Graphon estimation beyond binary edges: inference for decorated graphs with applications to multiplex and weighted networks

We introduce the first doubly non-parametric estimation method for decorated graphons, a generalisation of graphons that encodes edge weights, edge types, and other edge-level attributes in large networks. Graphons describe the limiting behaviour of large unlabelled networks through a symmetric measurable function governing the probability of edge formation, but the standard framework is restricted to binary edge information. Decorated graphons lift this restriction, yet no inference procedure has previously been available for them. The proposed estimator extends classical graphon estimation techniques to this enriched setting. We derive rates of convergence and show that, for compactly supported decorations, these rates agree with known non-parametric rates for estimating real-valued functions. Monte Carlo experiments confirm that the theoretical rates are attained in finite samples, and applications to synthetic and empirical networks show improved fit relative to binary-edge baselines. The method extends graphon-based inference to multiplex networks and attributed graphs simultaneously.