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Capturing dynamics of time-varying data via topology
1. | School of Information, University of Michigan, Ann Arbor, MI 48109, USA |
2. | Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA |
3. | Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, USA |
4. | Department of Mathematics, Statistics, and Computer Science, Macalester College, Saint Paul, MN 55105, USA |
One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time. A time-varying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic behavior. We provide an introduction to topological summaries of time-varying metric spaces including vineyards [
References:
[1] |
H. Adams and G. Carlsson,
Evasion paths in mobile sensor networks, International Journal of Robotics Research, 34 (2015), 90-104.
|
[2] |
H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, P. Shipman, S. Chepushtanova, E. Hanson, F. Motta and L. Ziegelmeier, Persistence images: A stable vector representation of persistent homology, J. Mach. Learn. Res., 18 (2017), Paper No. 8, 35 pp. http://jmlr.org/papers/v18/16-337.html. |
[3] |
H. Adams, D. Ghosh, C. Mask, W. Ott and K. Williams, Efficient evader detection in mobile sensor networks, arXiv preprint, arXiv: 2101.09813. |
[4] |
P. Arora, D. Deepali and S. Varshney,
Analysis of K-means and K-medoids algorithm for big data, Procedia Computer Science, 78 (2016), 507-512.
doi: 10.1016/j.procs.2016.02.095. |
[5] |
A. Banman and L. Ziegelmeier, Mind the gap: A study in global development through persistent homology, in Research in Computational Topology, Springer, 2018,125–144.
doi: 10.1007/978-3-319-89593-2_8. |
[6] |
D. Bhaskar, A. Manhart, J. Milzman, J. T. Nardini, K. M. Storey, C. M. Topaz and L. Ziegelmeier, Analyzing collective motion with machine learning and topology, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 123125, 12 pp.
doi: 10.1063/1.5125493. |
[7] |
P. Bubenik,
Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res., 16 (2015), 77-102.
|
[8] |
D. Burago, Y. Burago and S. Ivanov, A course in Metric Geometry, vol. 33, American Mathematical Society, Providence, 2001.
doi: 10.1090/gsm/033. |
[9] |
G. Carlsson,
Topology and data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 255-308.
doi: 10.1090/S0273-0979-09-01249-X. |
[10] |
G. Carlsson and V. de Silva,
Zigzag persistence, Found. Comput. Math., 10 (2010), 367-405.
doi: 10.1007/s10208-010-9066-0. |
[11] |
G. Carlsson, V. de Silva, S. Kališnik and D. Morozov,
Parametrized homology via zigzag persistence, Algebr. Geom. Topol., 19 (2019), 657-700.
doi: 10.2140/agt.2019.19.657. |
[12] |
G. Carlsson, V. de Silva and D. Morozov, Zigzag persistent homology and real-valued functions, in Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, ACM, 2009,247–256.
doi: 10.1145/1542362.1542408. |
[13] |
G. Carlsson, G. Singh and A. Zomorodian, Computing multidimensional persistence, Algorithms and computation, 730–739, Lecture Notes in Comput. Sci., 5878, Springer, Berlin, 2009.
doi: 10.1007/978-3-642-10631-6_74. |
[14] |
G. Carlsson and A. Zomorodian,
The theory of multidimensional persistence, Discrete Comput. Geom., 42 (2009), 71-93.
doi: 10.1007/s00454-009-9176-0. |
[15] |
A. Cerri, B. D. Fabio, M. Ferri, P. Frosini and C. Landi,
Betti numbers in multidimensional persistent homology are stable functions, Math. Methods Appl. Sci., 36 (2013), 1543-1557.
doi: 10.1002/mma.2704. |
[16] |
W. Chachólski, M. Scolamiero and F. Vaccarino,
Combinatorial presentation of multidimensional persistent homology, J. Pure Appl. Algebra, 221 (2017), 1055-1075.
doi: 10.1016/j.jpaa.2016.09.001. |
[17] |
F. Chazal, V. de Silva and S. Oudot,
Persistence stability for geometric complexes, Geometriae Dedicata, 174 (2014), 193-214.
