Fig. 7

Boxplot of squared errors of the four estimators of the regression slope parameter is given, where the intercept term of the regression model is zero. On each of 100 Monte Carlo samples, a random graph of \(n=800\) nodes is generated for which the latent position of the i-th node is given by \({\textbf{x}}_i=(t_i^2, 2t_i(1-t_i),(1-t_i)^2)\) where \(t_i \sim ^{iid} U[0,1]\). Response \(y_i\) is generated at the i-th node via the regression model \(y_i=\beta t_i+\epsilon _i\), \(\epsilon _i \sim ^{iid} N(0,\sigma ^2_{\epsilon })\) where \(\beta =5.0\), \(\sigma _{\epsilon }=0.1\). The naive estimator was computed by plugging-in the pre-images of the projections of the optimally rotated adjacency spectral estimates of latent positions. In order to compute \(\hat{\beta }_{adj,\sigma }\), we plug-in the sum of sample variances obtained from another set of Monte Carlo samples where the graphs are generated from the same model. We obtain \(\sum _{i=1}^n \hat{\Gamma }_i\), by using delta method on the asymptotic variance (see 2) of the optimally rotated adjacency spectral estimates of the latent positions, and thus compute \(\hat{\beta }_{adj,\hat{\sigma }}\)