# Beyond Topological Persistence: Starting from Networks

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## Abstract

**:**

## 1. Introduction

## 2. Persistence via the Poset of Subobjects

**Definition**

**1**

- $p({u}_{1},{v}_{1})\le p({u}_{2},{v}_{1})$ and $p({u}_{2},{v}_{2})\le p({u}_{2},{v}_{1})$, that is to say p is non-decreasing in the first argument, and non-increasing in the second.
- $p({u}_{2},{v}_{1})-p({u}_{1},{v}_{1})\le p({u}_{2},{v}_{2})-p({u}_{1},{v}_{2})$.

**Definition**

**2.**

- $p({u}_{1},{v}_{1})\le p({u}_{2},{v}_{1})$ and $p({u}_{2},{v}_{2})\le p({u}_{2},{v}_{1})$.
- $p({u}_{2},{v}_{1})-p({u}_{1},{v}_{1})\le p({u}_{2},{v}_{2})-p({u}_{1},{v}_{2})$.

**Definition**

**3**

**Definition**

**4**

**Proposition**

**1.**

**Proof.**

**Natural pseudodistance.**It is possible to define the natural pseudodistance on ${\mathbf{C}}_{m}$. In the following, we will adopt some finiteness assumptions, to ensure stability of all persistence functions we will consider. Note that, whenever we refer to categories such as $\mathbf{Set},\mathbf{Poset},\mathbf{Graph},\mathbf{Digraph}$, we always refer to the finite version—finite sets, finite posets, and finite simple graphs and digraphs.

**Finiteness assumptions.**From now on, we assume that every object in $\mathbf{C}$ has only a finite number of distinct subobjects (to ensure tameness in all constructions). Furthermore, we will only consider filtrations F that admit a colimit $F(\infty )$ in ${\mathbf{C}}_{m}$. As every object has only a finite number of distinct subobjects, this means that $F(x\le {x}^{\prime})$ is an isomorphism for sufficiently large $x,{x}^{\prime}$. This will allow us to define the natural pseudodistance [13,20,21].

**Definition**

**5.**

**Stability and universality**For applications, a wished-for quality is stability. A categorical persistence function p on $\mathbf{C}$ is said to be stable if, given filtrations ${F}_{1},{F}_{2}$ in $\mathbf{C}$, for the induced ${p}_{{F}_{1}},{p}_{{F}_{2}}$ and the corresponding persistence diagrams $D{F}_{1},D{F}_{2}$ the inequality

#### 2.1. Preliminaries on Posets

**Definition**

**6.**

**Theorem**

**1**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

- $\mathcal{D}F=\mathcal{D}H$, $\mathcal{D}{F}^{\prime}=\mathcal{D}{H}^{\prime}$,
- $d\left(\mathcal{D}H,\mathcal{D}{H}^{\prime}\right)=\delta \left(H,{H}^{\prime}\right)$,where d is the bottleneck distance between persistence diagrams. Therefore, d is universal with respect to the monic persistence function induced by M.

