Decorated graphons for temporal network estimation

We propose a unified nonparametric framework for modeling time-evolving networks using decorated graphons (also known as probability-graphons): symmetric functions that assign to each node pair a probability distribution over binary edge time series.

This generalizes the static decorated-graphon construction to dynamic graphs while preserving node exchangeability and allowing temporal dynamics such as memory and periodicity.

Models in which edges evolve independently given the latent variables, such as autoregressive and Markov edge processes, arise as special cases.

We develop a two-stage estimation procedure that separates temporal modeling from network structure.

Because the network stage requires only mild regularity conditions on the edge-process estimator, a broad class of temporal edge models can be used in the first stage.

We establish nonparametric convergence rates in both block-model and Hölder-smooth regimes, and make explicit how the rate depends on the number of observed time steps and on the quality of the edge-level estimation.

We illustrate the method on simulated data and a hospital contact network, recovering latent community structure and time-varying interaction patterns.

The framework gives a nonparametric baseline for dynamic network analysis with explicit convergence guarantees.