doi: 10.1007/s10711-013-9937-z. |
[18] |
D. Cohen-Steiner, H. Edelsbrunner and J. Harer,
Stability of persistence diagrams, Discrete Comput. Geom., 37 (2007), 103-120.
doi: 10.1007/s00454-006-1276-5. |
[19] |
D. Cohen-Steiner, H. Edelsbrunner and D. Morozov, Vines and vineyards by updating persistence in linear time, in Computational Geometry (SCG'06), ACM, 2006,119–126.
doi: 10.1145/1137856.1137877. |
[20] |
P. Corcoran and C. B. Jones,
Modelling topological features of swarm behaviour in space and time with persistence landscapes, IEEE Access, 5 (2017), 18534-18544.
doi: 10.1109/ACCESS.2017.2749319. |
[21] |
D. B. Damiano and M. R. McGuirl,
A topological analysis of targeted in-111 uptake in SPECT images of murine tumors, J. Math. Biol., 76 (2018), 1559-1587.
doi: 10.1007/s00285-017-1184-8. |
[22] |
V. de Silva and R. Ghrist,
Coordinate-free coverage in sensor networks with controlled boundaries via homology, The International Journal of Robotics Research, 25 (2006), 1205-1222.
doi: 10.1177/0278364906072252. |
[23] |
V. de Silva and R. Ghrist,
Coverage in sensor networks via persistent homology, Algebr. Geom. Topol., 7 (2007), 339-358.
doi: 10.2140/agt.2007.7.339. |
[24] |
T. K. Dey and C. Xin, Computing bottleneck distance for 2-d interval decomposable modules, arXiv preprint, arXiv: 1803.02869. |
[25] |
M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes,
Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302.
doi: 10.1103/PhysRevLett.96.104302. |
[26] |
H. Edelsbrunner and J. L. Harer, Computational Topology: An Introduction, American Mathematical Society, Providence, 2010.
doi: 10.1090/mbk/069. |
[27] |
H. Edelsbrunner, D. Morozov and A. Patel, The stability of the apparent contour of an orientable 2-manifold, Topological Methods in Data Analysis and Visualization. Mathematics and Visualization., 27–41, Math. Vis., Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-15014-2_3. |
[28] |
B. T. Fasy, J. Kim, F. Lecci, C. Maria, D. L. Millman and V. Rouvreau, Tda: Statistical tools for topological data analysis, https://cran.r-project.org/web/packages/TDA/index.html. |
[29] |
M. Feng and M. A. Porter,
Persistent homology of geospatial data: A case study with voting, SIAM Rev., 63 (2021), 67-99.
doi: 10.1137/19M1241519. |
[30] |
M. Feng and M. A. Porter,
Spatial applications of topological data analysis: Cities, snowflakes, random structures, and spiders spinning under the influence, Phys. Rev. Research, 2 (2020), 033426.
doi: 10.1103/PhysRevResearch.2.033426. |
[31] |
R. Ghrist,
Barcodes: The persistent topology of data, ull. Amer. Math. Soc. (N.S.), 45 (2008), 61-75.
doi: 10.1090/S0273-0979-07-01191-3. |
[32] |
C. Giusti, L. Papadopoulos, E. T. Owens, K. E. Daniels and D. S. Bassett,
Topological and geometric measurements of force-chain structure, Physical Review E, 94 (2016), 032909.
doi: 10.1103/PhysRevE.94.032909. |
[33] |
I. T. Jolliffe, Principal Component Analysis, Springer Verlag, 1986.
doi: 10.1007/978-1-4757-1904-8. |
[34] |
T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, vol. 157, pringer-Verlag, New York, 2004.
doi: 10.1007/b97315. |
[35] |
L. Kaufman and P. Rousseeuw, Clustering by Means of Medoids, North-Holland, 1987. |
[36] |
W. Kim and F. Mémoli, Stable signatures for dynamic metric spaces via zigzag persistent homology, arXiv preprint, arXiv: 1712.04064. |
[37] |
W. Kim and F. Mémoli,
Spatiotemporal persistent homology for dynamic metric spaces, Discrete Comput. Geom., 66 (2021), 831-875.