**Proof.**

#### 2.2. Weakly Directed Properties

**Definition**

**7.**

**Definition**

**8.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 3. Non-Simplicial Graph Persistence

**Definition**

**9.**

**Definition**

**10.**

**Remark**

**1.**

**Remark**

**2.**

**Definition**

**11.**

**Remark**

**3.**

#### 3.1. Clique Communities

**Definition**

**12.**

**Proposition**

**6.**

**Proof.**

**Remark**

**4.**

**Proposition**

**7.**

**Proof.**

#### 3.2. Blocks

**Proposition**

**8.**

**Proof.**

#### 3.3. Edge-Blocks

**Proposition**

**9.**

**Proof.**

#### 3.4. Strong Components in Digraphs

**Proposition**

**10.**

**Proof.**

**Remark**

**5.**

## 4. Conclusions and Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Zomorodian, A.; Carlsson, G. Computing persistent homology. Discret. Comput. Geom.
**2005**, 33, 249–274. [Google Scholar] [CrossRef][Green Version] - Kurlin, V. A fast persistence-based segmentation of noisy 2D clouds with provable guarantees. Pattern Recognit. Lett.
**2016**, 83, 3–12. [Google Scholar] [CrossRef][Green Version] - Rieck, B.; Fugacci, U.; Lukasczyk, J.; Leitte, H. Clique community persistence: A topological visual analysis approach for complex networks. IEEE Trans. Vis. Comput. Graph.
**2018**, 24, 822–831. [Google Scholar] [CrossRef] [PubMed] - Port, A.; Gheorghita, I.; Guth, D.; Clark, J.M.; Liang, C.; Dasu, S.; Marcolli, M. Persistent topology of syntax. Math. Comput. Sci.
**2018**, 12, 33–50. [Google Scholar] [CrossRef][Green Version] - Guerra, M.; De Gregorio, A.; Fugacci, U.; Petri, G.; Vaccarino, F. Homological scaffold via minimal homology bases. Sci. Rep.
**2021**, 11, 5355. [Google Scholar] [CrossRef] [PubMed] - Ferri, M. Persistent topology for natural data analysis—A survey. In Towards Integrative Machine Learning and Knowledge Extraction; Springer: Berlin/Heidelberg, Germany, 2017; pp. 117–133. [Google Scholar]
- Pal, S.; Moore, T.J.; Ramanathan, R.; Swami, A. Comparative topological signatures of growing collaboration networks. In International Workshop on Complex Networks; Springer: Berlin/Heidelberg, Germany, 2017; pp. 201–209. [Google Scholar]
- Lee, H.; Chung, M.K.; Kang, H.; Choi, H.; Kim, Y.K.; Lee, D.S. Abnormal hole detection in brain connectivity by kernel density of persistence diagram and Hodge Laplacian. In Proceedings of the 2018 IEEE 15th International Symposium on Biomedical Imaging (ISBI 2018), Washington, DC, USA, 4–7 April 2018; pp. 20–23. [Google Scholar]
- Expert, P.; Lord, L.D.; Kringelbach, M.L.; Petri, G. Topological Neuroscience. Netw. Neurosci.
**2019**, 3, 653–655. [Google Scholar] [CrossRef] - Hess, K. Topological adventures in neuroscience. In Topological Data Analysis; Springer: Berlin/Heidelberg, Germany, 2020; pp. 277–305. [Google Scholar]
- Turner, K. Rips filtrations for quasimetric spaces and asymmetric functions with stability results. Algebr. Geom. Topol.
**2019**, 19, 1135–1170. [Google Scholar] [CrossRef][Green Version] - Bergomi, M.G.; Vertechi, P. Rank-based persistence. Theory Appl. Categ.
**2020**, 35, 228–260. [Google Scholar] - Lesnick, M. The Theory of the Interleaving Distance on Multidimensional Persistence Modules. Found. Comput. Math.
**2015**, 15, 613–650. [Google Scholar] [CrossRef][Green Version] - De Silva, V.; Munch, E.; Stefanou, A. Theory of interleavings on categories with a flow. Theory Appl. Categ.
**2018**, 33, 583–607. [Google Scholar] - Patel, A. Generalized persistence diagrams. J. Appl. Comput. Topol.
**2018**, 1, 397–419. [Google Scholar] [CrossRef][Green Version] - McCleary, A.; Patel, A. Bottleneck stability for generalized persistence diagrams. Proc. Am. Math. Soc.
**2020**, 148, 3149–3161. [Google Scholar] [CrossRef][Green Version] - McCleary, A.; Patel, A. Edit Distance and Persistence Diagrams Over Lattices. arXiv
**2020**, arXiv:2010.07337. [Google Scholar] - Kim, W.; Mémoli, F. Generalized persistence diagrams for persistence modules over posets. J. Appl. Comput. Topol.
**2021**, 5. [Google Scholar] [CrossRef] - Bubenik, P.; Scott, J.A. Categorification of persistent homology. Discret. Comput. Geom.
**2014**, 51, 600–627. [Google Scholar] [CrossRef][Green Version] - Frosini, P.; Mulazzani, M. Size homotopy groups for computation of natural size distances. Bull. Belg. Math. Soc.
**1999**, 6, 455–464. [Google Scholar] [CrossRef] - D’Amico, M.; Frosini, P.; Landi, C. Natural pseudo-distance and optimal matching between reduced size functions. Acta Appl. Math.
**2010**, 109, 527–554. [Google Scholar] [CrossRef] - Stong, R.E. Finite topological spaces. Trans. Am. Math. Soc.
**1966**, 123, 325–340. [Google Scholar] [CrossRef] - Raptis, G. Homotopy Theory of Posets. Homol. Homotopy Appl.
**2010**, 12, 211–230. [Google Scholar] [CrossRef][Green Version] - Palla, G.; Derényi, I.; Farkas, I.; Vicsek, T. Uncovering the overlapping community structure of complex networks in nature and society. Nature
**2005**, 435, 814. [Google Scholar] [CrossRef] [PubMed][Green Version] - Toivonen, R.; Onnela, J.P.; Saramäki, J.; Hyvönen, J.; Kaski, K. A model for social networks. Phys. A Stat. Mech. Appl.
**2006**, 371, 851–860. [Google Scholar] [CrossRef][Green Version] - Kumpula, J.M.; Onnela, J.P.; Saramäki, J.; Kaski, K.; Kertész, J. Emergence of communities in weighted networks. Phys. Rev. Lett.
**2007**, 99, 228701. [Google Scholar] [CrossRef] [PubMed][Green Version] - Palla, G.; Barabási, A.L.; Vicsek, T. Quantifying social group evolution. Nature
**2007**, 446, 664. [Google Scholar] [CrossRef][Green Version] - Fortunato, S. Community detection in graphs. Phys. Rep.
**2010**, 486, 75–174. [Google Scholar] [CrossRef][Green Version] - Balinski, M. On the Graph Structure of Convex Polyhedra in n–Space. Pac. J. Math.
**1961**, 11, 431–434. [Google Scholar] [CrossRef][Green Version] - Harary, F. The maximum connectivity of a graph. Proc. Natl. Acad. Sci. USA
**1962**, 48, 1142. [Google Scholar] [CrossRef][Green Version] - Bondy, A.; Murty, U. Graph Theory; Graduate Texts in Mathematics; Springer: London, UK, 2011. [Google Scholar]
- Baez, J.C.; Dolan, J. Higher-dimensional algebra and topological quantum field theory. J. Math. Phys.
**1995**, 36, 6073–6105. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**A weighted graph (

