Topological Data Analysis for Characterizing Phase Transitions in High-Dimensional Random Graph Models Using Persistent Homology and Machine Learning Techniques

Project Abstract

This study develops a novel framework combining topological data analysis (TDA), persistent homology, and machine learning to characterize phase transitions in high-dimensional random graph models. We address the challenge that traditional statistical indicators often fail to capture global structural changes in complex networks as they undergo critical transitions, especially in high-dimensional regimes where connectivity patterns become intricate and multi-scale. Our approach constructs affine-invariant filtrations of graphs via edge- and vertex-weighted simplicial complexes derived from multi-parameter random graph ensembles, enabling robust extraction of topological signatures across scales. We compute persistent features such as Betti curves, persistence landscapes, and entropy-based summaries, alongside new topological descriptors tailored for high-dimensional graphs, to encode changes in connected components, cycles, and higher-dimensional holes as a function of model parameters like edge probability, degree constraints, and higher-order interaction terms. We integrate supervised and unsupervised learning to map topological summaries to phase domains and critical points, exploiting kernelized and deep learning architectures to handle the nonlinearity and high dimensionality of the feature space. Our methodology also incorporates stability analysis with respect to perturbations in data and parameter settings, ensuring robustness against sampling noise and finite-size effects. The research investigates several random graph models, including Erd?s–Rényi, stochastic block models, and sparse high-order interaction networks generated via hypergraphs, to evaluate the universality and limitations of topological indicators in signaling phase transitions. Key theoretical contributions include the development of multiscale topological descriptors that capture when the network undergoes fragmentation, emerging giant components, and the formation of cohesive higher-order motifs. We derive asymptotic relationships between topological invariants and phase transition thresholds, and establish conditions under which Persistent Homology features exhibit sharp or smooth transitions across parameter sweeps. Empirically, we demonstrate that combining TDA features with machine learning classifiers enhances precision and recall in identifying critical regions, outperforming baseline graph statistics in noisy high-dimensional settings. The work includes a comprehensive computational pipeline with scalable algorithms for large graphs, leveraging parallel processing and optimized persistent homology libraries. We perform a thorough sensitivity analysis to assess the impact of filtration choices, noise, and sample size on the stability and interpretability of the topological signals. Potential applications span network science, materials physics, and data-driven design of complex systems where phase-transition-like behavior governs functionality. The results offer a principled, geometry-aware lens to detect and quantify critical phenomena in high-dimensional networks, providing robust tools for researchers seeking to understand and predict phase transitions through the geometry of data.

Project Overview

What This Project Is About

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In simple terms, this project looks at how complicated networks (graphs) behave when they grow very large or have many connections. It combines a mathematical way of studying shapes (topology) with modern data analysis techniques to spot smooth changes, called phase transitions, as the network changes. It also uses machine learning to recognize patterns that indicate those transitions.

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The Problem It Addresses

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Many real-world networks can change abruptly as their connections increase or decrease, but it’s often hard to predict when these changes happen just by counting links. This project fills that gap by using a shape-based view of data to detect subtle signs of transitions, which can improve understanding in fields like physics, biology, and social networks.

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Objectives of the Project

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1. Identify how high-dimensional networks transition from one state to another.
2. Apply persistent homology to capture stable topological features across scales.
3. Combine topological features with simple machine learning models to classify phases.
4. Evaluate which features best indicate a phase transition.
5. Provide a framework that is accessible to students with minimal topology background.

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What You Will Do Step by Step

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1. Study basic graph concepts and introduce the idea of higher-dimensional connectivity.
2. Generate or collect synthetic and real network data with varying connection strengths.
3. Compute persistent homology features from network representations.
4. Train simple machine learning models to detect phases using topological features.
5. Validate findings with known transition points and perform sensitivity checks.
6. Interpret results with clear, non-technical explanations.
7. Document methodology and provide reproducible code.

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Expected Outcome

The project should deliver a practical method to detect phase transitions in large networks using topology-informed features, plus a straightforward demonstration using examples. It will offer insights into which topological signals are most informative and provide guidance for future studies and potential applications in data science and network science.