Joint Estimation of Sparse Multilayer Networks via Graph Limits

This paper introduces the multi-network histogram, a nonparametric estimator for sparse multilayer networks whose layers share vertices but may have different graphon structures and sparsity levels. Its scaled set of graphons assigns each layer a separate sparsity parameter while shared latent variables align the vertex partitions. The estimator maximizes a joint profile likelihood over common block assignments, then estimates layer-specific block densities with a shared, data-driven bandwidth; a homogeneous variant pools layers with the same graphon using sparsity-proportional weights. Theoretical analysis derives weighted mean integrated squared-error bounds and shows that joint information favors smaller bandwidths, allowing finer estimates for sparse layers. In simulations averaged over 100 replications, with 400 vertices and homogeneous graphon function 1, the homogeneous estimator’s WMSE multiplied by 100 fell from 2.798 with 5 layers to 1.523 with 10 layers, while the single-layer network histogram rose from 11.439 to 11.523. On Indian village socioeconomic networks, the method analyzed 12 layers across 231 households, with edge densities ranging from 0.0021 to 0.0198. It selected bandwidth 23 for the joint estimator, preserving ten interpretable groups, whereas fitting the sparsest layer independently collapsed to bandwidth 231 and a single block. The results show that shared vertex structure can substantially improve graphon resolution without imposing a common interaction pattern or common sparsity level.