Sublinear-Time Algorithms for Diagonally Dominant Systems and Applications to the Friedkin-Johnsen Model

2025-09-16

AI Summary

This paper studies sublinear-time solvers for diagonally dominant linear systems $Sz = b$, where $|S_{ii}| geq delta + sum_{j e i} |S_{ij}|$ for $delta geq 0$. We propose the first randomized algorithm applicable to both strictly and weakly diagonally dominant matrices, which reads only a small number of entries from $S$ and $b$, and outputs an estimate $hat{z}_u$ of any solution coordinate $z^*_u$ with additive error $varepsilon$ or $varepsilon |z^*|_infty$. Our key contribution is a novel probabilistic recurrence-based analysis framework that establishes an optimal linear dependence on $S_{max} = max_{i,j} |S_{ij}|$. The algorithm runs in time $O!left( frac{|b|_infty^2 S_{max}}{delta^3 varepsilon^2} log frac{|b|_infty}{delta varepsilon} ight)$. As an application, we achieve efficient sublinear-time opinion estimation in the Friedkin-Johnsen model.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study sublinear-time algorithms for solving linear systems $Sz = b$, where $S$ is a diagonally dominant matrix, i.e., $|S_{ii}| geq δ+ sum_{j e i} |S_{ij}|$ for all $i in [n]$, for some $δgeq 0$. We present randomized algorithms that, for any $u in [n]$, return an estimate $z_u$ of $z^*_u$ with additive error $varepsilon$ or $varepsilon lVert z^* Vert_infty$, where $z^*$ is some solution to $Sz^* = b$, and the algorithm only needs to read a small portion of the input $S$ and $b$. For example, when the additive error is $varepsilon$ and assuming $δ>0$, we give an algorithm that runs in time $Oleft( frac{|b|_infty^2 S_{max}}{δ^3 varepsilon^2} log frac{| b |_infty}{δvarepsilon} ight)$, where $S_{max} = max_{i in [n]} |S_{ii}|$. We also prove a matching lower bound, showing that the linear dependence on $S_{max}$ is optimal. Unlike previous sublinear-time algorithms, which apply only to symmetric diagonally dominant matrices with non-negative diagonal entries, our algorithm works for general strictly diagonally dominant matrices ($δ> 0$) and a broader class of non-strictly diagonally dominant matrices $(δ= 0)$. Our approach is based on analyzing a simple probabilistic recurrence satisfied by the solution. As an application, we obtain an improved sublinear-time algorithm for opinion estimation in the Friedkin--Johnsen model.

Problem

• Solving linear systems with diagonally dominant matrices
• Developing sublinear-time algorithms for matrix solutions
• Improving opinion estimation in Friedkin-Johnsen model

Innovation

• Randomized sublinear-time algorithms for diagonally dominant systems
• General applicability to strictly and non-strictly dominant matrices
• Probabilistic recurrence analysis for solution estimation