📊 statistics

Statistical Analysis of Network Collections Using Persistent Homology and Functional Data Analysis

This paper introduces functional topological data analysis (funTDA), a novel framework that integrates functional and topological data analysis to enable statistical inference, including mean/variance computation, principal component analysis, and hypothesis testing, on collections of networks by overcoming the challenges of non-Euclidean structures and varying node correspondences.

Original authors: Catherine Higgins, Hulin Wu, Michelle Carey

Published 2026-07-28

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CC BY 4.0

Original authors: Catherine Higgins, Hulin Wu, Michelle Carey

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery, but instead of looking at fingerprints or footprints, you are looking at the invisible connections between things. In the world of science, these connections are called networks. Think of a network like a giant web of friends on social media, a map of how genes talk to each other inside your body, or even the way words link together in a story. Usually, scientists study just one of these webs at a time. But what if you wanted to compare hundreds of different webs to see how they change? Maybe you want to know if the "friendship web" of a happy person looks different from that of a sad person, or if the "gene web" of someone with the flu looks different from someone who is healthy.

The problem is that these webs are messy. They don't sit neatly on a ruler like a stick of wood. One web might have 100 dots (nodes), while another has 1,000. Some dots are connected by strong, thick lines, while others have weak, thin lines. Some lines go one way, and others go both ways. Because they are so different, you can't just use a standard ruler or a basic calculator to measure them. It's like trying to compare the shape of a cloud to the shape of a mountain using only a tape measure; the tools just don't fit. This is the puzzle that statisticians have been trying to solve: How do you measure the "shape" of a messy, changing web when every web is unique?

This paper introduces a clever new toolkit called funTDA (which stands for Functional Topological Data Analysis) to solve this puzzle. The authors, Catherine Higgins, Hulin Wu, and Michelle Carey, propose a way to turn these messy webs into smooth, wiggly lines that mathematicians know how to handle. They do this by looking at the "holes" and "loops" inside the webs. Imagine blowing up a balloon; if you poke a hole in it, the shape changes. In a network, a "loop" is a circle of connections where you can start at one point, travel around, and come back to where you began without retracing your steps. The new method tracks how these loops appear and disappear as you slowly turn up the "volume" of the connections.

Instead of trying to match every single dot in one web to a dot in another (which is like trying to match every single grain of sand on two different beaches), this method ignores the specific names of the dots and focuses on the overall shape. It turns the web into a "persistence diagram," which is like a map showing when loops are born and when they die. Then, it transforms that map into a "persistence landscape," which is essentially a series of hills and valleys. Once the messy web is turned into these smooth hills, the authors can use standard statistical tools—like finding the average hill or seeing how much the hills wiggle—to compare different groups of webs.

The paper tests this idea with computer simulations first. They created thousands of fake networks with different levels of connectedness (some sparse, some dense) and asked if their new method could tell them apart. The results were promising: the method successfully separated the different types of networks, grouping similar ones together and keeping different ones apart. They also compared their method to older techniques that try to force the webs into a standard shape. The older methods struggled when the webs had different numbers of dots or were directed (one-way streets), but the new funTDA method handled these differences easily.

Finally, the authors applied their method to real-world data. They looked at word networks from novels by Jane Austen and Charles Dickens to see if the way words connected in their stories had different topological "shapes." They also looked at gene networks from people exposed to the H3N2 flu virus, comparing those who got sick (symptomatic) to those who didn't (asymptomatic). In both cases, the method found statistically significant differences. For the flu study, it suggested that the gene networks of sick people looked topologically different from those of healthy people, even though the genes themselves were the same. The paper doesn't claim to have solved every problem in network science, but it suggests that this new way of looking at the "shape" of connections is a powerful and flexible tool for understanding complex systems, from literature to biology.

Technical Summary: Statistical Analysis of Network Collections Using Persistent Homology and Functional Data Analysis

1. Problem Statement

The statistical analysis of collections of networks, where each network serves as an independent observational unit, presents significant challenges for conventional statistical methodologies. Traditional approaches often rely on graph-theoretic features (e.g., degree distributions, centrality measures) that reduce networks to vectors in Euclidean space. However, this dimensionality reduction frequently results in substantial information loss regarding global network structure and fails to capture structural equivalence.

A primary obstacle is the lack of a natural Euclidean embedding for networks, particularly when:

Vertex Correspondence is Absent: Networks in a sample may possess different numbers of vertices or arbitrary vertex orderings (e.g., gene regulatory networks where different individuals activate distinct gene subsets).
Structural Heterogeneity: Networks vary in size, directionality (directed vs. undirected), and weighting (weighted vs. unweighted).
Computational Complexity: Existing methods that attempt to handle unlabeled graphs, such as quotient-space approaches based on graph alignment (e.g., Jain and Obermayer, 2009; Guo et al., 2021), require solving NP-hard permutation problems to find optimal alignments or Fréchet means. This limits their scalability to small networks. Similarly, Laplacian-based methods (e.g., Severn et al., 2022) are restricted to networks sharing a common vertex set and suffer from high-dimensional covariance estimation issues.

