  Algorithms for node clustering typically focus on finding homophilous
structure in graphs. That is, they find sets of similar nodes with many edges
within, rather than across, the clusters. However, graphs often also exhibit
heterophilous structure, as exemplified by (nearly) bipartite and tripartite
graphs, where most edges occur across the clusters. Grappling with such
structure is typically left to the task of graph simplification. We present a
probabilistic model based on non-negative matrix factorization which unifies
clustering and simplification, and provides a framework for modeling arbitrary
graph structure. Our model is based on factorizing the process of taking a
random walk on the graph. It permits an unconstrained parametrization, allowing
for optimization via simple gradient descent. By relaxing the hard clustering
to a soft clustering, our algorithm relaxes potentially hard clustering
problems to a tractable ones. We illustrate our algorithm's capabilities on a
synthetic graph, as well as simple unsupervised learning tasks involving
bipartite and tripartite clustering of orthographic and phonological data.
