Artificial Intelligence integrated framework for stability of functions in persistent homology
DOI:
https://doi.org/10.59461/ijdiic.v4i2.195Keywords:
Artificial intelligence, Persitent homology, Integrated Framework, StabilityAbstract
Explaining the spatial properties in point set topological spaces is a tough task. Experts in TDA have sought to discover if it is possible to find a strong intuition about the geometry and topology in big datasets, easily seen when dealing with all of them at the same time. This point also notes that the estimates will stay useful if we can detect whether the constants remain stable as the data changes, for instance, the Hausdorff distance function when the data exhibits noise, or even when a little noise is added to the point-cloud datapoints. This could happen if these properties are considered in topologically invariant compact subsets of X, which requires very stringent and restrictive assumptions to obtain well-defined shapes that can be drawn from the data in the compact subsets. The aim of this study is to outline factors that make functions in persistent homology stable. The results show that factors like triangularability affect stability of functions. Moreover, we have given an integrated artificial intelligence (AI) framework for stability of functions, to trace the accuracy levels of algorithms in cyber-threat identification and cyber-threat attacks using Persistent Homology.
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F. Verhulst, Henri Poincaré. Boston, MA: Springer US, 2012. doi: 10.1007/978-1-4614-2407-9.
H. Poincaré, and J. Stillwell, “Papers on Topology: Analysis Situs and its Five Supplements,” Am Math Soc Math Soc, vol. 37, p. 228, 2010.
L. E. J. Brouwer, “Beweis der Invarianz der Dimensionenzahl,” Math Ann, vol. 70, no. 2, pp. 161–165, Jun. 1911, doi: 10.1007/BF01461154.
E. Betti, “Sopra gli spazi di un numero qualunque di dimensioni,” Ann di Mat Pura ed Appl, vol. 4, no. 1, pp. 140–158, Jul. 1870, doi: 10.1007/BF02420029.
J. W. Alexander, “A Proof of the Invariance of Certain Constants of Analysis Situs,” Trans Am Math Soc, vol. 16, no. 2, p. 148, Apr. 1915, doi: 10.2307/1988715.
P. Hilton, “A Brief, Subjective History of Homology and Homotopy Theory in This Century,” Math Mag, vol. 61, no. 5, pp. 282–291, Dec. 1988, doi: 10.1080/0025570X.1988.11977391.
B. Srinivasan and J. D. Sally, Eds., Emmy Noether in Bryn Mawr. New York, NY: Springer New York, 1983. doi: 10.1007/978-1-4612-5547-5.
G. Carlsson, “Topology and data,” Bull Am Math Soc, vol. 46, no. 2, pp. 255–308, Jan. 2009, doi: 10.1090/S0273-0979-09-01249-X.
T. Davies, “Persistence-Based Summaries for Data Analysis with Applications to Cyber Security,” Univ Southampton, Dr Thesis, p. 164, 2023.
P. Bubenik, M. Hull, D. Patel, and B. Whittle, “Persistent homology detects curvature,” Inverse Probl, vol. 36, no. 2, p. 025008, Feb. 2020, doi: 10.1088/1361-6420/ab4ac0.
J. Franklin, “The elements of statistical learning: data mining, inference and prediction,” Math Intell, vol. 27, no. 2, pp. 83–85, Mar. 2005, doi: 10.1007/BF02985802.
Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature, vol. 521, no. 7553, pp. 436–444, May 2015, doi: 10.1038/nature14539.
P. Bubenik and P. Dłotko, “A persistence landscapes toolbox for topological statistics,” J Symb Comput, vol. 78, pp. 91–114, Jan. 2017, doi: 10.1016/j.jsc.2016.03.009.
F. Chazal, D. Cohen‐Steiner, L. J. Guibas, F. Mémoli, and S. Y. Oudot, “Gromov‐Hausdorff Stable Signatures for Shapes using Persistence,” Comput Graph Forum, vol. 28, no. 5, pp. 1393–1403, Jul. 2009, doi: 10.1111/j.1467-8659.2009.01516.x.
M. K. Chung et al., “A Unified Statistical Approach to Deformation-Based Morphometry,” Neuroimage, vol. 14, no. 3, pp. 595–606, Sep. 2001, doi: 10.1006/nimg.2001.0862.
