Estimation of graph parameters based on a collection of graphs is essential for a wide range of graph inference tasks. Often, this problem is especially difficult because the sample size is relatively small as compared to the number of parameters to estimate. While using the element-wise sample mean of the adjacency matrices is a common approach, this method does not exploit any underlying structural properties of the graphs. Another challenge for this estimation problem is that, in practice, graphs are generally observed with edge contamination. We consider a weighted latent position graph model contaminated via an edge weight gross error model and propose an estimation methodology based on robust Lq estimation followed by low-rank adjacency spectral decomposition. We demonstrate that, under appropriate conditions, our estimator both maintains Lq robustness and wins the bias-variance trade-off by exploiting low-rank graph structure. We illustrate the improvement offered by our estimator via both simulations and a human connectome data experiment.
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