  This paper develops an approximation to the (effective) $p$-resistance and
applies it to multi-class clustering. Spectral methods based on the graph
Laplacian and its generalization to the graph $p$-Laplacian have been a
backbone of non-euclidean clustering techniques. The advantage of the
$p$-Laplacian is that the parameter $p$ induces a controllable bias on cluster
structure. The drawback of $p$-Laplacian eigenvector based methods is that the
third and higher eigenvectors are difficult to compute. Thus, instead, we are
motivated to use the $p$-resistance induced by the $p$-Laplacian for
clustering. For $p$-resistance, small $p$ biases towards clusters with high
internal connectivity while large $p$ biases towards clusters of small
"extent," that is a preference for smaller shortest-path distances between
vertices in the cluster. However, the $p$-resistance is expensive to compute.
We overcome this by developing an approximation to the $p$-resistance. We prove
upper and lower bounds on this approximation and observe that it is exact when
the graph is a tree. We also provide theoretical justification for the use of
$p$-resistance for clustering. Finally, we provide experiments comparing our
approximated $p$-resistance clustering to other $p$-Laplacian based methods.
