  Network-valued data are encountered in a wide range of applications and pose
challenges in learning due to their complex structure and absence of vertex
correspondence. Typical examples of such problems include classification or
grouping of protein structures and social networks. Various methods, ranging
from graph kernels to graph neural networks, have been proposed that achieve
some success in graph classification problems. However, most methods have
limited theoretical justification, and their applicability beyond
classification remains unexplored. In this work, we propose methods for
clustering multiple graphs, without vertex correspondence, that are inspired by
the recent literature on estimating graphons -- symmetric functions
corresponding to infinite vertex limit of graphs. We propose a novel graph
distance based on sorting-and-smoothing graphon estimators. Using the proposed
graph distance, we present two clustering algorithms and show that they achieve
state-of-the-art results. We prove the statistical consistency of both
algorithms under Lipschitz assumptions on the graph degrees. We further study
the applicability of the proposed distance for graph two-sample testing
problems.
