relation to multi-agent consensus problems.
Source: §9 · p. 25The state vector need not converge, so the distributed approach instead computes its time average.
Source: §3.2 · p. 10
Abstract
from arXiv · showhide
In the search engine of Google, the PageRank algorithm plays a crucial role in ranking the search results. The algorithm quantifies the importance of each web page based on the link structure of the web. We first provide an overview of the original problem setup. Then, we propose several distributed randomized schemes for the computation of the PageRank, where the pages can locally update their values by communicating to those connected by links. The main objective of the paper is to show that these schemes asymptotically converge in the mean-square sense to the true PageRank values. A detailed discussion on the close relations to the multi-agent consensus problems is also given.
1 Introduction
PageRank ranks pages by link-based importance, but the web’s scale makes centralized computation costly. The paper proposes randomized, fully distributed updates and establishes mean-square convergence to true PageRank values, relating the scheme to consensus while distinguishing their objectives.
1 Introduction
- The web is reported to contain over 8 billion pages, while centralized power-method computation reportedly takes about a week.The paper presents this scale and computation time as motivation for more efficient methods.Source: §1 · p. 2
- Pages can update locally by exchanging values with pages connected through incoming or outgoing links, with update times chosen independently at random.The scheme has no fixed update order or centralized agent, and each page’s computation is mild.Source: §1 · p. 2
- The proposed distributed algorithm converges in the mean-square sense to the true PageRank values by computing time averages at each page.The approach preserves and exploits the stochasticity of the original problem’s matrix.Source: §1 · p. 2
- The update recursion resembles consensus, but PageRank seeks a specific stochastic-matrix eigenvector rather than agreement among agents.The paper positions its approach in relation to consensus, agreement, and flocking research.Source: §1 · pp. 2–3
- The paper extends its basic scheme to allow multiple pages to update simultaneously and to reduce communication, then compares it with asynchronous iteration and gives numerical examples.The paper also discusses the relation between distributed PageRank updates and multi-agent consensus.Source: §1 · p. 3
2 The PageRank problem
PageRank assigns each page a normalized value based on contributions from pages linking to it. The section defines the link-matrix formulation, modifies it to ensure a unique target eigenvector, and describes the power method and an illustrative four-page network.
2 The PageRank problem
- A page’s value reflects contributions from pages linking to it, with each contribution weighted by the linking page’s outgoing-link count and values normalized to sum to one.The resulting value vector is defined through the link-matrix eigenvector formulation.Source: §2 · pp. 3–4
- Because the web is not strongly connected and has dangling nodes, the setup adds artificial links and modifies the link matrix to obtain a positive stochastic matrix with a unique dominant eigenvalue.The paper uses m = 0.15 for the modified matrix, following.Source: §2 · pp. 4–5
- The power method iterates the modified link matrix from a probability-vector initial state and converges to the PageRank vector; its asymptotic rate depends on the second-to-largest eigenvalue magnitude.The implementation can use the sparse link matrix A rather than the dense modified matrix M.Source: §2 · p. 5
- In the four-page example, the values are 0.119, 0.331, 0.260, and 0.289 for pages 1–4, respectively; page 2 ranks highest.Pages 3 and 4 have equal incoming-link counts, but page 4’s value is larger because it receives more contribution from page 3 than it gives back.Source: §2 · p. 6, Figure 1 · p. 6
3 A distributed randomized approach
The paper develops randomized local PageRank updates in which pages exchange values with pages linked to them. The scheme uses stochastic link matrices and time averaging to obtain the PageRank vector in the mean-square sense.
3 A distributed randomized approach
- Each update is initiated by a randomly selected page, which exchanges values with pages connected by incoming and outgoing links.The selection process is i.i.d.; the paper says equal update probabilities permit implementation without a centralized decision maker or fixed order.Source: §3 · p. 6
3.1 Distributed link matrices and their average
- Each distributed link matrix retains the selected page’s row and column from A and sets the other diagonal entries to 1 − a_iℓ; the remaining entries are zero.The resulting matrices are stochastic, a property the paper identifies as critical for convergence.Source: §3.1 · p. 7
- The original link matrix A and the average distributed link matrix share an eigenvector for eigenvalue 1, corresponding to the PageRank vector.This shared eigenvector motivates using the distributed matrices despite their different structures.Source: §3.1 · p. 8
3.2 Mean-square convergence of the distributed update scheme
- The scheme computes the PageRank vector through randomly switching among distributed link matrices, even though A does not explicitly appear in the scheme.The parameter ˆm is chosen so the average modified matrices share the relevant eigenvector with the original PageRank matrix.Source: §3.2 · pp. 9–10
- The time average converges to the PageRank vector in the mean-square sense, although the instantaneous state need not converge.The paper reports convergence of order 1/k, with dependence on n through the parameter ˆm.Source: §3.2 · p. 10
- Exact implementation of local computations by web pages is outside the paper’s scope.The authors note that regulations may be needed for page owners to cooperate.Source: §3.2 · p. 10
4 Relations to consensus problems
The paper relates distributed PageRank to multi-agent consensus: both use neighbor communication and randomized updates, but they pursue different objectives. It also identifies stochasticity as an important technical connection while distinguishing the convergence guarantees and matrix structures.
