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We propose a class of methods for graphon estimation based on exploitingconnections with nonparametric regression. The idea is to construct an orderingof the nodes in the network, similar in spirit to Chan and Airoldi (2014).However, rather than only considering orderings based on the empirical degreeas in Chan and Airoldi (2014), we use the nearest neighbor algorithm which isan approximating solution to the traveling salesman problem. This in turn canhandle general distances $\hat{d}$ between the nodes, something that allows usto incorporate rich information of the network. Once an ordering isconstructed, we formulate a 2D grid graph denoising problem that we solvethrough fused lasso regularization. For particular choices of the metric$\hat{d}$, we show that the corresponding two-step estimator can attaincompetitive rates when the true model is the stochastic block model, and whenthe underlying graphon is piecewise H\"{o}lder or it has bounded variation.

Graphon estimation via nearest neighbor algorithm and 2D fused lasso denoising