In [1] the Asymptotic Significance was proposed as a new optimization function for community detection: the main idea is to reduce the resolution limit intrinsically present in the modularity [2] by considering the distance between a null-model discounting the information of the number of internal (to the communities) links and one that does not. Nevertheless the chosen target function does not discount the information contained in the degree sequence. In the present work we extend the Asymptotic Significance approach by adding the information of the degree sequence: results are presented for the standard real network benchmarks (Zachary Karate Club, bottleneck dolphins, …) as well as for synthetic benchmarks, as Fortunato-Barthelemy rings [2]. Community partitions found by the degree sequence corrected asymptotic significance are in substantial agreement with results from other community detections. Moreover the extra information due to the degree sequence permits to highlight non-trivial structure that cannot be observed by the Asymptotic Significance alone. Moreover, our proposal permits to observe the small modules of a Fortunato-Barthelemy ring in most of the cases studied.