4 Relations to consensus problems
- For a strongly connected graph, the paper’s random communication-pattern scheme obtains consensus.Its average matrix is stochastic and primitive, supporting the stated consensus result.Source: §4 · p. 12
- Consensus values converge with probability one directly, while the PageRank scheme relies on time averages and targets column-stochastic link matrices rather than row-stochastic ones.The comparison also notes that consensus imposes no restriction on the common value.Source: §4 · p. 12
- Both problems use randomized communication among neighboring agents, but consensus seeks a common value whereas PageRank computes a specific eigenvector.The paper presents consensus research as a particular motivation for its distributed PageRank approach.Source: §4 · pp. 11–12
- The paper identifies stochasticity of the distributed link matrices as technically important and notes that the coefficient of ergodicity used for PageRank convergence is also useful for consensus.It presents the distributed PageRank approach as motivated by advances in consensus research.Source: §4 · p. 12
- The paper notes that few works on distributed PageRank exploit the network-of-agents viewpoint.It frames pages as agents that compute locally and communicate with neighboring agents.Source: §4 · p. 12
5 Extensions to simultaneous updates
The paper extends distributed PageRank updates so multiple pages can initiate randomized updates simultaneously. It establishes that time averages recover the PageRank value, with convergence depending on the update probability but not on the number of pages.
5 Extensions to simultaneous updates
- Multiple pages can initiate updates simultaneously through randomized, locally made decisions, extending the earlier scheme in which one page updates at a time.Each page's update decision is made with probability α, shared globally across pages.Source: §5 · p. 13
5.1 Distributed link matrices and their average
- The average link matrix is stochastic and shares the original link matrix's eigenvector for eigenvalue 1.The average matrix is expressed as A + (1 − α)^2(I − A).Source: §5.1 · pp. 14, 16
5.2 Mean-square convergence of the distributed update scheme
- Higher update probability α and larger m yield faster convergence in the mean, while α also governs update frequency and communication load.The scheme is fully decentralized when α < 1 and reduces to the centralized scheme when α = 1.Source: §5.2 · pp. 17–18
- For any update probability α∈(0, 1], time averages obtain x* in the mean-square sense; convergence is order 1/k, depends on α, and is independent of n.The update probability also affects the rate of convergence in the mean.Source: §5.2 · p. 18
6 Update termination in PageRank computation
The section introduces update termination to reduce computation and communication by letting pages stop after their time averages meet a convergence test. It shows that the remaining pages converge in the mean-square sense, with approximation accuracy tied to the terminated pages' estimates.
6 Update termination in PageRank computation
- Update termination reduces computation and communication because a page transmits its estimate only once after its values meet a convergence threshold.The algorithm uses a relative error δ and a window of Ns steps to test whether its time average has varied little enough.Source: §6 · p. 18, §6.1 · pp. 18–19
6.1 Convergence properties for the distributed scheme
- The terminated-page scheme's distributed link matrices are nonnegative but no longer stochastic, so the state can remain nonnegative without being a probability vector.The convergence analysis uses stability of the remaining-pages submatrix.Source: §6.1 · pp. 19, 21
- After some pages terminate, the remaining pages' time averages converge in the mean-square sense to an equilibrium whose approximation level matches the terminated pages' δ-level estimates.The result is established under fixed update sample paths up to the termination time.Source: §6.1 · pp. 19, 21
6.2 Distributed algorithm with update termination
- Pages test their time-average history against δ over Ns steps, then fix and transmit the estimate when the test is met; smaller δ or larger Ns improves accuracy but delays termination.The algorithm ends when every page has terminated its updates.Source: §6.2 · p. 22
7 Discussion on asynchronous iteration methods
The paper contrasts its randomized distributed PageRank updates with asynchronous iteration and describes a power-method update scheme in which pages update selectively. The scheme converges almost surely to the PageRank vector from any probability-vector initial state.
