Personalized PageRank Estimation in Undirected Graphs

Given an undirected graph $G=(V, E)$, the Personalized PageRank (PPR) of $t\in V$ with respect to $s\in V$, denoted $π(s,t)$, is the probability that an $α$-discounted random walk starting at $s$ terminates at $t$. We study the time complexity of estimating $π(s,t)$ with constant relative error and constant failure probability, whenever $π(s,t)$ is above a given threshold parameter $δ\in(0,1)$. We consider common graph-access models and furthermore study the single source, single target, and single node (PageRank centrality) variants of the problem. We provide a complete characterization of PPR estimation in undirected graphs by giving tight bounds (up to logarithmic factors) for all problems and model variants in both the worst-case and average-case setting. This includes both new upper and lower bounds. Tight bounds were recently obtained by Bertram, Jensen, Thorup, Wang, and Yan for directed graphs. However, their lower bound constructions rely on asymmetry and therefore do not carry over to undirected graphs. At the same time, undirected graphs exhibit additional structure that can be exploited algorithmically. Our results resolve the undirected case by developing new techniques that capture both aspects, yielding tight bounds.

💡 Research Summary

This paper provides a comprehensive study of the computational complexity of estimating Personalized PageRank (PPR) in undirected graphs. For a given undirected graph G = (V, E), the PPR value π(s, t) is the probability that an α‑discounted random walk starting at source s terminates at target t. The authors focus on algorithms that, for any threshold δ ∈ (0,1], return an estimate ˆπ(s, t) with constant relative error when π(s, t) > δ and with additive error O(δ) otherwise, while guaranteeing a constant failure probability.

Four query variants are considered: (1) single‑pair (estimate π(s, t) for a given pair), (2) single‑source (estimate π(s, t) for all t given a source s), (3) single‑target (estimate π(s, t) for all s given a target t), and (4) single‑node (estimate the global PageRank score π(t)). The analysis is performed under the standard adjacency‑list access model, supplemented by three optional queries: JUMP (return a uniformly random vertex), NEIGH‑SORTED (return the i‑th neighbor of a vertex when neighbors are ordered by increasing degree), and ADJ (test adjacency of two vertices). All 2⁴ combinations of optional queries are examined.

A key structural property of undirected graphs—reversibility—states that d(s)·π(s, t) = d(t)·π(t, s), where d(v) denotes the degree of v. This symmetry eliminates the extreme asymmetries possible in directed graphs and enables new algorithmic techniques.

Upper bounds.

• Single‑node: The classic backward‑Monte‑Carlo algorithm runs in O(√m) time. By exploiting JUMP, NEIGH‑SORTED, and ADJ, the authors achieve O(√n) time: low‑degree neighbors are handled via backward sampling, while high‑degree neighbors are hit efficiently by random jumps and adjacency checks.
• Single‑target (worst‑case): The standard backward push algorithm costs O(n/δ). With NEIGH‑SORTED, a hybrid deterministic‑then‑random push reduces the cost to O(n/√δ).
• Single‑target (average‑case): Ignoring neighbors whose degree exceeds Θ(1/δ) introduces at most O(δ) additive error. The remaining work is O(1/δ² + d), where d = m/n is the average degree.
• Single‑pair (average‑case): A bidirectional estimator combines forward Monte‑Carlo walks from s with the hybrid backward push from t. Balancing the two parts yields O((1/δ)^{2/3} + d) time; with NEIGH‑SORTED and ADJ the logarithmic factors disappear.
• Single‑source and other variants: The paper matches known bounds of O(min{m, 1/δ}) for worst‑case source estimation and O(min{m, d/δ, (1/δ)² + d}) for average‑case target estimation, showing these are optimal under the respective models.

Lower bounds.
The authors construct undirected hard instances parameterized by n, m, d, and δ that respect the optional query capabilities. For the single‑pair problem they prove a lower bound of Ω(min{n, 1/δ}) even in the strongest model, matching the upper bound up to polylogarithmic factors. For single‑node they establish Ω(√m) in the basic model and Ω(min{d, √n}) when all three optional queries are available, again tight with their algorithms. Similar tight lower bounds are given for the other variants, demonstrating that no algorithm can asymptotically beat the presented complexities.

Implications.
The results give a complete “complexity landscape” for PPR estimation in undirected graphs. They show that the reversibility property can be leveraged to obtain strictly better runtimes than in directed graphs for many settings, especially when the richer query set is available. Practically, the work informs the design of graph APIs: supporting JUMP, degree‑sorted neighbor access, and adjacency tests can dramatically reduce the cost of PPR‑based analytics on massive undirected networks.

In summary, the paper delivers tight (up to polylogarithmic factors) upper and lower bounds for all natural PPR estimation problems on undirected graphs, introduces novel hybrid algorithms that exploit graph symmetry, and clarifies which access primitives are most valuable for efficient implementation.