2. Methodology: Functional Topological Data Analysis (funTDA)

To address these limitations, the authors propose functional topological data analysis (funTDA), a framework integrating Persistent Homology (from Topological Data Analysis) with Functional Data Analysis (FDA). The methodology proceeds through the following pipeline:

2.1. From Networks to Persistence Diagrams

Filtration Construction: For each network $G_g$ , a geodesic distance matrix $\Gamma_g$ is computed, representing the shortest directed path lengths between vertices.
Topological Evolution: A filtration is constructed by increasing a threshold parameter $\epsilon$ . As $\epsilon$ increases, edges are added, and the network evolves.
Feature Tracking: The framework tracks the "birth" ( $b_j$ $b_{j}$ ) and "death" ( $d_j$ $d_{j}$ ) of topological features, specifically focusing on 1-dimensional homology ( $H_1$ ), which corresponds to loops or cycles in the network.
- A feature is "born" when a loop forms.
- It "dies" when the loop is filled by higher-order simplices (e.g., a 2-simplex).
Persistence Diagrams: The evolution is summarized as a set of birth-death pairs $P_g = \{(b_j, d_j)\}$ , forming a persistence diagram. This representation captures global structural characteristics without requiring node correspondence or equal network sizes.

2.2. From Diagrams to Functional Representations

Direct statistical analysis of persistence diagrams is difficult due to their variable cardinality and lack of vector space structure. The funTDA framework transforms these diagrams into Persistence Landscapes:

Each birth-death pair $(b_j, d_j)$ is mapped to a piecewise linear triangular function $f_{(b_j, d_j)}(x)$ .
The persistence landscape $F_g(x)$ is constructed by ordering these triangular functions at each point $x$ and selecting the $k$ -th largest value.
Typically, the first layer ( $k=1$ ) is retained, representing the most persistent (significant) topological features.
This transformation embeds the topological features into a Hilbert space, converting the network sample into a sample of functions $\{F_g\}_{g=1}^N$ .

2.3. Statistical Inference via Functional Data Analysis

Once represented as functions, standard FDA techniques are applied:

Smoothing: Observed landscape values are modeled using B-spline basis expansions to handle discretization and noise.
Descriptive Statistics: Pointwise means and covariance functions are estimated to characterize the central tendency and variability of the network sample.
Functional Principal Component Analysis (FPCA): Used to identify dominant modes of variation across the network collection. Bivariate score plots of FPC scores allow for visual exploration of clustering and group separation.
Hypothesis Testing:
- Functional T-test: Compares two groups of networks by testing if their mean landscape functions differ.
- Functional F-test: Extends this to $k$ groups.
- Global Testing: To address the multiple testing problem inherent in pointwise comparisons, the supremum of the pointwise test statistics ( $T_{max}$ or $F_{max}$ ) is used. Significance is assessed via permutation procedures to approximate the null distribution.

3. Key Contributions

Framework Integration: The paper introduces funTDA, a novel pipeline that bridges topological data analysis and functional data analysis to enable statistical inference on network collections.
Handling Heterogeneity: Unlike Laplacian-based methods, funTDA does not require a common vertex set or node correspondence, making it applicable to networks of varying sizes and structures.
Scalability: By avoiding the NP-hard graph alignment problems inherent in quotient-space methods, funTDA offers a computationally tractable alternative for larger networks and samples.
Global Structure Capture: The method prioritizes global topological features (loops) over local node-level statistics, providing a more robust summary of network topology.
Versatility: The framework is applicable to directed/undirected and weighted/unweighted networks.

4. Results

The methodology was evaluated through simulation studies and two real-world applications:

4.1. Simulation Studies

Differentiation of Connectivity: Using Erdős–Rényi random graphs with varying edge probabilities ( $p=0.1, 0.5, 0.9$ ), FPCA score plots clearly separated networks based on their underlying connectivity structure.
Hypothesis Testing: Functional t-tests and F-tests successfully distinguished between networks generated from different distributions (yielding $p < 0.05$ ) while failing to find significant differences between networks from the same distribution (yielding $p > 0.05$ ). This held true for both small ( $N=5$ ) and large ( $N=20$ ) sample sizes and varying vertex counts.
Comparison with Existing Methods: In comparisons with Quotient Space methods and Laplacian-based approaches (Euclidean, Square Root, Procrustes), funTDA demonstrated comparable or superior ability to distinguish network structures in bivariate score plots, without the computational burden of graph alignment.

4.2. Real-World Applications

Literary Networks: Networks constructed from pairwise word co-occurrences in novels by Jane Austen and Charles Dickens. FPCA and hypothesis testing revealed distinct topological differences between the authors' works.
Gene Regulatory Networks: Networks derived from gene expression data of 17 individuals exposed to H3N2 influenza (9 symptomatic, 8 asymptomatic). The framework successfully identified statistically significant differences in the topological structure of gene regulatory networks between symptomatic and asymptomatic groups.

5. Significance and Claims

The authors position funTDA as a necessary evolution in network science, addressing the growing need to analyze collections of networks rather than single instances. The paper claims that by embedding networks into a functional space via persistence landscapes, researchers can:

Perform standard statistical operations (means, variances, PCA, hypothesis testing) on network data that previously lacked a Euclidean structure.
Conduct inference on networks with varying sizes and without predefined node correspondence, a common scenario in biological and social data.
Avoid the computational intractability of graph matching required by previous quotient-space approaches.

The paper concludes that funTDA provides a robust, scalable, and statistically rigorous framework for exploratory data analysis and inference on network samples, with an implementation made available via an open-source repository.