We initiate a study of solving a row/column diagonally dominant (RDD/CDD) linear system $Mx=b$ in sublinear time, with the goal of estimating $t^{\top}x^*$ for a given vector $t\in R^n$ and a specific solution $x^*$. This setting naturally generalizes the study of sublinear-time solvers for symmetric diagonally dominant (SDD) systems [AKP19] to the asymmetric case. Our first contributions are characterizations of the problem's mathematical structure. We express a solution $x^*$ via a Neumann series, prove its convergence, and upper bound the truncation error on this series through a novel quantity of $M$, termed the maximum $p$-norm gap. This quantity generalizes the spectral gap of symmetric matrices and captures how the structure of $M$ governs the problem's computational difficulty. For systems with bounded maximum $p$-norm gap, we develop a collection of algorithmic results for locally approximating $t^{\top}x^*$ under various scenarios and error measures. We derive these results by adapting the techniques of random-walk sampling, local push, and their bidirectional combination, which have proved powerful for special cases of solving RDD/CDD systems, particularly estimating PageRank and effective resistance on graphs. Our general framework yields deeper insights, extended results, and improved complexity bounds for these problems. Notably, our perspective provides a unified understanding of Forward Push and Backward Push, two fundamental approaches for estimating random-walk probabilities on graphs. Our framework also inherits the hardness results for sublinear-time SDD solvers and local PageRank computation, establishing lower bounds on the maximum $p$-norm gap or the accuracy parameter. We hope that our work opens the door for further study into sublinear solvers, local graph algorithms, and directed spectral graph theory.

Personalized PageRank (PPR) is an extensively studied and applied node proximity measure in graphs. For a pair of nodes $s$ and $t$ on a graph $G=(V,E)$, the PPR value $π(s,t)$ is defined as the probability that an $α$-discounted random walk from $s$ terminates at $t$, where the walk terminates with probability $α$ at each step. We study the classic Single-Source PPR query, which asks for PPR approximations from a given source node $s$ to all nodes in the graph. Specifically, we aim to provide approximations with absolute error guarantees, ensuring that the resultant PPR estimates $\hatπ(s,t)$ satisfy $\max_{t\in V}\big|\hatπ(s,t)-π(s,t)\big|\le\varepsilon$ for a given error bound $\varepsilon$. We propose an algorithm that achieves this with high probability, with an expected running time of - $\widetilde{O}\big(\sqrt{m}/\varepsilon\big)$ for directed graphs, where $m=|E|$; - $\widetilde{O}\big(\sqrt{d_{\mathrm{max}}}/\varepsilon\big)$ for undirected graphs, where $d_{\mathrm{max}}$ is the maximum node degree in the graph; - $\widetilde{O}\left(n^{γ-1/2}/\varepsilon\right)$ for power-law graphs, where $n=|V|$ and $γ\in\left(\frac{1}{2},1\right)$ is the extent of the power law. These sublinear bounds improve upon existing results. We also study the case when degree-normalized absolute error guarantees are desired, requiring $\max_{t\in V}\big|\hatπ(s,t)/d(t)-π(s,t)/d(t)\big|\le\varepsilon_d$ for a given error bound $\varepsilon_d$, where the graph is undirected and $d(t)$ is the degree of node $t$. We give an algorithm that provides this error guarantee with high probability, achieving an expected complexity of $\widetilde{O}\left(\sqrt{\sum_{t\in V}π(s,t)/d(t)}\big/\varepsilon_d\right)$. This improves over the previously known $O(1/\varepsilon_d)$ complexity.

We revisit ApproxContributions, the classic local graph exploration algorithm proposed by Andersen, Borgs, Chayes, Hopcroft, Mirrokni, and Teng (WAW ’07, Internet Math. ’08) for computing an є-approximation of the PageRank contribution vector for a target node t on a graph with n nodes and m edges. We give a worst-case complexity bound of it as O(nπ(t)/є·min(Δin,Δout,√m)), where π(t) is the PageRank score of t, and Δin and Δout are the maximum in-degree and out-degree of the graph, resp. We also give a lower bound of Ω(min(Δin/δ,Δout/δ,√m/δ,m)) for detecting t’s δ-contributing set, showing that the simple ApproxContributions algorithm is already optimal.

Personalized PageRank (PPR) is a traditional measure for node proximity on large graphs. For a pair of nodes$\boldsymbol{s}$and$\boldsymbol{t}$, the PPR value$\boldsymbol{\pi_{s}(t)}$equals the probability that an$\boldsymbol{\alpha }$-discounted random walk from$\boldsymbol{s}$terminates at$\boldsymbol{t}$and reflects the importance between$\boldsymbol{s}$and$\boldsymbol{t}$in a bidirectional way. As a generalization of Google's celebrated PageRank centrality, PPR has been extensively studied and has found multifaceted applications in many fields, such as network analysis, graph mining, and graph machine learning. Despite numerous studies devoted to PPR over the decades, efficient computation of PPR remains a challenging problem, and there is a dearth of systematic summaries and comparisons of existing algorithms. In this paper, we recap several frequently used techniques for PPR computation and conduct a comprehensive survey of various recent PPR algorithms from an algorithmic perspective. We classify these approaches based on the types of queries they address and review their methodologies and contributions. We also discuss some representative algorithms for computing PPR on dynamic graphs and in parallel or distributed environments.