Skip to main content

A functorial Dowker theorem and persistent homology of asymmetric networks

Abstract

We study two methods for computing network features with topological underpinnings: the Rips and Dowker persistent homology diagrams. Our formulations work for general networks, which may be asymmetric and may have any real number as an edge weight. We study the sensitivity of Dowker persistence diagrams to asymmetry via numerous theoretical examples, including a family of highly asymmetric cycle networks that have interesting connections to the existing literature. In particular, we characterize the Dowker persistence diagrams arising from asymmetric cycle networks. We investigate the stability properties of both the Dowker and Rips persistence diagrams, and use these observations to run a classification task on a dataset comprising simulated hippocampal networks. Our theoretical and experimental results suggest that Dowker persistence diagrams are particularly suitable for studying asymmetric networks. As a stepping stone for our constructions, we prove a functorial generalization of a theorem of Dowker, after whom our constructions are named.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Notes

  1. 1.

    A thread with ideas towards the proof of Theorem 2 was discussed in https://ncatlab.org/nlab/show/Dowker%27s+theorem (Accessed 24 Apr 2017), but the proposed strategy was incomplete. We have inserted an addendum in https://ncatlab.org/nlab/show/Dowker%27s+theorem proposing a complete proof with a slightly different construction.

References

  1. Adamaszek, M., Adams, H.: The Vietoris–Rips complexes of a circle. Pac. J. Math. 290(1), 1–40 (2017)

    MathSciNet  Article  Google Scholar 

  2. Adamaszek, M., Adams, H., Frick, F., Peterson, C., Previte-Johnson, C.: Nerve complexes of circular arcs. Discrete Comput. Geom. 56(2), 251–273 (2016)

    MathSciNet  Article  Google Scholar 

  3. Acemoglu, D., Ozdaglar, A., Tahbaz-Salehi, A.: Systemic risk and stability in financial networks. Am. Econ. Rev. 105(2), 564–608 (2015)

    Article  Google Scholar 

  4. Atkin, R.H.: From cohomology in physics to q-connectivity in social science. Int. J. Man Mach. Stud. 4(2), 139–167 (1972)

    MathSciNet  Article  Google Scholar 

  5. Atkin, R.: Mathematical Structure in Human Affairs. Crane, Russak, New York (1975)

    Google Scholar 

  6. Barmak, J.A.: Algebraic Topology of Finite Topological Spaces and Applications, vol. 2032. Springer, Belrin (2011)

    MATH  Google Scholar 

  7. Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Volume 33 of AMS Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  8. Burghelea, D., Dey, T.K.: Topological persistence for circle-valued maps. Discrete Comput. Geom. 50(1), 69–98 (2013)

    MathSciNet  Article  Google Scholar 

  9. Björner, A.: Topological methods. Handb. Comb. 2, 1819–1872 (1995)

    MathSciNet  MATH  Google Scholar 

  10. Björner, A., Korte, B., Lovász, L.: Homotopy properties of greedoids. Adv. Appl. Math. 6(4), 447–494 (1985)

    MathSciNet  Article  Google Scholar 

  11. Bauer, U., Lesnick, M.: Induced matchings of barcodes and the algebraic stability of persistence. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, p. 355. ACM, London (2014)

  12. Barabási, A.-L., Oltvai, Z.N.: Network biology: understanding the cell’s functional organization. Nat. Rev. Genet. 5(2), 101–113 (2004)

    Article  Google Scholar 

  13. Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)

    MathSciNet  Article  Google Scholar 

  14. Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y: Proximity of persistence modules and their diagrams. In: Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, pp. 237–246. ACM, London (2009)

  15. Chazal, F., Cohen-Steiner, D., Guibas, L.J., Mémoli, F., Oudot, S.Y.: Gromov–Hausdorff stable signatures for shapes using persistence. In: Computer Graphics Forum, vol. 28, pp. 1393–1403. Wiley Online Library, New York (2009)

    Article  Google Scholar 

  16. Chowdhury, S., Dai, B., Mémoli, F.: The importance of forgetting: limiting memory improves recovery of topological characteristics from neural data (2017). arXiv preprint arXiv:1710.11279

  17. Chowdhury, S., Dai, B., Mémoli, F.: Topology of stimulus space via directed network persistent homology. Cosyne Abstracts 2017 (2017)

  18. Carlsson, G., De Silva, V.: Zigzag persistence. Found. Comput. Math. 10(4), 367–405 (2010)

    MathSciNet  Article  Google Scholar 

  19. Chazal, F., De Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules. Springer, Berlin (2016)

    Book  Google Scholar 

  20. Chazal, F., De Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geom. Dedic. 173(1), 193–214 (2014)

    MathSciNet  Article  Google Scholar 

  21. Carstens, C.J., Horadam, K.J.: Persistent homology of collaboration networks. Math. Problems Eng. 2013, 815035 (2013)

    MathSciNet  Article  Google Scholar 

  22. Curto, C., Itskov, V.: Cell groups reveal structure of stimulus space. PLoS Comput. Biol. 4(10), e1000205 (2008)

    MathSciNet  Article  Google Scholar 

  23. Carlsson, G., Mémoli, F.: Persistent clustering and a theorem of J. Kleinberg (2008). arXiv preprint arXiv:0808.2241

  24. Carlsson, G., Mémoli, F.: Characterization, stability and convergence of hierarchical clustering methods. J. Mach. Learn. Res. 11, 1425–1470 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Chowdhury, S., Mémoli, F.: Metric structures on networks and applications. In: 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1470–1472 (2015)

  26. Chowdhury, S., Mémoli, F.: Distances between directed networks and applications. In: 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 6420–6424. IEEE, Washington (2016)

  27. Chowdhury, S., Mémoli, F.: Persistent homology of directed networks. In: 2016 50th Asilomar Conference on Signals, Systems and Computers, pp. 77–81. IEEE, Washington (2016)

  28. Chowdhury, S., Mémoli, F.: Distances and isomorphism between networks and the stability of network invariants (2017). arXiv preprint arXiv:1708.04727

  29. Chowdhury, S., Mémoli, F.: Persistent path homology of directed networks. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1152–1169. SIAM, Philodelphia (2018)

    Chapter  Google Scholar 

  30. Carlsson, G.E., Mémoli, F., Ribeiro, A., Segarra, S.: Hierarchical quasi-clustering methods for asymmetric networks. In: Proceedings of the 31th International Conference on Machine Learning, pp. 352–360 (2014)

  31. Chazal, F., Oudot, S.Y: Towards persistence-based reconstruction in Euclidean spaces. In: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, pp. 232–241. ACM, New York (2008)

  32. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007)

    MathSciNet  Article  Google Scholar 

  33. Collins, A., Zomorodian, A., Carlsson, G., Guibas, L.J.: A barcode shape descriptor for curve point cloud data. Comput. Graph. 28(6), 881–894 (2004)