doi: 10.1007/s00454-019-00168-w. |
[38] |
M. Lesnick,
The theory of the interleaving distance on multidimensional persistence modules, Found. Comput. Math., 15 (2015), 613-650.
doi: 10.1007/s10208-015-9255-y. |
[39] |
M. Maechler, Finding groups in data: Cluster analysis extended rousseeuw et al, https://cran.r-project.org/web/packages/cluster/cluster.pdf. |
[40] |
A. McCleary and A. Patel,
Bottleneck stability for generalized persistence diagrams, Proc. Amer. Math. Soc., 148 (2020), 3149-3161.
doi: 10.1090/proc/14929. |
[41] |
A. McCleary and A. Patel, Edit distance and persistence diagrams over lattices, arXiv preprint, arXiv: 2010.07337. |
[42] |
E. Miller, Data structures for real multiparameter persistence modules, arXiv preprint, arXiv: 1709.08155. |
[43] |
N. Milosavljević, D. Morozov and P. Škraba, Zigzag persistent homology in matrix multiplication time, in Computational geometry (SCG'11), 2011,216–225.
doi: 10.1145/1998196.1998229. |
[44] | |
[45] |
D. Morozov, Dionysus, http://www.mrzv.org/software/dionysus/. |
[46] |
J. R. Munkres, Topology, Prentice-Hall Englewood Cliffs, NJ, 1975. |
[47] |
C. Nilsen, J. Paige, O. Warner, B. Mayhew, R. Sutley, M. Lam, A. J. Bernoff and C. M. Topaz, Social aggregation in pea aphids: Experiment and random walk modeling, PLoS ONE, 8 (2013), e83343.
doi: 10.1371/journal.pone.0083343. |
[48] |
N. Otter, M. A. Porter, U. Tillmann, P. Grindrod and H. A. Harrington,
A roadmap for the computation of persistent homology, EPJ Data Science, 6 (2017), 17.
|
[49] |
S. Y. Oudot, Persistence Theory: From Quiver Representations to Data Analysis, vol. 209, American Mathematical Society Providence, RI, 2015.
doi: 10.1090/surv/209. |
[50] |
H.-S. Park and C.-H. Jun,
A simple and fast algorithm for $k$-medoids clustering, Expert Systems with Applications, 36 (2009), 3336-3341.
doi: 10.1016/j.eswa.2008.01.039. |
[51] |
A. Patel,
Generalized persistence diagrams, J. Appl. Comput. Topol., 1 (2018), 397-419.
doi: 10.1007/s41468-018-0012-6. |
[52] |
V. Puuska,
Erosion distance for generalized persistence modules, Homology Homotopy Appl., 22 (2020), 233-254.
doi: 10.4310/HHA.2020.v22.n1.a14. |
[53] |
M. Scolamiero, W. Chachólski, A. Lundman, R. Ramanujam and S. Öberg,
Multidimensional persistence and noise, Found. Comput. Math., 17 (2017), 1367-1406.
doi: 10.1007/s10208-016-9323-y. |
[54] |
B. J. Stolz, H. A. Harrington and M. A. Porter, Persistent homology of time-dependent functional networks constructed from coupled time series, Chaos, 27 (2017), 047410, 17 pp.
doi: 10.1063/1.4978997. |
[55] |
C. M. Topaz, L. Ziegelmeier and T. Halverson, Topological data analysis of biological aggregation models, PloS One, 10 (2015), e0126383.
doi: 10.1371/journal.pone.0126383. |
[56] |
M. Ulmer, L. Ziegelmeier and C. M. Topaz, A topological approach to selecting models of biological experiments, PloS One, 14 (2019), e0213679.
doi: 10.1371/journal.pone.0213679. |
[57] |
T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
[58] |
T. Vicsek and A. Zafeiris,
Collective motion, Physics Reports, 517 (2012), 71-140.
doi: 10.1016/j.physrep.2012.03.004. |
[59] |
X. Zhu, Persistent homology: An introduction and a new text representation for natural language processing, in Twenty-Third International Joint Conference on Artificial Intelligence, 2013. |
[60] |
A. Zomorodian and G. Carlsson,
Computing persistent homology, Discrete Comput. Geom., 33 (2005), 249-274.
