Functional-Edged Network Modeling
Haijie Xu, Chen Zhang
TL;DR
The paper introduces Functional Edged Network (FEN), a framework for modeling networks whose edges are functions over a domain, by representing the network as an adjacency functional tensor $\mathcal{X}$ and decomposing it through a symmetric, community-aware Tucker model $\mathcal{X}=\mathbf{B}\times_1\boldsymbol{\Phi}\times_2\boldsymbol{\Phi}\times_3\mathcal{G}$. It handles irregular observations via a masking operator and smooths the functional basis, then estimates the model with a Riemannian conjugate gradient method on the Tucker low-rank manifold, including a SHOSVD-based retraction to enforce symmetry. The authors provide theoretical results on convergence and estimation error bounds and validate the approach through comprehensive simulations and real-world metro data from Hong Kong and Singapore, showing improvements over PCA-based, dynamic-network, and tensor-completion baselines, especially under substantial missingness. The work advances functional data analysis in networks by integrating functional edges, community structure, tensor completion, and non-Euclidean optimization, with practical implications for urban mobility and other domains with irregularly observed functional relationships.
Abstract
Contrasts with existing works which all consider nodes as functions and use edges to represent the relationships between different functions. We target at network modeling whose edges are functional data and transform the adjacency matrix into a functional adjacency tensor, introducing an additional dimension dedicated to function representation. Tucker functional decomposition is used for the functional adjacency tensor, and to further consider the community between nodes, we regularize the basis matrices to be symmetrical. Furthermore, to deal with irregular observations of the functional edges, we conduct model inference to solve a tensor completion problem. It is optimized by a Riemann conjugate gradient descent method. Besides these, we also derive several theorems to show the desirable properties of the functional edged network model. Finally, we evaluate the efficacy of our proposed model using simulation data and real metro system data from Hong Kong and Singapore.