Effective Resistances in Non-Expander Graphs

TL;DR

This paper establishes lower bounds for approximating effective resistances in general graphs using sublinear algorithms, and provides a specialized approximation algorithm for degree-2 graphs, highlighting the complexity landscape.

Contribution

It proves new lower bounds on query complexity for approximating effective resistance in general graphs and introduces a sublinear algorithm for degree-2 graphs.

Findings

Requires rac{n}{ ext{queries}} to approximate effective resistance within 1.01 for general graphs.

Lower bounds depend on graph degree and parameters, indicating inherent complexity.

Provides a sublinear time rac{m}{ ext{queries}} approximation algorithm for degree-2 graphs.

Abstract

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.