doi: 10.1007/s00454-004-1146-y. |
show all references
References:
[1] |
H. Adams and G. Carlsson,
Evasion paths in mobile sensor networks, International Journal of Robotics Research, 34 (2015), 90-104.
|
[2] |
H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, P. Shipman, S. Chepushtanova, E. Hanson, F. Motta and L. Ziegelmeier, Persistence images: A stable vector representation of persistent homology, J. Mach. Learn. Res., 18 (2017), Paper No. 8, 35 pp. http://jmlr.org/papers/v18/16-337.html. |
[3] |
H. Adams, D. Ghosh, C. Mask, W. Ott and K. Williams, Efficient evader detection in mobile sensor networks, arXiv preprint, arXiv: 2101.09813. |
[4] |
P. Arora, D. Deepali and S. Varshney,
Analysis of K-means and K-medoids algorithm for big data, Procedia Computer Science, 78 (2016), 507-512.
doi: 10.1016/j.procs.2016.02.095. |
[5] |
A. Banman and L. Ziegelmeier, Mind the gap: A study in global development through persistent homology, in Research in Computational Topology, Springer, 2018,125–144.
doi: 10.1007/978-3-319-89593-2_8. |
[6] |
D. Bhaskar, A. Manhart, J. Milzman, J. T. Nardini, K. M. Storey, C. M. Topaz and L. Ziegelmeier, Analyzing collective motion with machine learning and topology, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 123125, 12 pp.
doi: 10.1063/1.5125493. |
[7] |
P. Bubenik,
Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res., 16 (2015), 77-102.
|
[8] |
D. Burago, Y. Burago and S. Ivanov, A course in Metric Geometry, vol. 33, American Mathematical Society, Providence, 2001.
doi: 10.1090/gsm/033. |
[9] |
G. Carlsson,
Topology and data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 255-308.
doi: 10.1090/S0273-0979-09-01249-X. |
[10] |
G. Carlsson and V. de Silva,
Zigzag persistence, Found. Comput. Math., 10 (2010), 367-405.
doi: 10.1007/s10208-010-9066-0. |
[11] |
G. Carlsson, V. de Silva, S. Kališnik and D. Morozov,
Parametrized homology via zigzag persistence, Algebr. Geom. Topol., 19 (2019), 657-700.
doi: 10.2140/agt.2019.19.657. |
[12] |
G. Carlsson, V. de Silva and D. Morozov, Zigzag persistent homology and real-valued functions, in Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, ACM, 2009,247–256.
doi: 10.1145/1542362.1542408. |
[13] |
G. Carlsson, G. Singh and A. Zomorodian, Computing multidimensional persistence, Algorithms and computation, 730–739, Lecture Notes in Comput. Sci., 5878, Springer, Berlin, 2009.
doi: 10.1007/978-3-642-10631-6_74. |
[14] |
G. Carlsson and A. Zomorodian,
The theory of multidimensional persistence, Discrete Comput. Geom., 42 (2009), 71-93.
doi: 10.1007/s00454-009-9176-0. |
[15] |
A. Cerri, B. D. Fabio, M. Ferri, P. Frosini and C. Landi,
Betti numbers in multidimensional persistent homology are stable functions, Math. Methods Appl. Sci., 36 (2013), 1543-1557.
doi: 10.1002/mma.2704. |
[16] |
W. Chachólski, M. Scolamiero and F. Vaccarino,
Combinatorial presentation of multidimensional persistent homology, J. Pure Appl. Algebra, 221 (2017), 1055-1075.
doi: 10.1016/j.jpaa.2016.09.001. |
[17] |
F. Chazal, V. de Silva and S. Oudot,
Persistence stability for geometric complexes, Geometriae Dedicata, 174 (2014), 193-214.
doi: 10.1007/s10711-013-9937-z. |
[18] |
D. Cohen-Steiner, H. Edelsbrunner and J. Harer,
Stability of persistence diagrams, Discrete Comput. Geom., 37 (2007), 103-120.