    Article  Google Scholar 

  34. Carlsson, G., Zomorodian, A., Collins, A., Guibas, L.J.: Persistence barcodes for shapes. Int. J. Shape Model. 11(02), 149–187 (2005)

    Article  Google Scholar 

  35. Dey, T.K., Fan, F., Wang, Y.: Computing topological persistence for simplicial maps. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, pp. 345. ACM, London (2014)

  36. Dłotko, P., Hess, K., Levi, R., Nolte, M., Reimann, M., Scolamiero, M., Turner, K., Muller, E., Markram, H.: Topological analysis of the connectome of digital reconstructions of neural microcircuits (2016). arXiv preprint arXiv:1601.01580

  37. Dantchev, S., Ivrissimtzis, I.: Efficient construction of the Čech complex. Comput. Graph. 36(6), 708–713 (2012)

    Article  Google Scholar 

  38. Dabaghian, Y., Mémoli, F., Frank, L., Carlsson, G.: A topological paradigm for hippocampal spatial map formation using persistent homology. PLoS Comput. Biol. 8(8), e1002581 (2012)

    Article  Google Scholar 

  39. Dowker, C.H.: Homology groups of relations. Ann. Math. 56, 84–95 (1952)

    MathSciNet  Article  Google Scholar 

  40. De Silva, V., Carlsson, G.: Topological estimation using witness complexes. In: Proceedings of the Symposium on Point-Based Graphics, pp. 157–166 (2004)

  41. Elliott, M., Golub, B., Jackson, M.O.: Financial networks and contagion. Am Econ Rev 104(10), 3115–3153 (2014)

    Article  Google Scholar 

  42. Edelsbrunner, H., Harer, J.: Persistent homology—a survey. Contemp Math 453, 257–282 (2008)

    MathSciNet  Article  Google Scholar 

  43. Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Soc, Providence (2010)

    MATH  Google Scholar 

  44. Efrat, A., Itai, A., Katz, M.J.: Geometry helps in bottleneck matching and related problems. Algorithmica 31(1), 1–28 (2001)

    MathSciNet  Article  Google Scholar 

  45. Edelsbrunner, H., Jabłoński, G., Mrozek, M.: The persistent homology of a self-map. Found. Comput. Math. 15(5), 1213–1244 (2015)

    MathSciNet  Article  Google Scholar 

  46. Easley, D., Kleinberg, J.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, New York (2010)

    Book  Google Scholar 

  47. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002)

    MathSciNet  Article  Google Scholar 

  48. Edelsbrunner, H., Morozov, D.: Persistent homology: theory and practice. In: European Congress of Mathematics Kraków, 2–7 July, 2012, pp. 31–50 (2014)

  49. Engström, A.: Complexes of directed trees and independence complexes. Discrete Math. 309(10), 3299–3309 (2009)

    MathSciNet  Article  Google Scholar 

  50. Edelsbrunner, H., Wagner, H.: Topological data analysis with Bregman divergences (2017)

  51. Frosini, P., Landi, C.: Size theory as a topological tool for computer vision. Pattern Recogn. Image Anal. 9(4), 596–603 (1999)

    Google Scholar 

  52. Frosini, P.: Measuring shapes by size functions. In: Intelligent Robots and Computer Vision X: Algorithms and Techniques, pp. 122–133. International Society for Optics and Photonics (1992)

  53. Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. 45(1), 61–75 (2008)

    MathSciNet  Article  Google Scholar 

  54. Ghrist, R.: Elementary Applied Topology. Createspace, Scotts Valley (2014)

    Google Scholar 

  55. Giusti, C., Pastalkova, E., Curto, C., Itskov, V.: Clique topology reveals intrinsic geometric structure in neural correlations. Proc. Natl. Acad. Sci. 112(44), 13455–13460 (2015)

    MathSciNet  Article  Google Scholar 

  56. Horak, D., Maletić, S., Rajković, M.: Persistent homology of complex networks. J. Stat. Mech. Theory Exp. 2009(03), P03034 (2009)

    MathSciNet  Article  Google Scholar 

  57. Hiraoka, Y., Nakamura, T., Hirata, A., Escolar, E.G., Matsue, K., Nishiura, Y.: Hierarchical structures of amorphous solids characterized by persistent homology. Proc. Natl. Acad. Sci. 113(26), 7035–7040 (2016)

    Article  Google Scholar 

  58. Huson, D.H., Rupp, R., Scornavacca, C.: Phylogenetic Networks: Concepts, Algorithms and Applications. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  59. Johnson, J.: Hypernetworks in the Science of Complex Systems, vol. 3. World Scientific, Singapore (2013)

    MATH  Google Scholar 

  60. Khalid, A., Kim, B.S., Chung, M.K., Ye, J.C., Jeon, D.: Tracing the evolution of multi-scale functional networks in a mouse model of depression using persistent brain network homology. NeuroImage 101, 351–363 (2014)

    Article  Google Scholar 

  61. Kumar, R., Novak, J., Tomkins, A.: Structure and evolution of online social networks. In: Link Mining: Models, Algorithms, and Applications, pp. 337–357. Springer, Berlin (2010)

    Chapter  Google Scholar 

  62. Kalton, N.J., Ostrovskii, M.I.: Distances between Banach spaces. In: Fliess, M. (ed.) Forum Mathematicum, vol. 11, pp. 17–48. Walter de Gruyter, Berlin (1999)

    Google Scholar 

  63. Kozlov, D.: Combinatorial Algebraic Topology, vol. 21. Springer, Berlin (2007)

    MATH  Google Scholar 

  64. Krishnamoorthy, B., Provan, S., Tropsha, A.: A topological characterization of protein structure. In: Pardalos, P.M., Boginski, V.L., Alkis, V. (eds.) Data Mining in Biomedicine, pp. 431–455. Springer, Berlin (2007)

    Chapter  Google Scholar 

  65. Lee, H., Chung, M.K., Kang, H., Kim, B.-N., Lee, D.S.: Computing the shape of brain networks using graph filtration and Gromov–Hausdorff metric. In: Medical Image Computing and Computer-Assisted Intervention—MICCAI 2011, pp. 302–309. Springer, Berlin (2011)

    Google Scholar 

  66. Lefschetz, S.: Algebraic Topology, vol. 27. American Mathematical Soc, Providence (1942)

    MATH  Google Scholar 

  67. Masys, A.J.: Networks and Network Analysis for Defence and Security. Springer, Berlin (2014)

    Book  Google Scholar 

  68. Mac Lane, S.: Categories for the Working Mathematician, vol. 5. Springer, Berlin (2013)

    MATH  Google Scholar 

  69. Munkres, J.R.: Elements of Algebraic Topology, vol. 7. Addison-Wesley, Reading (1984)

    MATH  Google Scholar 

  70. Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)

