  Network data is prevalent in many contemporary big data applications in which
a common interest is to unveil important latent links between different pairs
of nodes. Yet a simple fundamental question of how to precisely quantify the
statistical uncertainty associated with the identification of latent links
still remains largely unexplored. In this paper, we propose the method of
statistical inference on membership profiles in large networks (SIMPLE) in the
setting of degree-corrected mixed membership model, where the null hypothesis
assumes that the pair of nodes share the same profile of community memberships.
In the simpler case of no degree heterogeneity, the model reduces to the mixed
membership model for which an alternative more robust test is also proposed.
Both tests are of the Hotelling-type statistics based on the rows of empirical
eigenvectors or their ratios, whose asymptotic covariance matrices are very
challenging to derive and estimate. Nevertheless, their analytical expressions
are unveiled and the unknown covariance matrices are consistently estimated.
Under some mild regularity conditions, we establish the exact limiting
distributions of the two forms of SIMPLE test statistics under the null
hypothesis and contiguous alternative hypothesis. They are the chi-square
distributions and the noncentral chi-square distributions, respectively, with
degrees of freedom depending on whether the degrees are corrected or not. We
also address the important issue of estimating the unknown number of
communities and establish the asymptotic properties of the associated test
statistics. The advantages and practical utility of our new procedures in terms
of both size and power are demonstrated through several simulation examples and
real network applications.