doi: 10.1007/s00454-006-1276-5. |
[19] |
D. Cohen-Steiner, H. Edelsbrunner and D. Morozov, Vines and vineyards by updating persistence in linear time, in Computational Geometry (SCG'06), ACM, 2006,119–126.
doi: 10.1145/1137856.1137877. |
[20] |
P. Corcoran and C. B. Jones,
Modelling topological features of swarm behaviour in space and time with persistence landscapes, IEEE Access, 5 (2017), 18534-18544.
doi: 10.1109/ACCESS.2017.2749319. |
[21] |
D. B. Damiano and M. R. McGuirl,
A topological analysis of targeted in-111 uptake in SPECT images of murine tumors, J. Math. Biol., 76 (2018), 1559-1587.
doi: 10.1007/s00285-017-1184-8. |
[22] |
V. de Silva and R. Ghrist,
Coordinate-free coverage in sensor networks with controlled boundaries via homology, The International Journal of Robotics Research, 25 (2006), 1205-1222.
doi: 10.1177/0278364906072252. |
[23] |
V. de Silva and R. Ghrist,
Coverage in sensor networks via persistent homology, Algebr. Geom. Topol., 7 (2007), 339-358.
doi: 10.2140/agt.2007.7.339. |
[24] |
T. K. Dey and C. Xin, Computing bottleneck distance for 2-d interval decomposable modules, arXiv preprint, arXiv: 1803.02869. |
[25] |
M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes,
Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302.
doi: 10.1103/PhysRevLett.96.104302. |
[26] |
H. Edelsbrunner and J. L. Harer, Computational Topology: An Introduction, American Mathematical Society, Providence, 2010.
doi: 10.1090/mbk/069. |
[27] |
H. Edelsbrunner, D. Morozov and A. Patel, The stability of the apparent contour of an orientable 2-manifold, Topological Methods in Data Analysis and Visualization. Mathematics and Visualization., 27–41, Math. Vis., Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-15014-2_3. |
[28] |
B. T. Fasy, J. Kim, F. Lecci, C. Maria, D. L. Millman and V. Rouvreau, Tda: Statistical tools for topological data analysis, https://cran.r-project.org/web/packages/TDA/index.html. |
[29] |
M. Feng and M. A. Porter,
Persistent homology of geospatial data: A case study with voting, SIAM Rev., 63 (2021), 67-99.
doi: 10.1137/19M1241519. |
[30] |
M. Feng and M. A. Porter,
Spatial applications of topological data analysis: Cities, snowflakes, random structures, and spiders spinning under the influence, Phys. Rev. Research, 2 (2020), 033426.
doi: 10.1103/PhysRevResearch.2.033426. |
[31] |
R. Ghrist,
Barcodes: The persistent topology of data, ull. Amer. Math. Soc. (N.S.), 45 (2008), 61-75.
doi: 10.1090/S0273-0979-07-01191-3. |
[32] |
C. Giusti, L. Papadopoulos, E. T. Owens, K. E. Daniels and D. S. Bassett,
Topological and geometric measurements of force-chain structure, Physical Review E, 94 (2016), 032909.
doi: 10.1103/PhysRevE.94.032909. |
[33] |
I. T. Jolliffe, Principal Component Analysis, Springer Verlag, 1986.
doi: 10.1007/978-1-4757-1904-8. |
[34] |
T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, vol. 157, pringer-Verlag, New York, 2004.
doi: 10.1007/b97315. |
[35] |
L. Kaufman and P. Rousseeuw, Clustering by Means of Medoids, North-Holland, 1987. |
[36] |
W. Kim and F. Mémoli, Stable signatures for dynamic metric spaces via zigzag persistent homology, arXiv preprint, arXiv: 1712.04064. |
[37] |
W. Kim and F. Mémoli,
Spatiotemporal persistent homology for dynamic metric spaces, Discrete Comput. Geom., 66 (2021), 831-875.
doi: 10.1007/s00454-019-00168-w. |
[38] |
M. Lesnick,
The theory of the interleaving distance on multidimensional persistence modules, Found. Comput. Math., 15 (2015), 613-650.
