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

• URL: http://arxiv.org/abs/2509.13112v1
• Date: Tue, 16 Sep 2025 14:13:31 GMT
• Title: Sublinear-Time Algorithms for Diagonally Dominant Systems and Applications to the Friedkin-Johnsen Model
• Authors: Weiming Feng, Zelin Li, Pan Peng,
• Abstract summary: We study sublinear-time algorithms for solving linear systems $Sz = b$, where $S$ is a diagonally dominant matrix.<n>We present randomized algorithms that, for any $u in [n]$, return an estimate $z_u$ of $z*_u$ with additive error.<n>We also prove a matching lower bound, showing that the linear dependence on $S_max$ is optimal.
• Score: 5.101318208537081
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study sublinear-time algorithms for solving linear systems $Sz = b$, where $S$ is a diagonally dominant matrix, i.e., $|S_{ii}| \geq \delta + \sum_{j \ne i} |S_{ij}|$ for all $i \in [n]$, for some $\delta \geq 0$. We present randomized algorithms that, for any $u \in [n]$, return an estimate $z_u$ of $z^*_u$ with additive error $\varepsilon$ or $\varepsilon \lVert z^*\rVert_\infty$, 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 $\delta>0$, we give an algorithm that runs in time $O\left( \frac{\|b\|_\infty^2 S_{\max}}{\delta^3 \varepsilon^2} \log \frac{\| b \|_\infty}{\delta \varepsilon} \right)$, 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 ($\delta > 0$) and a broader class of non-strictly diagonally dominant matrices $(\delta = 0)$. 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.

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