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\cite{bickel2009nonparametric} developed a general framework to establishconsistency of community detection in stochastic block model (SBM). In mostapplications of this framework, the community label is discrete. For example,in \citep{bickel2009nonparametric,zhao2012consistency} the degree corrected SBMis assumed to have a discrete degree parameter. In this paper, we generalizethe method of \cite{bickel2009nonparametric} to give consistency analysis ofmaximum likelihood estimator (MLE) in SBM with continuous community label. Weshow that there is a standard procedure to transform the $||\cdot||_2$ errorbound to the uniform error bound. We demonstrate the application of our generalresults by proving the uniform consistency (strong consistency) of the MLE inthe exponential network model with interaction effect. Unfortunately, in thecontinuous parameter case, the condition ensuring uniform consistency weobtained is much stronger than that in the discrete parameter case, namely$n\mu_n^5/(\log n)^{8}\rightarrow\infty$ versus $n\mu_n/\logn\rightarrow\infty$. Where $n\mu_n$ represents the average degree of thenetwork. But continuous is the limit of discrete. So it is not surprising as weshow that by discretizing the community label space into sufficiently small(but not too small) pieces and applying the MLE on the discretized communitylabel space, uniform consistency holds under almost the same condition as indiscrete community label space. Such a phenomenon is surprising since thediscretization does not depend on the data or the model. This reminds us of thethresholding method.

Uniform Consistency in Stochastic Block Model with Continuous Community Label