Inference on vertex-aligned graphs is of wide theoretical and practical importance. There are, however, few flexible and tractable statistical models for correlated graphs, and even fewer comprehensive approaches to parametric inference on data arising from such graphs. In this paper, we consider the correlated Bernoulli random graph model (allowing different Bernoulli coefficients and edge correlations for different pairs of vertices), and we introduce a new variance-reducing technique—called balancing—that can refine estimators for model parameters. Specifically, we construct a disagreement statistic and show that it is complete and sufficient; balancing can be interpreted as Rao-Blackwellization with this disagreement statistic. We show that for unbiased estimators of functions of model parameters, balancing generates uniformly minimum variance unbiased estimators (UMVUEs). However, even when unbiased estimators for model parameters do not exist—which, as we prove, is the case with both the heterogeneity correlation and the total correlation parameters—balancing is still useful, and lowers mean squared error. In particular, we demonstrate how balancing can improve the efficiency of the alignment strength estimator for the total correlation, a parameter that plays a critical role in graph matchability and graph matching runtime complexity.
This material is based on research sponsored by the Air Force Research Laboratory and Defense Advanced Research Projects Agency (DARPA) under agreement number FA8750-20-2-1001 and FA8750-18-2-0035. This work is also supported in part by the D3M program of DARPA. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory and DARPA or the U.S. Government. The authors also gratefully acknowledge the support of NIH grant BRAIN U01-NS108637. The authors also thank an anonymous referee for very helpful and thoughtful comments that added much to the quality of the exposition.
"On a complete and sufficient statistic for the correlated Bernoulli random graph model." Electron. J. Statist. 15 (1) 2336 - 2359, 2021. https://doi.org/10.1214/21-EJS1839