  Distance plays a fundamental role in measuring similarity between objects.
Various visualization techniques and learning tasks in statistics and machine
learning such as shape matching, classification, dimension reduction and
clustering often rely on some distance or similarity measure. It is of
tremendous importance to have a distance that can incorporate the underlying
structure of the object. In this paper, we focus on proposing such a distance
between network objects. Our key insight is to define a distance based on the
long term diffusion behavior of the whole network. We first introduce a dynamic
system on graphs called Laplacian flow. Based on this Laplacian flow, a new
version of diffusion distance between networks is proposed. We will demonstrate
the utility of the distance and its advantage over various existing distances
through explicit examples. The distance is also applied to subsequent learning
tasks such as clustering network objects.
