Fast Query of Biharmonic Distance in Networks

Changan Liu, Ahad N. Zehmakan, Zhongzhi Zhang

Abstract:

The biharmonic distance (BD) is a fundamental metric that measures the distance of two nodes in a graph. It has found applications in network coherence, machine learning, and computational
graphics, among others. In spite of BD’s importance, efficient algorithms for the exact computation or approximation of this metric
on large graphs remain notably absent. In this work, we provide
several algorithms to estimate BD, building on a novel formulaion of hthis metric. These algorithms enjoy locality property (that
is, they only read a small portion of the input graph) and at the
same time possess provable performance guarantees. In particular,
our main algorithms approximate the BD between any node pair
with an arbitrarily small additive error $\epsilon$ in $O(\frac{1}{\epsilon^2}poly(\log\frac{n}{\epsilon}))$ time .
Furthermore, we perform an extensive empirical study on several
benchmark networks, validating the performance and accuracy of
our algorithms