Effective resistances are ubiquitous in graph algorithms and network analysis. In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair $s$ and $t$. We consider the classical adjacency list model for local algorithms. While recent works have provided sublinear time algorithms for expander graphs, we prove several lower bounds for general graphs of $n$ vertices and $m$ edges: 1.It needs $\Omega(n)$ queries to obtain $1.01$-approximations of the effective resistance of an adjacent pair $s$ and $t$, even for graphs of degree at most 3 except $s$ and $t$. 2.For graphs of degree at most $d$ and any parameter $\ell$, it needs $\Omega(m/\ell)$ queries to obtain $c \cdot \min\{d, \ell\}$-approximations where $c>0$ is a universal constant. Moreover, we supplement the first lower bound by providing a sublinear time $(1+\epsilon)$-approximation algorithm for graphs of degree 2 except the pair $s$ and $t$. One of our technical ingredients is to bound the expansion of a graph in terms of the smallest non-trivial eigenvalue of its Laplacian matrix after removing edges. We discover a new lower bound on the eigenvalues of perturbed graphs (resp. perturbed matrices) by incorporating the effective resistance of the removed edge (resp. the leverage scores of the removed rows), which may be of independent interest.

翻译：有效电阻在图算法和网络分析中普遍存在。本文研究了用于近似相邻顶点对$s$和$t$之间有效电阻的次线性时间算法。我们采用经典的邻接表模型进行局部算法设计。尽管近期工作已为扩展图提供了次线性时间算法，但我们针对具有$n$个顶点和$m$条边的通用图证明了若干下界：1. 即使除$s$和$t$外其他顶点度数均不超过3，若要获得相邻顶点对$s$和$t$有效电阻的1.01倍近似值，仍需$\Omega(n)$次查询。2. 对于度数不超过$d$的图及任意参数$\ell$，若要获得$c \cdot \min\{d, \ell\}$倍近似值（其中$c>0$为普适常数），需$\Omega(m/\ell)$次查询。此外，我们通过提供针对除顶点对$s$和$t$外度数均为2的图的$(1+\epsilon)$倍近似次线性时间算法，对第一个下界进行了补充。本文的技术核心之一在于通过移除边后拉普拉斯矩阵的最小非平凡特征值来界定图的扩张性。我们发现了一个关于扰动图（相应地，扰动矩阵）特征值的新下界，该下界通过融入被移除边的有效电阻（或被移除行的杠杆分数）推导得出，这一结果可能具有独立的研究价值。