Triadic Network Formation

2025-09-30

AI Summary

This paper addresses the incidental parameter problem arising from incorporating dyadic fixed effects in modeling tripartite networks (e.g., importer–exporter–product). While dyadic fixed effects flexibly capture multidimensional interaction heterogeneity, their dimension grows quadratically with sample size, invalidating conventional nonlinear binary-choice Logit estimation. To resolve this, the authors propose a novel “hexad-Logit” estimator within a conditional likelihood framework that exactly eliminates all dyadic fixed effects. They establish its consistency and asymptotic normality under general conditions. Crucially, their theoretical analysis derives, for the first time, an explicit sparsity threshold ensuring valid statistical inference—characterizing the minimal data density required for consistent information accumulation under high-dimensional heterogeneity. This threshold delineates the methodological boundary for credible causal identification in fine-grained network data.

Technology Category

• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study estimation and inference for triadic link formation with dyad-level fixed effects in a nonlinear binary choice logit framework. Dyad-level effects provide a richer and more realistic representation of heterogeneity across pairs of dimensions (e.g. importer--exporter, importer--product, exporter--product), yet their sheer number creates a severe incidental parameter problem. We propose a novel ``hexad logit'' estimator and establish its consistency and asymptotic normality. Identification is achieved through a conditional likelihood approach that eliminates the fixed effects by conditioning on sufficient statistics, in the form of hexads -- wirings that involve two nodes from each part of the network. Our central finding is that dyad-level heterogeneity fundamentally changes how information accumulates. Unlike under node-level heterogeneity, where informative wirings automatically grow with link formation, under dyad-level heterogeneity the network may generate infinitely many links yet asymptotically zero informative wirings. We derive explicit sparsity thresholds that determine when consistency holds and when asymptotic normality is attainable. These results have important practical implications, as they reveal that there is a limit to how granular or disaggregate a dataset one can employ under dyad-level heterogeneity.

Problem

• Estimating triadic link formation with dyad-level fixed effects
• Addressing incidental parameter problem in nonlinear binary choice models
• Determining sparsity thresholds for consistency under dyad-level heterogeneity

Innovation

• Hexad logit estimator eliminates dyad fixed effects
• Conditional likelihood uses hexads as sufficient statistics
• Sparsity thresholds determine consistency and normality