doi: 10.1007/s10208-015-9255-y. |
[39] |
M. Maechler, Finding groups in data: Cluster analysis extended rousseeuw et al, https://cran.r-project.org/web/packages/cluster/cluster.pdf. |
[40] |
A. McCleary and A. Patel,
Bottleneck stability for generalized persistence diagrams, Proc. Amer. Math. Soc., 148 (2020), 3149-3161.
doi: 10.1090/proc/14929. |
[41] |
A. McCleary and A. Patel, Edit distance and persistence diagrams over lattices, arXiv preprint, arXiv: 2010.07337. |
[42] |
E. Miller, Data structures for real multiparameter persistence modules, arXiv preprint, arXiv: 1709.08155. |
[43] |
N. Milosavljević, D. Morozov and P. Škraba, Zigzag persistent homology in matrix multiplication time, in Computational geometry (SCG'11), 2011,216–225.
doi: 10.1145/1998196.1998229. |
[44] | |
[45] |
D. Morozov, Dionysus, http://www.mrzv.org/software/dionysus/. |
[46] |
J. R. Munkres, Topology, Prentice-Hall Englewood Cliffs, NJ, 1975. |
[47] |
C. Nilsen, J. Paige, O. Warner, B. Mayhew, R. Sutley, M. Lam, A. J. Bernoff and C. M. Topaz, Social aggregation in pea aphids: Experiment and random walk modeling, PLoS ONE, 8 (2013), e83343.
doi: 10.1371/journal.pone.0083343. |
[48] |
N. Otter, M. A. Porter, U. Tillmann, P. Grindrod and H. A. Harrington,
A roadmap for the computation of persistent homology, EPJ Data Science, 6 (2017), 17.
|
[49] |
S. Y. Oudot, Persistence Theory: From Quiver Representations to Data Analysis, vol. 209, American Mathematical Society Providence, RI, 2015.
doi: 10.1090/surv/209. |
[50] |
H.-S. Park and C.-H. Jun,
A simple and fast algorithm for $k$-medoids clustering, Expert Systems with Applications, 36 (2009), 3336-3341.
doi: 10.1016/j.eswa.2008.01.039. |
[51] |
A. Patel,
Generalized persistence diagrams, J. Appl. Comput. Topol., 1 (2018), 397-419.
doi: 10.1007/s41468-018-0012-6. |
[52] |
V. Puuska,
Erosion distance for generalized persistence modules, Homology Homotopy Appl., 22 (2020), 233-254.
doi: 10.4310/HHA.2020.v22.n1.a14. |
[53] |
M. Scolamiero, W. Chachólski, A. Lundman, R. Ramanujam and S. Öberg,
Multidimensional persistence and noise, Found. Comput. Math., 17 (2017), 1367-1406.
doi: 10.1007/s10208-016-9323-y. |
[54] |
B. J. Stolz, H. A. Harrington and M. A. Porter, Persistent homology of time-dependent functional networks constructed from coupled time series, Chaos, 27 (2017), 047410, 17 pp.
doi: 10.1063/1.4978997. |
[55] |
C. M. Topaz, L. Ziegelmeier and T. Halverson, Topological data analysis of biological aggregation models, PloS One, 10 (2015), e0126383.
doi: 10.1371/journal.pone.0126383. |
[56] |
M. Ulmer, L. Ziegelmeier and C. M. Topaz, A topological approach to selecting models of biological experiments, PloS One, 14 (2019), e0213679.
doi: 10.1371/journal.pone.0213679. |
[57] |
T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
[58] |
T. Vicsek and A. Zafeiris,
Collective motion, Physics Reports, 517 (2012), 71-140.
doi: 10.1016/j.physrep.2012.03.004. |
[59] |
X. Zhu, Persistent homology: An introduction and a new text representation for natural language processing, in Twenty-Third International Joint Conference on Artificial Intelligence, 2013. |
[60] |
A. Zomorodian and G. Carlsson,
Computing persistent homology, Discrete Comput. Geom., 33 (2005), 249-274.
