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Stable Homology-Based Cycle Centrality Measures

John Rick D. Manzanares, Paul Samuel P. Ignacio

TL;DR

The paper addresses identifying influential cycles in weighted graphs by extending centrality to cycle generators via persistence and merge histories. It defines three monotone centrality functions $J_1,J_2,J_3$ that accumulate cycle persistence $P_{\\epsilon}(\\sigma)$ and merge information $M_r[\\sigma,\\epsilon]$ across filtrations, with stability bounds expressed through $p$-centrality norms and the bottleneck distance $d_B(D,D')$. The authors provide an algorithm for computing first-order and higher-order merge clusters and demonstrate robustness under perturbations, plus applications to fractal-like point clouds where centrality aligns with persistence and reveals additional structure. The results suggest that homology-based cycle centrality offers a robust, multiscale descriptor that complements traditional persistence-based summaries and can enhance analysis of complex networks and spatial data.

Abstract

Network centrality measures play a crucial role in understanding graph structures, assessing the importance of nodes, paths, or cycles based on directed or reciprocal interactions encoded by vertices and edges. Estrada and Ross extended these measures to simplicial complexes to account for higher-order connections. In this work, we introduce novel centrality measures by leveraging algebraically-computable topological signatures of cycles and their homological persistence. We apply tools from algebraic topology to extract multiscale signatures within cycle spaces of weighted graphs, tracking homology generators persisting across a weight-induced filtration of simplicial complexes built over point clouds. This approach incorporates persistent signatures and merge information of homology classes along the filtration, quantifying cycle importance not only by geometric and topological significance but also by homological influence on other cycles. We demonstrate the stability of these measures under small perturbations using an appropriate metric to ensure robustness in practical applications. Finally, we apply these measures to fractal-like point clouds, revealing their capability to detect information consistent with, and possibly overlooked by, common topological summaries.

Stable Homology-Based Cycle Centrality Measures

TL;DR

The paper addresses identifying influential cycles in weighted graphs by extending centrality to cycle generators via persistence and merge histories. It defines three monotone centrality functions that accumulate cycle persistence and merge information across filtrations, with stability bounds expressed through -centrality norms and the bottleneck distance . The authors provide an algorithm for computing first-order and higher-order merge clusters and demonstrate robustness under perturbations, plus applications to fractal-like point clouds where centrality aligns with persistence and reveals additional structure. The results suggest that homology-based cycle centrality offers a robust, multiscale descriptor that complements traditional persistence-based summaries and can enhance analysis of complex networks and spatial data.

Abstract

Network centrality measures play a crucial role in understanding graph structures, assessing the importance of nodes, paths, or cycles based on directed or reciprocal interactions encoded by vertices and edges. Estrada and Ross extended these measures to simplicial complexes to account for higher-order connections. In this work, we introduce novel centrality measures by leveraging algebraically-computable topological signatures of cycles and their homological persistence. We apply tools from algebraic topology to extract multiscale signatures within cycle spaces of weighted graphs, tracking homology generators persisting across a weight-induced filtration of simplicial complexes built over point clouds. This approach incorporates persistent signatures and merge information of homology classes along the filtration, quantifying cycle importance not only by geometric and topological significance but also by homological influence on other cycles. We demonstrate the stability of these measures under small perturbations using an appropriate metric to ensure robustness in practical applications. Finally, we apply these measures to fractal-like point clouds, revealing their capability to detect information consistent with, and possibly overlooked by, common topological summaries.
Paper Structure (11 sections, 10 theorems, 36 equations, 13 figures, 2 algorithms)

This paper contains 11 sections, 10 theorems, 36 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial Complex
  4. Simplicial Homology
  5. Filtrations and Persistent Homology
  6. Algorithm for Class Merges
  7. Cycle Centrality
  8. Centrality Measures
  9. Stability of Centrality Measures
  10. Application to Fractal-Like Point Clouds
  11. Conclusion

Key Result

Lemma 2.15

Suppose that $[\sigma]$ is a $k$th homology class in a filtered simplicial complex $\mathscr{C}$. If $[\sigma]$ is not $k$-near to any homology class $[\nu]$, then the reduction algorithm produces a unique class representative for $[\sigma]$.

Figures (13)

  • Figure 1: Rips complex of a point cloud
  • Figure 2: Two (red and blue) homologous 1-cycles
  • Figure 3: The $J_3$ centrality plot, with $f_{\sigma} = 1$, of dimension $1$ produced by the Rips filtration of the point cloud sampled around a wedge sum of two annuli.
  • Figure 4: Boxplots for the $1$-centrality distance between the centrality functions of the point cloud in Figure \ref{['fig:graph']} and its perturbations for all noise levels.
  • Figure 5: Boxplots of the difference between the $1$-centrality distance in Figure \ref{['fig:boxplot']} and corresponding bounds given by Proposition \ref{['thm:sharp']}.
  • ...and 8 more figures

Theorems & Definitions (33)

  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.8
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • Definition 2.13
  • ...and 23 more