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# Optimal Single Sample Tests for Structured versus Unstructured Network Data

*Proceedings of the 31st Conference On Learning Theory*, PMLR 75:1657-1690, 2018.

#### Abstract

We study the problem of testing, using only a single sample, between mean field distribu- tions (like Curie-Weiss, Erdős-Rényi) and structured Gibbs distributions (like Ising model on sparse graphs and Exponential Random Graphs). Our goal is to test without know- ing the parameter values of the underlying models: only the structure of dependencies is known. We develop a new approach that applies to both the Ising and Exponential Random Graph settings based on a general and natural statistical test. The test can dis- tinguish the hypotheses with high probability above a certain threshold in the (inverse) temperature parameter, and is optimal in that below the threshold no test can distinguish the hypotheses. The thresholds do not correspond to the presence of long-range order in the models. By aggregating information at a global scale, our test works even at very high temperatures. The proofs are based on distributional approximation and sharp concentration of quadratic forms, when restricted to Hamming spheres. The restriction to Hamming spheres is necessary, since otherwise any scalar statistic is useless without explicit knowledge of the temperature parameter. At the same time, this restriction changes the behavior of the functions under consideration, making it hard to directly apply standard methods (i.e., Stein’s method) for concentration of weakly dependent variables. Instead, we carry out an additional tensorization argument using a Markov chain that respects the symmetry of the Hamming sphere.