    MathSciNet  Article  Google Scholar 

  71. Dowker’s theorem. https://ncatlab.org/nlab/show/Dowker%27s+theorem. Accessed 24 Apr 2017

  72. O’Keefe, J., Dostrovsky, J.: The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat. Brain Res. 34(1), 171–175 (1971)

    Article  Google Scholar 

  73. Pessoa, L.: Understanding brain networks and brain organization. Phys. Life Rev. 11(3), 400–435 (2014)

    Article  Google Scholar 

  74. Petri, G., Scolamiero, M., Donato, I., Vaccarino, F.: Topological strata of weighted complex networks. PLoS ONE 8(6), e66506 (2013)

    Article  Google Scholar 

  75. Robins, V.: Towards computing homology from finite approximations. Topol. Proc. 24, 503–532 (1999)

    MathSciNet  MATH  Google Scholar 

  76. Rubinov, M., Sporns, O.: Complex network measures of brain connectivity: uses and interpretations. NeuroImage 52(3), 1059–1069 (2010)

    Article  Google Scholar 

  77. Schmiedl, F.: Shape matching and mesh segmentation. Dissertation, Technische Universität München, München (2015)

  78. Sporns, O., Kötter, R.: Motifs in brain networks. PLoS Biol. 2(11), e369 (2004)

    Article  Google Scholar 

  79. Spanier, E.H.: Algebraic Topology, vol. 55. Springer, Berlin (1994)

    MATH  Google Scholar 

  80. Sporns, O.: Networks of the Brain. MIT Press, Cambridge (2011)

    MATH  Google Scholar 

  81. Sporns, O.: Discovering the Human Connectome. MIT Press, Cambridge (2012)

    Google Scholar 

  82. Turner, K.: Generalizations of the Rips filtration for quasi-metric spaces with persistent homology stability results (2016). arXiv preprint arXiv:1608.00365

  83. Tausz, A., Vejdemo-Johansson, M., Adams, H.: Javaplex: a research software package for persistent (co)homology (2011)

  84. Weinberger, S.: What is... persistent homology? Not. AMS 58(1), 36–39 (2011)

    MathSciNet  MATH  Google Scholar 

  85. Xia, K., Wei, G.-W.: Persistent homology analysis of protein structure, flexibility, and folding. Int. J. Numer. Methods Biomed. Eng. 30(8), 814–844 (2014)

    MathSciNet  Article  Google Scholar 

  86. Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This work was supported by NSF Grants IIS-1422400 and CCF-1526513. We thank Pascal Wild and Zhengchao Wan for pointing out errors on an early preprint, and also Guilherme Vituri, Osman Okutan, and Tim Porter for useful discussions. We are especially thankful to Henry Adams for numerous helpful observations and suggestions, especially regarding the material in Appendix B, and for suggesting the proof strategy for Theorem 49.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Facundo Mémoli.

Appendices

Appendix A: Proofs

Proof of Lemma 8

The first inequality holds by the Algebraic Stability Theorem. For the second inequality, note that the contiguous simplicial maps in the diagrams above induce chain maps between the corresponding chain complexes, and these in turn induce equal linear maps at the level of homology vector spaces. To be more precise, first consider the maps \(t_{\delta +\eta ,\delta '+\eta }\circ \varphi _\delta \) and \(\varphi _{\delta '}\circ s_{\delta ,\delta '}\). These simplicial maps induce linear maps \((t_{\delta +\eta ,\delta '+\eta }\circ \varphi _\delta )_\#, (\varphi _{\delta '}\circ s_{\delta ,\delta '})_\#: H_k(\mathfrak {F}^\delta ) \rightarrow H_k(\mathfrak {G}^{\delta '+\eta })\). Because the simplicial maps are assumed to be contiguous, we have:

$$\begin{aligned} (t_{\delta +\eta ,\delta '+\eta }\circ \varphi _\delta )_\# = (\varphi _{\delta '}\circ s_{\delta ,\delta '})_\#. \end{aligned}$$

By invoking functoriality of homology, we then have:

$$\begin{aligned} (t_{\delta +\eta ,\delta '+\eta })_\# \circ (\varphi _\delta )_\# = (\varphi _{\delta '})_\#\circ (s_{\delta ,\delta '})_\#. \end{aligned}$$

Analogous results hold for the other pairs of contiguous maps. Thus we obtain commutative diagrams upon passing to homology, and so \(\mathcal {H}_k(\mathfrak {F}), \mathcal {H}_k(\mathfrak {G})\) are \(\eta \)-interleaved for each \(k\in \mathbb {Z}_+\). Thus we get:

$$\begin{aligned} d_{{\text {I}}}(\mathcal {H}_k(\mathfrak {F}), \mathcal {H}_k(\mathfrak {G}))\le \eta . \end{aligned}$$

\(\square \)

Proof of Proposition 9

First we show that:

$$\begin{aligned} d_{\mathcal {N}}(X,Y)\ge & {} \tfrac{1}{2}\inf \{\max ({\text {dis}}(\varphi ),{\text {dis}}(\psi ),C_{X,Y}(\varphi ,\psi ),\\&C_{Y,X}(\psi ,\varphi )) : \varphi :X \rightarrow Y, \psi :Y \rightarrow X \text { any maps}\}. \end{aligned}$$

Let \(\eta = d_{\mathcal {N}}(X,Y)\), and let R be a correspondence such that \({\text {dis}}(R) = 2\eta \). We define maps \(\varphi :X\rightarrow Y\) and \(\psi :Y\rightarrow X\) as follows: for each \(x\in X\), set \(\varphi (x)=y\) for some y such that \((x,y)\in R\). Similarly, for each \(y\in Y\), set \(\psi (y)=x\) for some x such that \((x,y)\in R\). Thus for any \(x \in X, y\in Y\), we obtain \(|\omega _X(x,\psi (y)) - \omega _Y(\varphi (x),y)| \le 2\eta \) and \(|\omega _X(\psi (y),x) - \omega _Y(y,\varphi (x))| \le 2\eta \). So we have both \(C_{X,Y}(\varphi ,\psi ) \le 2\eta \) and \(C_{Y,X}(\psi ,\varphi ) \le 2\eta \). Also for any \(x,x' \in X\), we have \((x,\varphi (x)),(x',\varphi (x')) \in R\). Thus we also have

$$\begin{aligned} |\omega _X(x,x') - \omega _Y(\varphi (x),\varphi (x'))| \le 2\eta . \end{aligned}$$

So \({\text {dis}}(\varphi ) \le 2\eta \) and similarly \({\text {dis}}(\psi ) \le 2\eta \). This proves the “\(\ge \)” case.

Next we wish to show:

$$\begin{aligned} d_{\mathcal {N}}(X,Y)\le & {} \tfrac{1}{2}\inf \{\max ({\text {dis}}(\varphi ),{\text {dis}}(\psi ),C_{X,Y}(\varphi ,\psi ),\\&C_{Y,X}(\psi ,\varphi )) : \varphi :X \rightarrow Y, \psi :Y \rightarrow X \text { any maps}\}. \end{aligned}$$

Suppose \(\varphi , \psi \) are given, and \(\frac{1}{2}\max ({\text {dis}}(\varphi ),{\text {dis}}(\psi ),C_{X,Y}(\varphi ,\psi ),C_{Y,X}(\psi ,\varphi )) = \eta \).

