  We study the behavior of network diffusions based on the PageRank random walk
from a set of seed nodes. These diffusions are known to reveal small, localized
clusters (or communities) and also large macro-scale clusters by varying a
parameter that has a dual-interpretation as an accuracy bound and as a
regularization level. We propose a new method that quickly approximates the
result of the diffusion for all values of this parameter. Our method
efficiently generates an approximate $\textit{solution path}$ or
$\textit{regularization path}$ associated with a PageRank diffusion, and it
reveals cluster structures at multiple size-scales between small and large. We
formally prove a runtime bound on this method that is independent of the size
of the network, and we investigate multiple optimizations to our method that
can be more practical in some settings. We demonstrate that these methods
identify refined clustering structure on a number of real-world networks with
up to 2 billion edges.
