Higher-Order Network Structure Inference: A Topological Approach to Network Selection

Abstract

Thresholding---the pruning of nodes or edges based on their properties or weights---is an essential preprocessing tool for extracting interpretable structure from complex network data, yet existing methods face several key limitations. Threshold selection often relies on heuristic methods or trial and error due to large parameter spaces and unclear optimization criteria, leading to sensitivity where small parameter variations produce significant changes in network structure. Moreover, most approaches focus on pairwise relationships between nodes, overlooking critical higher-order interactions involving three or more nodes. We introduce a systematic thresholding algorithm that uses topological data analysis to identify optimal network parameters by accounting for higher-order structural relationships. Our method uses persistent homology to favor thresholds at which the network's topological representation is stable under small parameter changes, subject to minimum requirements for higher-order features. Hyperparameters allow users to specify minimum requirements for topological features, effectively constraining the parameter search to avoid spurious solutions. We demonstrate the approach using networks of concepts linked by co-occurrence in scientific abstracts. Across scientific fields, paired comparisons with ten alternative thresholds favor our method at the median for finite-feature density and normalized total persistence in homological dimensions one and two. Our approach allows researchers to incorporate domain-specific constraints and extends beyond network thresholding to general parameterization problems in data analysis.