Let \(R_X = \left\{ (x,\varphi (x) : x\in X\right\} \) and let \(R_Y = \left\{ (\psi (y),y) : y\in Y\right\} \). Then \(R = R_X \cup R_Y\) is a correspondence. We wish to show that for any \(z = (a,b), z' = (a',b') \in R\),

$$\begin{aligned}|\omega _X(a,a') - \omega _Y(b,b')| \le 2\eta .\end{aligned}$$

This will show that \({\text {dis}}(R) \le 2\eta \), and so \(d_{\mathcal {N}}(X,Y) \le \eta \).

To see this, let \(z,z' \in R\). Note that there are four cases: (1) \(z,z' \in R_X\), (2) \(z,z' \in R_Y\), (3) \(z \in R_X, z' \in R_Y\), and (4) \(z\in R_Y, z'\in R_X\). In the first two cases, the desired inequality follows because \({\text {dis}}(\varphi ), {\text {dis}}(\psi ) \le 2\eta \). The inequality follows in cases (3) and (4) because \(C_{X,Y}(\varphi ,\psi ) \le 2\eta \) and \(C_{Y,X}(\psi ,\varphi ) \le 2\eta \), respectively. Thus \(d_{\mathcal {N}}(X,Y) \le \eta \). \(\square \)

Proof of Proposition 22

It suffices to show that \(\Phi \) is a simplicial approximation to \(\mathcal {E}_{|\Sigma |}\), i.e. whenever \(\mathcal {E}_{|\Sigma |}(x) \in |{\sigma }|\) for some vertex \(x \in |\Sigma ^{(1)}|\) and some simplex \(\sigma \in |\Sigma |\), we also have \(|\Phi |(x) \in |{\sigma }|\) (Spanier 1994, §3.4). Here \(|\sigma |\) denotes the closed simplex of \(\sigma \); for any simplex \(\sigma =[v_0,\ldots , v_k]\), this is the collection of formal convex combinations \(\sum _{i=0}^ka_iv_i\) with \(a_i \ge 0\) for each \(0\le i \le k\) and \(\sum _{i=0}^ka_i =1\).

Let \(x = \sum _{i=0}^ka_i\sigma _i\) be a vertex in \(|\Sigma ^{(1)}|\), with each \(a_i > 0\). Then we have \(\mathcal {E}_{|\Sigma |}(x) = \sum _{i=0}^ka_i\mathcal {B}(\sigma _i) = \sum _{i=0}^ka_i\sum _{v\in \sigma _i}v/{{\text {card}}(\sigma _i)},\) a vertex in \(|\sigma _k|\).

Also we have \(|\Phi |(x) = \sum _{i=0}^ka_i\Phi (\sigma _i)\), a vertex in \(|{\sigma _k}|\). Thus \(\Phi \) is a simplicial approximation to \(\mathcal {E}_{|\Sigma |}\), and so we have \(|\Phi |\simeq \mathcal {E}_{|\Sigma |}\). \(\square \)

Proof of Proposition 39

Let \(\delta \in \mathbb {R}\). We first claim that \(\mathfrak {D}^{{\text {si}}}_\delta (X) = \mathfrak {D}^{{\text {so}}}_\delta (X^\top )\). Let \(\sigma \in \mathfrak {D}^{{\text {si}}}_\delta (X)\). Then there exists \(x'\) such that \(\omega _X(x,x')\le \delta \) for any \(x\in \sigma \). Thus \(\omega _{X^\top }(x',x)\le \delta \). So \(\sigma \in \mathfrak {D}^{{\text {so}}}_\delta (X^{\top })\). A similar argument shows the reverse containment. This proves our claim. Thus for \(\delta \le \delta ' \le \delta ''\), we obtain the following diagram:

figuren

Since the maps \(\mathfrak {D}^{{\text {si}}}_\delta \rightarrow \mathfrak {D}^{{\text {si}}}_{\delta '}\), \(\mathfrak {D}^{{\text {so}}}_\delta \rightarrow \mathfrak {D}^{{\text {so}}}_{\delta '}\) for \(\delta '\ge \delta \) are all inclusion maps, it follows that the diagrams commute. Thus at the homology level, we obtain, via functoriality of homology, a commutative diagram of vector spaces where the intervening vertical maps are isomorphisms. By the Persistence Equivalence Theorem (21), the diagrams \({\text {Dgm}}_k^{{\text {si}}}(X)\) and \({\text {Dgm}}_k^{{\text {so}}}{(X^\top )}\) are equal. By invoking Corollary 20, we obtain \({\text {Dgm}}_k^{\mathfrak {D}}(X)={\text {Dgm}}_k^{\mathfrak {D}}(X^\top )\). \(\square \)

Proof of Proposition 41

Let \(\delta \in \mathbb {R}\). For notational convenience, we write, for each \(k\in \mathbb {Z}_+\),

figureo

First note that pair swaps do not affect the entry of 0-simplices into the Dowker filtration. More precisely, for any \(x\in X\), we can unpack the definition of \(R_{\delta ,X}\) (Eq. 3) to obtain:

$$\begin{aligned}{}[x]\in \mathfrak {D}^{{\text {si}}}_\delta \iff \omega _X(x,x)\le \delta \iff \omega _X^{z,z'}(x,x)\le \delta \iff [x]\in \mathfrak {D}^{{\text {si}}}_{\delta ,S}. \end{aligned}$$

Thus for any \(\delta \in \mathbb {R}\), we have \(C_0^\delta = C_0^{\delta ,S}\). Since all 0-chains are automatically 0-cycles, we have \(\ker (\partial _0^\delta )=\ker (\partial _0^{\delta ,S})\).

Next we wish to show that \({\text {im}}(\partial _1^{\delta })={\text {im}}(\partial _1^{\delta ,S})\) for each \(\delta \in \mathbb {R}\). Let \(\gamma \in C_1^\delta \). We first need to show the forward inclusion, i.e. that \(\partial _1^\delta (\gamma ) \in {\text {im}}(\partial _1^{\delta ,S})\). It suffices to show this for the case that \(\gamma \) is a single 1-simplex \([x,x']\in \mathfrak {D}^{{\text {si}}}_\delta \); the case where \(\gamma \) is a linear combination of 1-simplices will then follow by linearity. Let \(\gamma =[x,x']\in \mathfrak {D}^{{\text {si}}}_\delta \) for \(x,x'\in X\). Then we have the following possibilities:

  1. (1)

    \(x'' \in X{\setminus }\{z,z'\}\) is a \(\delta \)-sink for \([x,x']\).

