Persistent Homology with Path-Representable Distances on Graph Data

TL;DR

This work investigates how distance definitions on graphs influence persistent homology in topological data analysis. By introducing path-representable distances and focusing on cost-dominated cases, it proves a 1D persistence barcode injection between such distances, extending beyond the classical $d_{ ext{weight}}$ vs $d_{ ext{edge}}$ comparison. The authors connect distance choices to a structured framework using path choices, domination notions, and minimum spanning trees, and show the injection holds for 1D PH but not in higher dimensions through counterexamples and experiments. The results illuminate how alternative graph distances reveal or hide topological features, with potential implications for graph-based data analysis and applications requiring robust topological summaries at multiple scales.

Abstract

Topological data analysis over graph has been actively studied to understand the underlying topological structure of data. However, limited research has been conducted on how different distance definitions impact persistent homology and the corresponding topological inference. To address this, we introduce the concept of path-representable distance in a general form and prove the main theorem for the case of cost-dominated distances. We found that a particular injection exists among the $1$-dimensional persistence barcodes of these distances with a certain condition. We prove that such an injection relation exists for $0$- and $1$-dimensional homology. For higher dimensions, we provide the counterexamples that show such a relation does not exist.

Figures (8)

• Figure 1: An example where $d_1 \leq d_2$ yet $\textbf{bcd}_1(d_1) \not\subseteq \textbf{bcd}_1(d_2)$, with $\textbf{bcd}_1(d_1) =\{ [4,5] \}$ and $\textbf{bcd}_1(d_2) =\emptyset$. The subgraph $G_{\epsilon}$ of $G$ is defined as $G_{\epsilon} = (V, E_{\epsilon})$, where $E_{\epsilon} = \{ e \in E \mid W_E(e) \leq \epsilon \}$.
• Figure 2: An example where $\textbf{bcd}_1(d_{\text{edge}})$ and $\textbf{bcd}_1(d_{\text{weight}})$ differ yet they form a hierarchical structure within $1$-dimensional hole structures. Bold lines represent newly formed edges, while dashed lines indicate edge that appear at different values.
• Figure 3: The graph $G=(V,E)$ and all of its possible path choice functions represented by their maximal paths.
• Figure 4: The weighted graph $G=(V,E,W_E)$ and all its possible path-representable distances.
• Figure 5: The injective functions $\varphi_{i,j} : \textbf{bcd}_1(d_i) \rightarrow \textbf{bcd}_1(d_j)$, as described in \ref{['Thm: main theorem_general']}, for the four cost-dominated path-representable distances $d_5, d_6, d_7, d_8$ in \ref{['fig:Example_path_repre_distance']}.

Theorems & Definitions (32)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Theorem 2.4
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Theorem 3.4