Title: Uniform Consistency in Stochastic Block Model with Continuous Community Label,
Abstract: \cite{bickel2009nonparametric} developed a general framework to establish
consistency of community detection in stochastic block model (SBM). In most
applications of this framework, the community label is discrete. For example,
in \citep{bickel2009nonparametric,zhao2012consistency} the degree corrected SBM
is assumed to have a discrete degree parameter. In this paper, we generalize
the method of \cite{bickel2009nonparametric} to give consistency analysis of
maximum likelihood estimator (MLE) in SBM with continuous community label. We
show that there is a standard procedure to transform the $||\cdot||_2$ error
bound to the uniform error bound. We demonstrate the application of our general
results by proving the uniform consistency (strong consistency) of the MLE in
the exponential network model with interaction effect. Unfortunately, in the
continuous parameter case, the condition ensuring uniform consistency we
obtained is much stronger than that in the discrete parameter case, namely
$n\mu_n^5/(\log n)^{8}\rightarrow\infty$ versus $n\mu_n/\log
n\rightarrow\infty$. Where $n\mu_n$ represents the average degree of the
network. But continuous is the limit of discrete. So it is not surprising as we
show that by discretizing the community label space into sufficiently small
(but not too small) pieces and applying the MLE on the discretized community
label space, uniform consistency holds under almost the same condition as in
discrete community label space. Such a phenomenon is surprising since the
discretization does not depend on the data or the model. This reminds us of the
thresholding method.