  2. (2)

    z (or \(z'\)) is the only \(\delta \)-sink for \([x,x']\), and \(x,x'\not \in \left\{ z,z'\right\} \).

  3. (3)

    z (or \(z'\)) is the only \(\delta \)-sink for \([x,x']\), and either x or \(x'\) belongs to \(\left\{ z,z'\right\} \).

  4. (4)

    z (or \(z'\)) is the only \(\delta \)-sink for \([x,x']\), and both \(x,x'\) belong to \(\left\{ z,z'\right\} \).

In cases (1), (2), and (4), the \((z,z')\)-pair swap has no effect on \([x,x']\), in the sense that we still have \([x,x']\in \mathfrak {D}^{{\text {si}}}_{\delta ,S}\). So \([x']-[x]=\partial _1^\delta (\gamma )=\partial _1^{\delta ,S}(\gamma )\in {\text {im}}(\partial _1^{\delta ,S})\). Next consider case (3), and assume for notational convenience that \([x,x']=[z,x']\) and \(z'\) is the only \(\delta \)-sink for \([z,x']\). By the definition of a \(\delta \)-sink, we have \(\overline{\omega }_X(z,z')\le \delta \) and \(\overline{\omega }_X(x',z')\le \delta \). Notice that we also have:

$$\begin{aligned}{}[z,z'],[z',x']\in \mathfrak {D}^{{\text {si}}}_\delta , \text { with }z'\text { as a }\delta \text {-sink}. \end{aligned}$$

After the \((z,z')\)-pair swap, we still have \(\overline{\omega }_X^{z,z'}(x',z')\le \delta \), but possibly \(\overline{\omega }_X^{z,z'}(z,z')> \delta \). So it might be the case that \([z,x']\not \in \mathfrak {D}^{{\text {si}}}_{\delta ,S}\). However, we now have:

$$\begin{aligned}&[z',x']\in \mathfrak {D}^{{\text {si}}}_{\delta ,S}, \text { with }z'\text { as a }\delta \text {-sink, and}\\&[z,z'] \in \mathfrak {D}^{{\text {si}}}_{\delta ,S}, \text { with }z\text { as a }\delta \text {-sink}. \end{aligned}$$

Then we have:

$$\begin{aligned} \partial _1^\delta (\gamma )&=\partial _1^\delta ([z,x'])=x'-z =z'-z + x'-z'\\&=\partial _1^{\delta }([z,z'])+\partial _1^{\delta }([z',x'])\\&=\partial _1^{\delta ,S}([z,z'])+\partial _1^{\delta ,S}([z',x'])\in {\text {im}}(\partial _1^{\delta ,S}), \end{aligned}$$

where the last equality is defined because we have checked that \([z,z'],[z',x']\in \mathfrak {D}^{{\text {si}}}_{\delta ,S}\). Thus \({\text {im}}(\partial _1^{\delta })\subseteq {\text {im}}(\partial _1^{\delta ,S})\), and the reverse inclusion follows by a similar argument.

Since \(\delta \in \mathbb {R}\) was arbitrary, this shows that \({\text {im}}(\partial _1^{\delta })= {\text {im}}(\partial _1^{\delta ,S})\) for each \(\delta \in \mathbb {R}\). Previously we had \(\ker (\partial _0^\delta )=\ker (\partial _0^{\delta ,S})\) for each \(\delta \in \mathbb {R}\). It then follows that \(H_0(\mathfrak {D}^{{\text {si}}}_\delta )=H_0(\mathfrak {D}^{{\text {si}}}_{\delta ,S})\) for each \(\delta \in \mathbb {R}\).

Next let \(\delta '\ge \delta \in \mathbb {R}\), and for any \(k\in \mathbb {Z}_+\), let \(f_k^{\delta ,\delta '}:C_k^\delta \rightarrow C_k^{\delta '}, g_k^{\delta ,\delta '}:C_k^{\delta ,S} \rightarrow C_k^{\delta ',S}\) denote the chain maps induced by the inclusions \(\mathfrak {D}^{{\text {si}}}_\delta \hookrightarrow \mathfrak {D}^{{\text {si}}}_{\delta '}, \mathfrak {D}^{{\text {si}}}_{\delta ,S} \hookrightarrow \mathfrak {D}^{{\text {si}}}_{\delta ',S}\). Since \(\mathfrak {D}^{{\text {si}}}_\delta \) and \(\mathfrak {D}^{{\text {si}}}_{\delta ,S}\) have the same 0-simplices at each \(\delta \in \mathbb {R}\), we know that \(f_0^{\delta ,\delta '}\equiv g_0^{\delta ,\delta '}\).

Let \(\gamma \in \ker (\partial _0^\delta )=\ker (\partial _0^{\delta ,S})\), and let \(\gamma +{\text {im}}(\partial _1^\delta ) \in H_0(\mathfrak {D}^{{\text {si}}}_\delta )\). Then observe that

$$\begin{aligned} (f_0^{\delta ,\delta '})_\#(\gamma + {\text {im}}(\partial _1^\delta ))&=f_0^{\delta ,\delta '}(\gamma ) + {\text {im}}(\partial _1^{\delta '}) \quad \text {(}f_0^{\delta ,\delta '}\text { is a chain map)} \\&=g_0^{\delta ,\delta '}(\gamma ) + {\text {im}}(\partial _1^{\delta '}) \quad \text {(}f_0^{\delta ,\delta '}\equiv g_0^{\delta ,\delta '}\text {)}\\&=g_0^{\delta ,\delta '}(\gamma ) + {\text {im}}(\partial _1^{\delta ',S}) \quad \text {(}{\text {im}}(\partial _1^{\delta '})={\text {im}}(\partial _1^{\delta ',S})\text {)}\\&=(g_0^{\delta ,\delta '})_\#(\gamma + {\text {im}}(\partial _1^{\delta ,S})).\quad \text {(}g_0^{\delta ,\delta '} \text {is a chain map)} \end{aligned}$$

Thus \((f_0^{\delta ,\delta '})_\#=(g_0^{\delta ,\delta '})_\#\) for each \(\delta '\ge \delta \in \mathbb {R}\). Since we also have \(H_0(\mathfrak {D}^{{\text {si}}}_\delta )=H_0(\mathfrak {D}^{{\text {si}}}_{\delta ,S})\) for each \(\delta \in \mathbb {R}\), we now apply the Persistence Equivalence Theorem (Theorem 21) to conclude the proof. \(\square \)

Appendix B: Higher dimensional Dowker persistence diagrams of cycle networks

The contents of this section rely on results in Adamaszek and Adams (2017) and Adamaszek et al. (2016). We introduce some minimalistic versions of definitions from the referenced papers to use in this section. The reader should refer to these papers for the original definitions.

