Doubly stochastic scaling unifies community detection

Graph partitioning, or community detection, has been widely investigated in network science. Yet, the correct community structure on a given network is essentially data-driven. Thus, instead of a formal definition, diverse measures have been conceived to capture intuitive desirable properties shared by most of the community structures. In this work, we propose a preprocessing based on a doubly stochastic scaling of network adjacency matrices, to highlight these desirable properties. By investigating a range of community detection measures, and carefully generalising them to doubly stochastic graphs, we show that such a scaling unifies a whole category of these measures—namely, the so-called linear criteria—onto two unique measures to set up. Finally, to help practitioners setting up these measures, we provide an extensive numerical comparison of the capacity of these measures to uncover community structures within stochastic block models, using the Louvain algorithm.