Community Detection: a Refined Axiomization and Beyond.

Borgs et al. [2016] investigate the essential requirements for community in preference networks, and defined six axioms on community detection rules (or community functions). Two of these axioms play critical roles in their system, however, it is coNP-complete to check whether they are satisfied. We propose efficiently-checkable alternatives to the two axioms, and prove that various desirable properties are preserved. Most strikingly, the new axioms enable to easily find constructive and consistent community functions. Actually, we find such a natural community function which is also enumerable and samplable, answering an open problem in the literature. We further discuss how to infer a corresponding preference network from a given collection of communities. We completely solve the problem when the simplest community rule is used, and uncover an inherent limitation of preference networks. Motivated by this observation, we extend the concept of preference to multi-preferences and prove that any collection of communities can be exactly generated by multi-preferences.