Metrics for Parametric Families of Networks

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.

Extracting Complex Topology from Multivariate Functional Approximation: Contours, Jacobi Sets, and Ridge-Valley Graphs

Implicit continuous models, such as functional models and implicit neural networks, are an increasingly popular method for replacing discrete data representations with continuous, high-order, and differentiable surrogates. These models offer new perspectives on the storage, transfer, and analysis of scientific data. In this paper, we introduce the first framework to directly extract complex topological features -- contours, Jacobi sets, and ridge-valley graphs -- from a type of continuous implicit model known as multivariate functional approximation (MFA). MFA replaces discrete data with continuous piecewise smooth functions. Given an MFA model as the input, our approach enables direct extraction of complex topological features from the model, without reverting to a discrete representation of the model. Our work is easily generalizable to any continuous implicit model that supports the queries of function values and high-order derivatives. Our work establishes the building blocks for performing topological data analysis and visualization on implicit continuous models.