**left**), the corresponding filtration (

**above**), and its persistent Betti numbers functions of degree 0 (

**middle**) and 1 (

**right**).

**Figure 2.**Two persistence diagrams $\mathcal{D}F$ (green), $\mathcal{D}{F}^{\prime}$ (red) and the Hasse diagrams of the corresponding poset filtrations $H,\phantom{\rule{4pt}{0ex}}{H}^{\prime}$; edges are marked with the value at which the relation arises.

**Figure 3.**From the top: the inclusion of poset $H\left(5\right)$ into poset $H(\infty )$ of Figure 2, the image under ${T}_{2}$, and the image under ${T}_{3}$ (Definition 11).

**Figure 4.**The weighted graph of Figure 1 (left), the corresponding filtration (above) and its 3-clique community number function.

**Figure 5.**The weighted graphs whose natural pseudodistance equals the bottleneck distance of the persistence diagrams of Figure 2, relative to the persistence function ${p}_{{c}^{2}}$. Above (resp. below), the weighted graph corresponding to the green (resp. red) persistence diagram and to the upper (resp. lower) Hasse diagram of Figure 2.

**Figure 6.**The weighted graph of Figure 1 (left), the corresponding filtration (above) and its 2-block number function.

**Figure 7.**The weighted graph of Figure 1 (left), the corresponding filtration (above) and its 2-edge-block number function.

**Figure 8.**Two orientations of the weighted graph of Figure 1, differing by the orientation of the edge with weight 2. For each, the weighted digraph (left), the corresponding filtration (above) and its persistent strong component number function.

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Bergomi, M.G.; Ferri, M.; Vertechi, P.; Zuffi, L. Beyond Topological Persistence: Starting from Networks. *Mathematics* **2021**, *9*, 3079.
https://doi.org/10.3390/math9233079

**AMA Style**

Bergomi MG, Ferri M, Vertechi P, Zuffi L. Beyond Topological Persistence: Starting from Networks. *Mathematics*. 2021; 9(23):3079.
https://doi.org/10.3390/math9233079

**Chicago/Turabian Style**

Bergomi, Mattia G., Massimo Ferri, Pietro Vertechi, and Lorenzo Zuffi. 2021. "Beyond Topological Persistence: Starting from Networks" *Mathematics* 9, no. 23: 3079.
https://doi.org/10.3390/math9233079