Given a metric space \((M,d_M)\) and \(m\in M\), we will write \(\overline{B(m,\varepsilon )}\) to denote a closed \(\varepsilon \)-ball centered at m, for any \(\varepsilon > 0\). For a subset \(X\subseteq M\) and some \(\varepsilon >0\), the Čech complex of X at resolution \(\varepsilon \) is defined to be the following simplicial complex:

$$\begin{aligned} \check{\mathbf{C}}(X,\varepsilon ):=\left\{ \sigma \subseteq X : \cap _{x\in \sigma }\overline{B(x,\varepsilon )} \ne \varnothing \right\} . \end{aligned}$$

In the setting of metric spaces, the Čech complex coincides with the Dowker source and sink complexes. We will be interested in the special case where the underlying metric space is the circle. We write \(S^1\) to denote the circle with unit circumference. Next, for any \(n\in \mathbb {N}\), we write \(\mathbb {X}_n:=\left\{ 0,\tfrac{1}{n},\tfrac{2}{n},\ldots , \tfrac{n-1}{n}\right\} \) to denote the collection of n equally spaced points on \(S^1\) with the restriction of the arc length metric on \(S^1\). Also let \(G_n\) denote the n-node cycle network with vertex set \(\mathbb {X}_n\) (in contrast with \(\mathbb {X}_n\), here \(G_n\) is equipped with the asymmetric weights defined in Sect. 6.1). The connection between \(\mathbb {X}_n\) and Dowker complexes of the cycle networks \(G_n\) is highlighted by the following observation:

Proposition 46

Let \(n\in \mathbb {N}\). Then for any \(\delta \in [0,1]\), we have \(\check{\mathbf{C}}(\mathbb {X}_n,\frac{\delta }{2}) = \mathfrak {D}^{{\text {si}}}_{n\delta ,G_n}.\)

The scaling factor arises because \(G_n\) has diameter \(\sim n\), whereas \(\mathbb {X}_n\subseteq S^1\) has diameter \(\sim 1/2\). This proposition provides a pedagogical step which helps us transport results from the setting of Adamaszek and Adams (2017) and Adamaszek et al. (2016) to that of the current paper.

Proof

For \(\delta =0\), both the Čech and Dowker complexes consist of the n vertices, and are equal. Similarly for \(\delta =1\), both \(\check{\mathbf{C}}(\mathbb {X}_n,1)\) and \(\mathfrak {D}^{{\text {si}}}_{n,G_n}\) are equal to the \((n-1)\)-simplex.

Now suppose \(\delta \in (0,1)\). Let \(\sigma \in \mathfrak {D}^{{\text {si}}}_{n\delta ,G_n}\). Then \(\sigma \) is of the form \([\tfrac{k}{n},\tfrac{k+1}{n},\ldots , \tfrac{\left\lfloor {k+n\delta }\right\rfloor }{n}]\) for some integer \(0\le k \le n-1\), where the \(n\delta \)-sink is \(\tfrac{\left\lfloor {k+n\delta }\right\rfloor }{n}\) and all the numerators are taken modulo n. We claim that \(\sigma \in \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{\delta }{2})\). To see this, observe that \(d_{S^1}(\tfrac{k}{n},\tfrac{\left\lfloor {k+n\delta }\right\rfloor }{n}) \le \delta \), and so \(\overline{B(\tfrac{k}{n},\tfrac{\delta }{2})} \cap \overline{B(\tfrac{\left\lfloor {k+n\delta }\right\rfloor }{n},\tfrac{\delta }{2})} \ne \varnothing \). Then we have \(\sigma \in \bigcap _{i=0}^{n\delta }\overline{B\left( \tfrac{\left\lfloor {k+i}\right\rfloor }{n},\tfrac{\delta }{2}\right) }\), and so \(\sigma \in \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{\delta }{2})\).

Now let \(\sigma \in \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{\delta }{2})\). Then \(\sigma \) is of the form \([\tfrac{k}{n},\tfrac{k+1}{n},\ldots , \tfrac{k+j}{n}]\) for some integer \(0\le k\le n-1\), where j is an integer such that \(\tfrac{j}{n} \le \delta \). In this case, we have \(\sigma = \mathbb {X}_n \cap _{i=0}^j\overline{B\left( \tfrac{k+i}{n},\delta \right) }\). Then in \(G_n\), after applying the scaling factor n, we have \(\sigma \in \mathfrak {D}^{{\text {si}}}_{n\delta ,G_n}\), with \(\tfrac{k+j}{n}\) as an \(n\delta \)-sink in \(G_n\). This shows equality of the two simplicial complexes.\(\square \)

Theorem 47

(Theorem 3.5, Adamaszek et al. 2016) Fix \(n\in \mathbb {N}\), and let \(0\le k \le n-2\) be an integer. Then,

$$\begin{aligned} \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k}{2n})\simeq {\left\{ \begin{array}{ll} \bigvee ^{n-k-1}S^{2l} &{} \text {if } \tfrac{k}{n} = \tfrac{l}{l+1},\\ S^{2l+1} &{}\text {or if } \tfrac{l}{l+1}< \tfrac{k}{n} < \tfrac{l+1}{l+2}, \end{array}\right. } \end{aligned}$$

for some \(l \in \mathbb {Z}_+\). Here \(\bigvee \) denotes the wedge sum, and \(\simeq \) denotes homotopy equivalence.

Theorem 37

(Even dimension) Fix \(n\in \mathbb {N}\), \(n \ge 3\). If \(l\in \mathbb {N}\) is such that n is divisible by \((l+1)\), and \(k:=\tfrac{nl}{l+1}\) is such that \(0\le k \le n-2\), then \({\text {Dgm}}^{\mathfrak {D}}_{2l}(G_n)\) consists of precisely the point \((\tfrac{nl}{l+1},\tfrac{nl}{l+1} + 1)\) with multiplicity \(\tfrac{n}{l+1} -1\). If l or k do not satisfy the conditions above, then \({\text {Dgm}}^{\mathfrak {D}}_{2l}(G_n)\) is trivial.

Proof of Theorem 37

Let \(l \in \mathbb {N}\) be such that \((l+1)\) divides n and \(0\le k\le n-2\). Then \(\mathfrak {D}^{{\text {si}}}_{k,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k}{2n})\) has the homotopy type of a wedge sum of \((n-k-1)\) copies of \(S^{2l}\), by Theorem 47. Here the equality follows from Proposition 46. Notice that \(n-k-1 = \tfrac{n}{l+1}-1\). Furthermore, by another application of Theorem 47, it is always possible to choose \(\varepsilon >0\) small enough so that \(\mathfrak {D}^{{\text {si}}}_{k-\varepsilon ,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k-\varepsilon }{2n})\) and \(\mathfrak {D}^{{\text {si}}}_{k+\varepsilon ,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k+\varepsilon }{2n})\) have the homotopy types of odd-dimensional spheres. Thus the inclusions \(\mathfrak {D}^{{\text {si}}}_{k-\varepsilon ,G_n} \subseteq \mathfrak {D}^{{\text {si}}}_{k,G_n} \subseteq \mathfrak {D}^{{\text {si}}}_{k+\varepsilon ,G_n}\) induce zero maps upon passing to homology. It follows that \({\text {Dgm}}^{\mathfrak {D}}_{2l}(G_n)\) consists of the point \((\tfrac{nl}{l+1},\tfrac{nl}{l+1} + 1)\) with multiplicity \(\tfrac{n}{l+1} -1\).

