Equivalences between network modularity and diverse low-dimensional representations

Modularity is a popular clustering objective in network science. Here, we equate normalized and generalized versions of the modularity with variants of $k$-means objective, spectral manifold learning objectives, and UMAP. These equivalences naturally lead to definitions of new representation objectives. As an example, we show that one of these objectives embeds brain-imaging data much better than UMAP. Together, our results unify outwardly distinct representations across unsupervised learning, network science, and imaging neuroscience.