7 Discussion on asynchronous iteration methods
- Pages independently initiate power-method updates according to i.i.d. random processes; pages not selected at a time step retain their values.The update matrices keep the original link-matrix rows for pages initiating updates and use identity rows for the others, so they are not stochastic.Source: §7 · pp. 22–23
- From any probability-vector initial state, the distributed update scheme converges almost surely to the PageRank vector.The result is stated for the update scheme in (43).Source: §7 · p. 23
- The scheme is a randomized version of asynchronous iteration [5, Section 6.2], whose general form handles communication delays and has exponential convergence.The discussion notes that the proof uses an argument similar to Lyapunov functions.Source: §7 · p. 23
- The proposed schemes preserve and exploit stochastic link matrices; unlike asynchronous iteration's incoming-link communication, a later extension communicates over outgoing links.The paper notes that incoming-link methods require popular pages to store data about many linking pages.Source: §7 · p. 23
8 Numerical example
In a 1,000-page randomly generated web example, the paper examines approximate PageRank estimates, their errors, and the update stopping behavior. Individual-value error falls rapidly, while total error decreases more slowly.
8 Numerical example
- The example uses 1,000 pages with randomly generated links; the first ten receive links from over 90% of pages, while other pages have 2–333 links.For the stopping experiment, each page initiates updates with probability α = 0.01; the parameters are N_s = 800 and δ = 0.01.Source: §8 · pp. 23–24
- For pages 21–30, updates stop by k = 4,500, with the time-average estimates showing fairly fast convergence.Figure 2 shows sample paths, stopping times, and true PageRank values.Source: §8 · p. 24, Figure 2 · p. 24
- The ℓ∞ error in individual estimates rapidly decreases and remains small, while the total ℓ1 error decreases more slowly.Figure 3 plots the two error norms.Source: §8 · p. 24, Figure 3 · p. 24
- At k = 8,000, the sum of the time-average estimates is about 0.989, close to 1, although the estimates are not normalized.Figure 4 plots the first 20 estimates against acceptable error ranges.Source: §8 · pp. 24–25, Figure 4 · p. 25
9 Conclusion
The paper concludes that its randomized distributed approach achieves mean-square convergence to PageRank values and is closely related to consensus-type problems.
9 Conclusion
- The proposed randomized distributed schemes are shown to converge in the mean-square sense to PageRank values.The paper also identifies a clear relation between the approach and consensus-type problems.Source: §9 · p. 25
- The authors identify communication delays, faster convergence, and implementation issues as subjects for future research.They also note that later papers address random link failures and computations based on aggregating groups of pages.Source: §9 · p. 25
A Appendix
The appendix provides preliminary material and proofs for two theorems.
A Appendix
- The appendix begins with preliminary material.It then presents proofs of Theorems 3.4 and 6.3.Source: A Appendix · p. 27
- The appendix contains a proof of Theorem 3.4.The proof follows preliminary material.Source: A Appendix · p. 27
- The appendix also contains a proof of Theorem 6.3.The proof follows preliminary material.Source: A Appendix · p. 27
A.1 Preliminaries
The preliminaries characterize weak ergodicity through convergence of product-matrix columns and use the coefficient of ergodicity to test it. A uniform upper bound below one on that coefficient guarantees weak ergodicity.
A.1 Preliminaries
- Weak ergodicity means that differences between corresponding entries of product-matrix columns tend to zero, so the columns coincide in the limit.The columns need not converge to a single vector.Source: §A.1 · p. 27
- The coefficient of ergodicity τ(P) lies in [0, 1] and is zero exactly when P has identical columns given by a probability vector.Source: §A.1 · p. 28
- A uniform bound τ(P(k)) ≤ τ0 < 1 guarantees that the backward products are weakly ergodic.Source: §A.1 · p. 28
A.2 The distributed update scheme
The analysis establishes weak ergodicity for the update matrices under arbitrary mode sequences and uses mean-square error bounds to show that time averaging recovers the PageRank value.
A.2 The distributed update scheme
- For any mode sequence, the update-matrix sequence is weakly ergodic.Source: §A.2 · p. 28
- The update states and PageRank vector are probability vectors, which keeps the error bounded during the convergence analysis.The proof uses the bound ∥e(ℓ)∥ ≤ 2.Source: §A.2 · pp. 28–29
- The proof uses independence of the random modes and the average matrix’s eigenvector property to bound error terms.Source: §A.2 · p. 29
- The PageRank value x* is obtained through the time average y.Source: §A.2 · p. 30
A.3 Proof of Theorem 6.3
The proof bounds the state’s deviation from the average-system equilibrium and controls the randomized update dynamics using a contraction bound. The resulting upper bound tends to zero, completing the convergence proof.
A.3 Proof of Theorem 6.3
- The error is bounded, with ∥e(k)∥1 ≤ ∥xN(k)∥1 + ∥exN∥1.Source: §A.3 · p. 30
- The randomized update’s nonnegative-state block has 1-norm at most 1 − m̂, providing a contraction bound.Source: §A.3 · p. 30
- The proof uses independent random modes and the average matrix bA to relate expected products of updates to the average-system equilibrium.Source: §A.3 · p. 31
- The bound on the proof’s right-hand side converges to zero as k → ∞, completing the convergence proof.Source: §A.3 · p. 32