If \(l \in \mathbb {N}\) does not satisfy the condition described above, then there does not exist an integer \(1\le j \le n-2\) such that \(j/n = l/(l+1)\). So for each \(1\le j \le n-2\), \(\mathfrak {D}^{{\text {si}}}_{j,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{j}{2n})\) has the homotopy type of an odd-dimensional sphere by Theorem 47, and thus does not contribute to \({\text {Dgm}}^{\mathfrak {D}}_{2l}(G_n)\). If l satisfies the condition but \(k \ge n-1\), then \(\check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k}{2n})\) is just the \((n-1)\)-simplex, hence contractible. \(\square \)

Theorem 37 gives a characterization of the even dimensional Dowker persistence diagrams of cycle networks. The most interesting case occurs when considering the 2-dimensional diagrams: we see that cycle networks of an even number of nodes have an interesting barcode, even if the bars are all short-lived. For dimensions 4, 6, 8, and beyond, there are fewer and fewer cycle networks with nontrivial barcodes (in the sense that only cycle networks with number of nodes equal to a multiple of 4, 6, 8, and so on have nontrivial barcodes). For a complete picture, it is necessary to look at odd-dimensional persistence diagrams. This is made possible by the next set of constructions.

We have already recalled the definition of a Rips complex of a metric space. To facilitate the assessment of the connection to Adamaszek and Adams (2017), we temporarily adopt the notation \(\mathbf{VR}(X,\varepsilon )\) to denote the Vietoris–Rips complex of a metric space \((X,d_X)\) at resolution \(\varepsilon >0\), i.e. the simplicial complex \(\left\{ \sigma \subseteq X : {\text {diam}}(\sigma ) \le \varepsilon \right\} \).

Theorem 48

(Theorem 9.3, Proposition 9.5, Adamaszek and Adams 2017) Let \(0< r < \tfrac{1}{2}\). Then there exists a map \(T_r: {\text {pow}}(S^1) \rightarrow {\text {pow}}(S^1)\) and a map \(\pi _r: S^1 \rightarrow S^1\) such that there is an induced homotopy equivalence

$$\begin{aligned} \mathbf{VR}(T_r(X), \tfrac{2r}{1+2r}) \xrightarrow {\simeq } \check{\mathbf{C}}(X,r). \end{aligned}$$

Next suppose \(X\subseteq S^1\) and let \(0< r \le r' < \tfrac{1}{2}\). Then there exists a map \(\eta : S^1 \rightarrow S^1\) such that the following diagram commutes:

figurep

Theorem 49

Consider the setup of Theorem 48. If \(\check{\mathbf{C}}(X,r)\) and \(\check{\mathbf{C}}(X,r')\) are homotopy equivalent, then the inclusion map between them is a homotopy equivalence.

Before providing the proof, we show how it implies Theorem 38.

Theorem 38

(Odd dimension) Fix \(n\in \mathbb {N}\), \(n\ge 3\). Then for \(l\in \mathbb {N}\), define \(M_l:=\left\{ m \in \mathbb {N}: \tfrac{nl}{l+1}< m < \tfrac{n(l+1)}{l+2}\right\} \). If \(M_l\) is empty, then \({\text {Dgm}}^{\mathfrak {D}}_{2l+1}(G_n)\) is trivial. Otherwise, we have:

$$\begin{aligned} {\text {Dgm}}^{\mathfrak {D}}_{2l+1}(G_n) = \left\{ \left( a_l,\left\lceil {\tfrac{n(l+1)}{l+2}}\right\rceil \right) \right\} , \end{aligned}$$

where \(a_l:=\min \left\{ m \in M_l\right\} .\) We use set notation (instead of multisets) to mean that the multiplicity is 1.

Proof of Theorem 38

By Proposition 46 and Theorem 47, we know that \(\mathfrak {D}^{{\text {si}}}_{k,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k}{2n}) \simeq S^1 \) for integers \(0< k < \tfrac{n}{2}\). Let \(b \in \mathbb {N}\) be the greatest integer less than n / 2. Then by Theorem 49, we know that each inclusion map in the following chain is a homotopy equivalence:

$$\begin{aligned} \mathfrak {D}^{{\text {si}}}_{1,G_n} \subseteq \cdots \subseteq \mathfrak {D}^{{\text {si}}}_{b,G_n} = \mathfrak {D}^{{\text {si}}}_{\left\lceil {n/2}\right\rceil ^-,G_n}. \end{aligned}$$

It follows that \({\text {Dgm}}^{\mathfrak {D}}_1(G_n) = \left\{ \left( 1,\left\lceil {\tfrac{n}{2}}\right\rceil \right) \right\} \). The notation in the last equality means that \(\mathfrak {D}^{{\text {si}}}_{b,G_n} = \mathfrak {D}^{{\text {si}}}_{\delta ,G_n}\) for all \(\delta \in [b,b+1)\), where \(b+1 = \left\lceil {n/2}\right\rceil \).

In the more general case, let \(l \in \mathbb {N}\) and let \(M_l\) be as in the statement of the result. Suppose first that \(M_l\) is empty. Then by Proposition 46 and Theorem 47, we know that \(\mathfrak {D}^{{\text {si}}}_{k,G_n}\) has the homotopy type of a wedge of even-dimensional spheres or an odd-dimensional sphere of dimension strictly different from \((2l+1)\), for any choice of integer k. Thus \({\text {Dgm}}^{\mathfrak {D}}_{2l+1}(G_n)\) is trivial.

Next suppose \(M_l\) is nonempty. By another application of Proposition 46 and Theorem 47, we know that \(\mathfrak {D}^{{\text {si}}}_{k,G_n} = \check{\mathbf{C}}(\mathbb {X}_n,\tfrac{k}{2n}) \simeq S^{2l+1} \) for integers \(\tfrac{nl}{l+1}< k < \tfrac{n(l+1)}{l+2}\). Write \(a_l:=\min \left\{ m\in M_l\right\} \) and \(b_l:=\max \left\{ m\in M_l\right\} \). Then by Theorem 49, we know that each inclusion map in the following chain is a homotopy equivalence:

$$\begin{aligned} \mathfrak {D}^{{\text {si}}}_{a_l,G_n} \subseteq \cdots \subseteq \mathfrak {D}^{{\text {si}}}_{b_l,G_n} = \mathfrak {D}^{{\text {si}}}_{\left\lceil {n(l+1)/(l+2)}\right\rceil ^-,G_n}. \end{aligned}$$

It follows that \({\text {Dgm}}^{\mathfrak {D}}_{2l+1}(G_n) = \left\{ \left( a_l,\left\lceil {\tfrac{n(l+1)}{l+2}}\right\rceil \right) \right\} \). \(\square \)

It remains to provide a proof of Theorem 49. For this, we need some additional machinery.

