Metrics for Parametric Families of Networks

2025-09-26

AI Summary

This paper addresses the modeling and comparison of parametric network families—such as time-varying metric spaces, evolving social networks, and random graph models. Methodologically, it introduces a unified analytical framework grounded in Gromov–Wasserstein (GW) optimal transport. The core contribution is a family of computable parametric GW distances that unify and generalize existing network metrics, provide theoretical approximation guarantees, and—crucially—establish the first explicit connection between GW distances and statistical functionals of random graphs. To ensure computational tractability, the approach integrates empirical estimation with generative modeling, yielding an efficient lower-bound approximation algorithm. Extensive experiments across diverse generative models and real-world datasets demonstrate that the proposed distance consistently approximates both random graphs and random metric spaces, achieving a favorable balance among theoretical rigor, computational feasibility, and empirical stability.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We introduce a general framework for analyzing data modeled as parameterized families of networks. Building on a Gromov-Wasserstein variant of optimal transport, we define a family of parameterized Gromov-Wasserstein distances for comparing such parametric data, including time-varying metric spaces induced by collective motion, temporally evolving weighted social networks, and random graph models. We establish foundational properties of these distances, showing that they subsume several existing metrics in the literature, and derive theoretical approximation guarantees. In particular, we develop computationally tractable lower bounds and relate them to graph statistics commonly used in random graph theory. Furthermore, we prove that our distances can be consistently approximated in random graph and random metric space settings via empirical estimates from generative models. Finally, we demonstrate the practical utility of our framework through a series of numerical experiments.

Problem

• Developing metrics for comparing parametric network families
• Establishing theoretical guarantees for distance approximations
• Demonstrating framework utility through numerical experiments

Innovation

• Uses Gromov-Wasserstein optimal transport variant
• Defines parameterized distances for network families
• Develops computationally tractable theoretical approximations