A classification of well-behaved graph clustering schemes

Community detection in graphs is a problem that is likely to be relevant whenever network data appears, and consequently the problem has received much attention with many different methods and algorithms applied. However, many of these methods are hard to study theoretically, and they optimise for somewhat different goals. A general and rigorous account of the problem and possible methods remains elusive. We study the class of all clustering methods that are monotone under addition of vertices and edges, phrasing this as a functoriality notion. We show that if additionally we require the methods to have no resolution limit in a strong sense, this is equivalent to a notion of representability, which requires them to be explainable and determined by a representing set of graphs. We show that representable clustering methods are always computable in polynomial time, and in any nowhere dense class they are computable in roughly quadratic time. Finally, we extend our definitions to the case of hierarchical clustering, and give a notion of representability for hierarchical clustering schemes.