Joint Estimation of Edge Probabilities for Multi-layer Networks via Neighborhood Smoothing

In this paper we focus on jointly estimating the edge probabilities for multi-layer networks. We define a novel multi-layer graphon, a ternary function in contrast to the bivariate graphon function in the literature by introducing an additional latent layer position parameter, which is model-free and covers a wide range of multi-layer networks. We develop a computationally efficient two-step neighborhood smoothing algorithm to estimate the edge probabilities of multi-layer networks, which requires little tuning and fully utilize the similarity across both network layers and nodes. Numerical experiments demonstrate the advantages of our method over the existing state-of-the-art ones. A real Worldwide Food Import/Export Network dataset example is analyzed to illustrate the better performance of the proposed method over benchmark methods in terms of link prediction.

💡 Research Summary

This paper addresses the problem of jointly estimating edge probabilities in multi‑layer networks, a setting that has become increasingly important as data sources capture multiple modalities of interaction. The authors introduce a novel ternary graphon model f(ξ_i, ξ_j, η_k) that extends the classical bivariate graphon by adding a latent layer position η_k. This formulation subsumes a variety of existing models: when η_k is constant it reduces to the ordinary graphon, when it varies over time it becomes a dynamic graphon, and when it takes only two values it can detect change‑points in temporal networks. The latent variables ξ_i and η_k are assumed i.i.d. Uniform