Moving PageRank (MPR) Methods

• Moving PageRank (MPR) is a method that incrementally updates PageRank scores in evolving graphs using algebraic, fluid-diffusion, and dynamical system techniques.
• It leverages local updates and partial inverses to rapidly converge to accurate scores without recomputing the entire eigenvector, even under structural changes.
• Empirical results demonstrate significant performance gains, with up to 30× reductions in operations and improved forecasting on large-scale networks.

Moving PageRank (MPR) comprises a class of methods that efficiently maintain up-to-date PageRank centrality scores under evolving graph topologies or dynamic node importance, without recomputing the entire eigenvector after each change. MPR approaches exploit incremental updates via algebraic, fluid-diffusion, or dynamical-system interpretations, thus allowing scalable, responsive computation for large-scale or rapidly changing networks. These techniques have precise algebraic foundations, yield geometrically fast convergence, and in many cases support strictly local operations, making them particularly suitable for real-time web, network, or data-stream contexts.

1. Algebraic and Algorithmic Foundations

The canonical PageRank eigenproblem seeks $x \in \mathbb{R}^n$ such that

$x = \alpha P x + (1-\alpha) v,$

where $P$ is a column-stochastic transition matrix, $\alpha\in (0,1)$ the damping factor, and $v$ the teleportation (personalization) vector. The corresponding linear system is

$(I - \alpha P)x = (1-\alpha) v,$

with solution $x = (1-\alpha) (I - \alpha P)^{-1} v$.

MPR builds upon the observation that $x$ responds linearly to changes in $P$ and $v$. For evolving graphs, consider $P \to P'$ at time $t$. Rather than solving afresh, one computes the updated PageRank vector by leveraging partial inverses, blockwise inversion, or explicit correction terms based on the structure of the changes. In fluid-diffusion MPR (Hong, 2012), the PageRank update is maintained via a history vector $H$ and a residual fluid vector $F$, with locally injected corrections as the graph changes, and convergence guaranteed by the contractive nature of the PageRank operator.

In the evolving teleportation MPR framework (Rossi et al., 2012), $v$ is replaced by a time-dependent vector $v(t)$, and PageRank evolution is described by the ODE

$x'(t) = (1 - \alpha) v(t) - (I - \alpha P) x(t),$

leading to a continuous-time generalization whose solution tracks both structural and extrinsic (teleportation) dynamics.

2. Fluid Diffusion Model and Online Updates

The fluid diffusion MPR model (Hong, 2012) contrasts with classical power iteration by distributing "fluid" $F$ across nodes, diffusing mass locally. The process at iteration $n$ consists of:

• Picking a node $i_n$,
• Adding $F[i_n]$ to $H[i_n]$,
• Pushing $d \cdot F[i_n]$ along $i_n$'s outgoing edges,
• Resetting $F[i_n]$ to zero.

Key algebraic identities include:

• $H_n + F_n = F_0 + d P H_n$,
• The error is $X - H_n = (I - d P)^{-1} F_n$,
• Under greedy scheduling, $\|F_n\|_1$ decreases by a factor $1 - (1-d)/N$, yielding exponential convergence.

Upon any graph update $P \to P'$, residual fluid is promptly adjusted by an injected correction $d(P'-P)H$, after which diffusion resumes under $P'$. This formulation admits provable convergence, strictly local edge-pushes, and circumvents the need for global synchronizations or full dense matrix multiplications.

3. Dynamical System and Evolving Teleportation

The evolving-teleportation perspective (Rossi et al., 2012) represents PageRank as the solution to a linear ODE, naturally accommodating time-varying teleportation:

$x'(t) = (1-\alpha) v(t) - (I - \alpha P)x(t).$

Discretization via forward-Euler yields

$$x(t+h) = x(t) + h[(1-\alpha) v(t) - (I - \alpha P)x(t)].$$

This approach generalizes PageRank to arbitrary temporal evolutions of external attention, with the ability to adapt to both sudden and gradual changes in $v(t)$. If $v(t)$ stabilizes, $x(t) \to x_\infty$ with the expected stationary solution.

Empirical studies demonstrate that this dynamic PageRank effectively highlights shifts due to exogenous temporal activity (e.g., trending topics), outperforms static PageRank in forecasting volatile node behavior, and smooths high-frequency fluctuations due to the intrinsic low-pass characteristics of $(I-\alpha P)$ (Rossi et al., 2012).

4. Explicit Algebraic Updates in Structured Subgraphs

For networks with specific topologies—such as lines and cliques—the MPR framework enables explicit incremental updates leveraging the algebraic structure of $(I-\alpha P)^{-1}$ (Engström et al., 2013). For a simple line of $n_L$ nodes, the inverse features a lower-triangular pattern permitting closed-form solutions for node ranks, and edge addition/removal corresponds to local modifications of the inverse entries. In complete graphs, blockwise inverse formulas yield updated PageRanks upon the addition or removal of nodes or links. When subgraphs are connected (e.g., a line joined to a clique by a single edge), hitting probabilities and block decompositions allow for localized update schemes. However, this approach is practical primarily in highly regular subgraphs due to the complexity of maintaining partial inverses in general graphs.

5. Empirical Performance and Computational Guarantees

Across synthetic and real-world datasets, MPR methods yield significant performance gains. Fluid-diffusion MPR variants achieve 5–30× reductions in edge-push operations compared to standard power iteration and 8–20× speedups to $10^{-6}$ error on massive web graphs (e.g., uk-2007-05 with over 1M nodes and 41M links), with sub-second response to batches of updates (Hong, 2012). Accuracy remains within prescribed tolerances, and the method scales to millions of nodes while supporting asynchronous, local operations.

Evolving teleportation MPR demonstrates improved prediction of next-period node activity, especially for unstable nodes, on networks like Wikipedia (4.1M nodes, 72.7M links) and Twitter (465K users, 835K follower links) (Rossi et al., 2012). Dynamic PageRank time series produce substantial improvements in forecast metrics (e.g., sMAPE reductions), and event detection analysis confirms correspondence to real-world events.

6. Constraints, Applicability, and Open Issues

While MPR frameworks allow scalable, responsive centrality tracking under dynamic conditions, some limitations remain:

• Closed-form incremental algebraic updates are available only for restricted subgraph classes (lines, cliques, stars); for large-scale heterogeneous graphs, general decompositions are unavailable (Engström et al., 2013).
• Maintaining partial inverses or histories for general graph evolutions introduces storage and computational overhead.
• For arbitrary structural changes, fallback to standard iterative or advanced numerical techniques is required.

A plausible implication is that practical deployment of MPR at web scale will rely on hybrid schemes—leveraging explicit updates in localized structured regions, dynamic diffusion for broad graph regions, and stochastic or sketch-based aggregation in highly dynamic contexts.

7. Relation to Broader Dynamic Network Analysis

The moving PageRank approach exemplifies a broader trend in network science toward incremental, online algorithms for dynamic graphs. By reformulating static centrality as a process amenable to local, asynchronous updates, MPR advances the state of scalable ranking under time-varying conditions. Its integration with streaming data, real-time event detection, and social-network analytics underscores its practical relevance and ongoing importance for computational network science.

Key References: (Hong, 2012, Rossi et al., 2012, Engström et al., 2013)