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

TL;DR

This work tackles joint edge-probability estimation in multi-layer networks by introducing a novel ternary graphon $f(\xi_i,\xi_j,\eta_k)$ that encodes layer-specific latent structure. It develops a scalable two-step neighborhood smoothing algorithm that adaptively selects similar layers and neighboring nodes to borrow strength across the multi-layer system, yielding a practical estimator with rigorous convergence guarantees. Theoretical results show enhanced rates as the number of layers $K$ grows (up to a feasible regime) and empirical results—both in simulations and on FAO trade data—demonstrate that the proposed method improves link prediction relative to existing approaches. Overall, the method offers a model-free, data-efficient framework for multi-layer network inference with clear applicability to real-world cross-layer link prediction tasks.

Abstract

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.

Figures (4)

• Figure 1: Estimated probability matrices for graphons 1–4, shown in 4 groups. For each group of images, The heatmap in the lower left corner shows the true $P$ values, while the one in the upper left displays estimates from our MNS method. The heatmaps on the right present estimates from the NS (lower triangle) method and the MultiNeSS method (upper triangle).
• Figure 2: Estimation errors (RMSE and MAE) for probability matrix. All results are multiplied by 100. Red line is for MNS method, green line is for MultiNeSS method and blue line is for NS. Lines marked with circles represent the changes in RMSE and lines marked with cross represent the changes in MAE.
• Figure 3: Estimation errors (RMSE and MAE) for probability matrix of MNS method. All results are multiplied by 100. Red line is for Graphon 1; blue line is for Graphon 2; green line is for Graphon 3 and orange line is for Graphon 4. Lines marked with circles represent the changes in RMSE and Lines marked with cross represent the changes in MAE.
• Figure 4: Receiver operating characteristic curves for link prediction on the overall FAO network; 10% of edges are missing at random. The blue dot-dash curve is for our method (MNS), the green dashed curve for NS and the red dotted line for MultiNeSS. The AUC values for the MNS, NS and MultiNeSS methods were 0.82, 0.78 and 0.72 respectively.

Theorems & Definitions (12)

• Definition 1: Multi-layer Graphon
• Definition 2: Piecewise Lipschitz Multi-graphon Family
• Theorem 1
• Theorem 2
• Lemma 1
• proof : Proof of Lemma \ref{['lemma1']}
• Lemma 2
• proof : Proof of Lemma \ref{['lemma2']}
• Lemma 3
• proof : Proof of Lemma \ref{['lemma3']}