doi: 10.1007/s00454-004-1146-y. |

















Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | |||||
Order Parameters | 0.63 | 0.51 | 0.61 | 0.59 | 0.35 | 0.35 | 0.21 | 0.17 |
Crocker Plots, |
1.00 | 1.00 | 0.67 | 0.67 | 0.44 | 0.43 | 0.42 | 0.43 |
Crocker Plots, |
1.00 | 1.00 | 0.67 | 0.77 | 0.45 | 0.43 | 0.39 | 0.43 |
Crocker Plots, |
0.98 | 0.99 | 0.71 | 0.67 | 0.36 | 0.35 | 0.37 | 0.33 |
Crocker Stacks, |
1.00 | 0.98 | 0.67 | 0.67 | 0.47 | 0.38 | 0.41 | 0.35 |
Crocker Stacks, |
1.00 | 1.00 | 0.67 | 0.67 | 0.49 | 0.46 | 0.41 | 0.41 |
Crocker Stacks, |
0.96 | 0.98 | 0.63 | 0.67 | 0.34 | 0.37 | 0.32 | 0.35 |
Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | |||||
Order Parameters | 0.63 | 0.51 | 0.61 | 0.59 | 0.35 | 0.35 | 0.21 | 0.17 |
Crocker Plots, |
1.00 | 1.00 | 0.67 | 0.67 | 0.44 | 0.43 | 0.42 | 0.43 |
Crocker Plots, |
1.00 | 1.00 | 0.67 | 0.77 | 0.45 | 0.43 | 0.39 | 0.43 |
Crocker Plots, |
0.98 | 0.99 | 0.71 | 0.67 | 0.36 | 0.35 | 0.37 | 0.33 |
Crocker Stacks, |
1.00 | 0.98 | 0.67 | 0.67 | 0.47 | 0.38 | 0.41 | 0.35 |
Crocker Stacks, |
1.00 | 1.00 | 0.67 | 0.67 | 0.49 | 0.46 | 0.41 | 0.41 |
Crocker Stacks, |
0.96 | 0.98 | 0.63 | 0.67 | 0.34 | 0.37 | 0.32 | 0.35 |
69 | 31 | 0 | |
64 | 36 | 0 | |
4 | 96 | 99 | |
1 | 99 | 0 | |
0 | 0 | 100 | |
0 | 0 | 100 |
69 | 31 | 0 | |
64 | 36 | 0 | |
4 | 96 | 99 | |
1 | 99 | 0 | |
0 | 0 | 100 | |
0 | 0 | 100 |
Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | |
stack | 1.00 | 0.67 | 0.47 | 0.41 |
1.00 | 0.67 | 0.44 | 0.42 | |
1.00 | 0.67 | 0.46 | 0.40 | |
1.00 | 0.67 | 0.49 | 0.40 | |
1.00 | 0.58 | 0.44 | 0.38 | |
0.99 | 0.67 | 0.39 | 0.36 | |
0.97 | 0.67 | 0.33 | 0.35 | |
0.92 | 0.73 | 0.41 | 0.28 | |
0.73 | 0.66 | 0.40 | 0.21 |
Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | |
stack | 1.00 | 0.67 | 0.47 | 0.41 |
1.00 | 0.67 | 0.44 | 0.42 | |
1.00 | 0.67 | 0.46 | 0.40 | |
1.00 | 0.67 | 0.49 | 0.40 | |
1.00 | 0.58 | 0.44 | 0.38 | |
0.99 | 0.67 | 0.39 | 0.36 | |
0.97 | 0.67 | 0.33 | 0.35 | |
0.92 | 0.73 | 0.41 | 0.28 | |
0.73 | 0.66 | 0.40 | 0.21 |
Order Parameters | (0.61) | (0.61) |
Crocker Plots | 0.67 | 0.71 |
Crocker Stacks | 0.67 | 0.63 |
Stacked Persistence Diagrams | 0.67 | 0.49 |
Order Parameters | (0.61) | (0.61) |
Crocker Plots | 0.67 | 0.71 |
Crocker Stacks | 0.67 | 0.63 |
Stacked Persistence Diagrams | 0.67 | 0.49 |
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