A class of network models recoverable by spectral clustering
Finding communities in networks is a problem that remains difficult, in spite of the amount of attention it has recently received. The Stochastic Block-Model (SBM) is a generative model for graphs with "communities" for which, because of its simplicity, the theoretical understanding has advanced fast in recent years. In particular, there have been various results showing that simple versions of spectral clustering using the Normalized Laplacian of the graph can recover the communities almost perfectly with high probability. Here we show that essentially the same algorithm used for the SBM and for its extension called Degree-Corrected SBM, works on a wider class of Block-Models, which we call Preference Frame Models, with essentially the same guarantees. Moreover, the parametrization we introduce clearly exhibits the free parameters needed to specify this class of models, and results in bounds that expose with more clarity the parameters that control the recovery error in this model class.
The authors present an analysis that establishes the applicability of the classical spectral clustering algorithm on a wide variety of generalized block stochastic models (various communities, sizes, transition probabilities, etc). Their results consider the weak recovery regime, where "most of the nodes" are clustered correctly; certainly a more realistic setup.The ideas presented seem novel, however I am not an expert in the area. The paper is well-written, however it is extremely heavy in notation, making the technical result inaccessible. One example where the technical discussion is confusing: It is not clear how parameter kappa should be thought of. For example, what is a maximum degree bound with a reasonable kappa, where the probability bounds are meaningful?Also, there are many relevant and important recent results on the subject, and a comparison against them will be very useful - Results by Mossel, Neeman, Sly et al. http://arxiv.org/abs/1407.1591 http://arxiv.org/abs/1404.6325 http://arxiv.org/abs/1311.4115 - Results by Abbe, Bandeira, Sandon et al. http://arxiv.org/abs/1506.03729 http://arxiv.org/abs/1503.00609 http://arxiv.org/abs/1405.3267 - Results by Lelarge, Massoulie et al. http://arxiv.org/abs/1506.08621 http://arxiv.org/abs/1406.6897 Finally, a minor comment: the authors mention at some point that: "there are re strong reasons to believe that the KPFM is the most general class for which community recovery by direct spectral clustering is possible." Why is this true? Overall, it seems that the contribution could be a valuable addition to the literature of community detection on SBMs, however it is not clear how it compares against the state of the art spectral methods for extended SBMs as noted above. Some typos: - [19,18,9].In particular -> space missing- overfill in line 090- "degree model"[8]-> space missing