Cyclic maps and winding fractions We introduce some more terms from Adamaszek and Adams (2017), but for efficiency, we try to minimize the scope of the definitions to only what is needed for our purpose. Recall that we write \(S^1\) to denote the circle with unit circumference. Thus we naturally identify any \(x\in S^1\) with a point in [0, 1). We fix a choice of \(0\in S^1\), and for any \(x,x' \in S^1\), the length of a clockwise arc from x to \(x'\) is denoted by \(\overrightarrow{d_{S^1}}(x,x')\). Then, for any finite subset \(X\subseteq S^1\) and any \(r \in (0,1/2)\), the directed Vietoris–Rips graph \(\overrightarrow{{\text {VR}}}(X,r)\) is defined to be the graph with vertex set X and edge set \(\{(x,x') : 0< \overrightarrow{d_{S^1}}(x,x') < r\}\). Next, let \(\overrightarrow{G}\) be a Vietoris–Rips graph such that the vertices are enumerated as \(x_0,x_1,\ldots , x_{n-1}\), according to the clockwise order in which they appear. A cyclic map between \(\overrightarrow{G}\) and a Vietoris–Rips graph \(\overrightarrow{H}\) is a map of vertices f such that for each edge \((x,x') \in \overrightarrow{G}\), we have either \(f(x)=f(x')\), or \((f(x),f(x')) \in \overrightarrow{H}\), and \(\sum _{i=0}^{n-1}\overrightarrow{d_{S^1}}(f(x_i),f(x_{i+1})) = 1\). Here \(x_n:=x_0\).

Next, the winding fraction of a Vietoris–Rips graph \(\overrightarrow{G}\) with vertex set \(V(\overrightarrow{G})\) is defined to be the infimum of numbers \(\tfrac{k}{n}\) such that there is an order-preserving map \(V(\overrightarrow{G}) \rightarrow \mathbb {Z}/n\mathbb {Z}\) such that each edge is mapped to a pair of numbers at most k apart. A key property of the winding fraction, denoted \({\text {wf}}\), is that if there is a cyclic map between Vietoris–Rips graphs \(\overrightarrow{G} \rightarrow \overrightarrow{H}\), then \({\text {wf}}(\overrightarrow{G}) \le {\text {wf}}(\overrightarrow{H})\).

Theorem 50

(Corollary 4.5, Proposition 4.9, Adamaszek and Adams 2017) Let \(X\subseteq S^1\) be a finite set and let \(0< r < \tfrac{1}{2}\). Then,

$$\begin{aligned} \mathbf{VR}(X,r) \simeq {\left\{ \begin{array}{ll} S^{2l+1} &{}: \tfrac{l}{2l+1}< {\text {wf}}(\overrightarrow{{\text {VR}}}(X,r)) < \tfrac{l+1}{2l+3} \text { for some } l\in \mathbb {Z}_+,\\ \bigvee ^j S^{2l} &{}: {\text {wf}}(\overrightarrow{{\text {VR}}}(X,r)) = \tfrac{l}{2l+1}, \text { for some } j\in \mathbb {N}. \end{array}\right. } \end{aligned}$$

Next let \(X' \subseteq S^1\) be another finite set, and let \(r \le r' < \tfrac{1}{2}\). Suppose \(f:\overrightarrow{{\text {VR}}}(X,r) \rightarrow \overrightarrow{{\text {VR}}}(X',r')\) is a cyclic map between Vietoris–Rips graphs and \(\tfrac{l}{2l+1}< {\text {wf}}(\overrightarrow{{\text {VR}}}(X,r)) \le {\text {wf}}(\overrightarrow{{\text {VR}}}(X',r')) < \tfrac{l+1}{2l+3}\). Then f induces a homotopy equivalence between \(\mathbf{VR}(X,r)\) and \(\mathbf{VR}(X',r')\).

We now have the ingredients for a proof of Theorem 49.

Proof of Theorem 49

Since the maps \(\pi _r\) and \(\pi _{r'}\) induce homotopy equivalences, it follows that

$$\begin{aligned} \mathbf{VR}(T_r(X),\tfrac{2r}{1+2r}) \simeq \mathbf{VR}(T_{r'}(X),\tfrac{2r'}{1+2r'}). \end{aligned}$$

By the characterization result in Theorem 50, we know that there exists \(l \in \mathbb {Z}_+\) such that

$$\begin{aligned} \tfrac{l}{2l+1}< {\text {wf}}(\overrightarrow{{\text {VR}}}(T_r(X),\tfrac{2r}{1+2r})) \le {\text {wf}}(\overrightarrow{{\text {VR}}}(T_{r'}(X),\tfrac{2r'}{1+2r'})) < \tfrac{l+1}{2l+3}. \end{aligned}$$

The map \(\eta \) in Theorem 48 appears in (Adamaszek and Adams 2017, Proposition 9.5) through an explicit construction. Moreover, it is shown that \(\eta \) induces a cyclic map \({\text {wf}}(\overrightarrow{{\text {VR}}}(T_r(X),\tfrac{2r}{1+2r})) \rightarrow {\text {wf}}(\overrightarrow{{\text {VR}}}(T_{r'}(X),\tfrac{2r'}{1+2r'}))\). Thus by Theorem 50, \(\eta \) induces a homotopy equivalence between \(\mathbf{VR}(T_r(X),\tfrac{2r}{1+2r})\) and \(\mathbf{VR}(T_{r'}(X),\tfrac{2r'}{1+2r'})\). Finally, the commutativity of the diagram in Theorem 48 shows that the inclusion \(\check{\mathbf{C}}(X,r) \subseteq \check{\mathbf{C}}(X,r')\) induces a homotopy equivalence. \(\square \)

Remark 51

The analogue of Theorem 49 for Čech complexes appears as Proposition 4.9 of Adamaszek and Adams (2017) for Vietoris–Rips complexes. We prove Theorem 49 by connecting Čech and Vietoris–Rips complexes using Proposition 9.5 of Adamaszek and Adams (2017). However, as remarked in §9 of Adamaszek and Adams (2017), one could prove Theorem 49 directly using a parallel theory of winding fractions for Čech complexes.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chowdhury, S., Mémoli, F. A functorial Dowker theorem and persistent homology of asymmetric networks. J Appl. and Comput. Topology 2, 115–175 (2018). https://doi.org/10.1007/s41468-018-0020-6

Download citation

Keywords

  • Directed networks
  • Functorial Dowker theorem
  • Cycle networks
  • Functorial Nerve Theorem

Mathematics Subject Classification

  • 55U99
  • 68U05
  • 55N35