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arxiv: 1907.08708 · v1 · pith:JQXGGS62new · submitted 2019-07-19 · 🧮 math.AT · cs.SI

Persistence Homology of Networks: Methods and Applications

Mehmet Emin Aktas , Esra Akbas , Ahmed El Fatmaoui This is my paper

Pith reviewed 2026-05-24 18:38 UTC · model grok-4.3

classification 🧮 math.AT cs.SI
keywords persistent homologycomplex networkstopological data analysisfiltrationnetwork miningcomputational topologygraph topology
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The pith

A unified framework organizes recent approaches to persistent homology on networks while highlighting conceptual distinctions between filtrations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper reviews the use of persistent homology to capture topological features like connected components and holes in networks such as social, citation, and biological graphs. It supplies background on the mathematics, surveys filtrations and algorithms for applying the tool to networks, and introduces a unified framework that groups these methods while stressing their differences. Readers in network science would care because standard graph measures stay local and ignore multi-scale topological structure. The review limits itself to approaches drawing attention in mathematics and data mining and ends by listing open directions.

Core claim

The authors develop a unified framework to describe recent approaches to persistent homology on networks, emphasize major conceptual distinctions among them, review different filtrations and algorithms, and point to applications in network mining problems.

What carries the argument

The unified framework that groups methods for defining persistent homology on networks by their choice of filtration and isolates the main conceptual distinctions among those choices.

If this is right

  • Persistent homology supplies global topological summaries that persist across scales and are missed by local graph measures.
  • Choice of filtration determines which topological features are tracked in a given network.
  • The same network can yield different persistence diagrams depending on the filtration chosen.
  • Algorithms differ in how they compute these diagrams from network data.
  • Applications include measuring similarity or distance between entire networks on the basis of topology.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework may help practitioners match a filtration to the scale and type of network they study.
  • The distinctions drawn could serve as a starting point for comparing computational cost across methods.
  • Future work could test whether the same distinctions apply when persistent homology is combined with other network descriptors.

Load-bearing premise

The selected recent approaches that receive significant attention in the mathematics and data mining communities are representative of the key advancements in applying persistent homology to networks.

What would settle it

Identification of a widely used method for persistent homology on networks that cannot be accommodated inside the proposed unified framework or that contradicts the stated conceptual distinctions.

Figures

Figures reproduced from arXiv: 1907.08708 by Ahmed El Fatmaoui, Esra Akbas, Mehmet Emin Aktas.

Figure 1
Figure 1. Figure 1: 0-,1-,2-, and 3-simplex from left to right [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Finite collection of simplices where (a) is a simplicial complex and, (b) and [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An example for constructing the clique complex of a graph [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A simplicial complex with labeled simplices To map an i-simplex to an (i − 1)-simplex, we define the boundary of an i-simplex as the sum of its (i −1)-dimensional faces. Formally speaking, for an i-simplex σ = [v0,..., vi ], its boundary is ∂iσ = X i j=0 [v0,..., vˆj ,..., vi ] where the hat indicates the v j is omitted. We can expand this definition to i-chains. For an i-chain c = ciσi , ∂i(c) = Pci ∂iσi.… view at source ↗
Figure 5
Figure 5. Figure 5: A filtration for δ = 0, 1, 2, 3, 4, 5, 6, 7 (from left to right) Example 1 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Results of the filtration in Figure [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring similarity or distance between networks based on topology. However, classical graph-theoretic measures are usually local and mainly based on differences between either node or edge measurements or correlations without considering the topology of networks such as the connected components or holes. In recent years, mathematical tools and deep learning based methods have become popular to extract the topological features of networks. Persistent homology (PH) is a mathematical tool in computational topology that measures the topological features of data that persist across multiple scales with applications ranging from biological networks to social networks. In this paper, we provide a conceptual review of key advancements in this area of using PH on complex network science. We give a brief mathematical background on PH, review different methods (i.e. filtrations) to define PH on networks and highlight different algorithms and applications where PH is used in solving network mining problems. In doing so, we develop a unified framework to describe these recent approaches and emphasize major conceptual distinctions. We conclude with directions for future work. We focus our review on recent approaches that get significant attention in the mathematics and data mining communities working on network data. We believe our summary of the analysis of PH on networks will provide important insights to researchers in applied network science.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript is a conceptual review of persistent homology (PH) applied to complex networks (e.g., social, citation, biological). It supplies brief mathematical background on PH, surveys filtrations and methods for defining PH on networks, discusses algorithms and applications in network mining, develops a unified framework for describing recent approaches while highlighting conceptual distinctions, and outlines future directions. The scope is restricted to methods receiving significant attention in the mathematics and data mining communities.

Significance. If the unified framework is coherent and the literature summary accurate, the review would provide a useful organizing reference for applied network scientists seeking to incorporate topological data analysis. The work contains no new theorems, empirical results, or parameter-dependent claims; its value lies in synthesis and clarification of distinctions among existing methods.

minor comments (1)
  1. The abstract states that the review 'develop[s] a unified framework'; the manuscript should make explicit (e.g., in an early section) whether this framework is a new synthesis or a reorganization of prior taxonomies.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report contains no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

Review paper with no internal derivations or load-bearing claims

full rationale

This is a survey paper whose stated purpose is to review external literature on persistent homology applied to networks, provide background, and organize existing methods into a unified descriptive framework. No new theorems, equations, empirical predictions, or fitted parameters are asserted anywhere in the manuscript. The scope limitation to 'approaches that get significant attention' is an explicit boundary condition, not a derived result. All technical content is attributed to prior work; no step reduces by construction to the paper's own inputs or self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper summarizing existing methods; no new free parameters, axioms, or invented entities are introduced.

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Reference graph

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