Robustness to Nonidentical Vertex Sets Across Layers
Investigate whether the multi-network histogram and related joint graphon-estimation methods remain valid when the vertex sets are not identical across network layers, including settings with inter-layer edges and supra-adjacency-based grouping.
References
An area of future work might be to perturb that assumption and assess which methods continue to work. Multilayer networks in general also concern edges between vertices in different layers, so grouping vertices using the supra-adjacency matrix may capture these inter-layer relationships.
Our theoretical guarantees are established under the oracle label and do not extend to the label estimated by Algorithm~\ref{alg:multi_nethist_combined_short}. In the single-layer network case, the minimax risk of graphon estimation separates into a nonparametric term and a clustering term of order $\log k/n$ from assigning $n$ vertices to $k$ groups . Within the graphon estimation framework, consistency of joint latent position estimation across multiple networks has only recently begun to be studied. establish consistency in a dense setting with a single common graphon and without a shared node set, and allow for a shared node set via ordinal embedding but likewise assume an aggregated graphon. Our setting is more challenging than theirs, as it allows layers to share a node set while having distinct graphons and sparsity levels that may decay at different orders. Establishing consistency guarantees for the estimated label under this more general setting is an important direction for future work.