F. Chazal, L. J. Guibas, S. Y. Oudot, and P. Skraba, “Persistence-Based Clustering in Riemannian Manifolds,” J ACM, vol. 60, no. 6, pp. 1–38, Nov. 2013, doi: 10.1145/2535927.
T. Florian, H. Carl, K. Hedvig, and K. Danica, “Topological constraints and kernel-based density estimation,” Adv Neural Inf Process Syst, vol. 25, 2012.
C. S. Pun, S. X. Lee, and K. Xia, “Persistent-homology-based machine learning: a survey and a comparative study,” Artif Intell Rev, vol. 55, no. 7, pp. 5169–5213, Oct. 2022, doi: 10.1007/s10462-022-10146-z.
Edelsbrunner, Letscher, and Zomorodian, “Topological Persistence and Simplification,” Discrete Comput Geom, vol. 28, no. 4, pp. 511–533, Nov. 2002, doi: 10.1007/s00454-002-2885-2.
E. Munch, M. Shapiro, and J. Harer, “Failure filtrations for fenced sensor networks,” Int J Rob Res, vol. 31, no. 9, pp. 1044–1056, Aug. 2012, doi: 10.1177/0278364912451671.
V. Snášel, J. Nowaková, F. Xhafa, and L. Barolli, “Geometrical and topological approaches to Big Data,” Futur Gener Comput Syst, vol. 67, pp. 286–296, Feb. 2017, doi: 10.1016/j.future.2016.06.005.
V. Divol and T. Lacombe, “Understanding the topology and the geometry of the space of persistence diagrams via optimal partial transport,” J Appl Comput Topol, vol. 5, no. 1, pp. 1–53, Mar. 2021, doi: 10.1007/s41468-020-00061-z.
P. J. Chocano, M. A. Morón, and F. R. Ruiz del Portal, “Computational approximations of compact metric spaces,” Phys D Nonlinear Phenom, vol. 433, p. 133168, May 2022, doi: 10.1016/j.physd.2022.133168.
W. S. Admass, Y. Y. Munaye, and A. A. Diro, “Cyber security: State of the art, challenges and future directions,” Cyber Secur Appl, vol. 2, p. 100031, 2024, doi: 10.1016/j.csa.2023.100031.
I. Jada and T. O. Mayayise, “The impact of artificial intelligence on organisational cyber security: An outcome of a systematic literature review,” Data Inf Manag, vol. 8, no. 2, p. 100063, Jun. 2024, doi: 10.1016/j.dim.2023.100063.
A. Berentsen, “Aleksander Berentsen Recommends ‘Bitcoin: A Peer-to-Peer Electronic Cash System’ by Satoshi Nakamoto,” in 21st Century Economics, Cham: Springer International Publishing, 2019, pp. 7–8. doi: 10.1007/978-3-030-17740-9_3.
J. Lewis, “Economic impact of cybercrime -No Slowing Down,” McAfee, St Cl, vol. 2, 2018.
A. Martin, J. Hernandez-Castro, and D. Camacho, “An in-Depth Study of the Jisut Family of Android Ransomware,” IEEE Access, vol. 6, pp. 57205–57218, 2018, doi: 10.1109/ACCESS.2018.2873583.
B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan, and A. Singh, “Confidence sets for persistence diagrams,” Ann Stat, vol. 42, no. 6, Dec. 2014, doi: 10.1214/14-AOS1252.
L. N. Tidjon, M. Frappier, and A. Mammar, “Intrusion Detection Systems: A Cross-Domain Overview,” IEEE Commun Surv Tutorials, vol. 21, no. 4, pp. 3639–3681, 2019, doi: 10.1109/COMST.2019.2922584.
A. Zomorodian and G. Carlsson, “Computing Persistent Homology,” Discrete Comput Geom, vol. 33, no. 2, pp. 249–274, Feb. 2005, doi: 10.1007/s00454-004-1146-y.
F. Chazal and B. Michel, “An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists,” Front Artif Intell, vol. 4, Sep. 2021, doi: 10.3389/frai.2021.667963.
M. E. Aktas, E. Akbas, and A. El Fatmaoui, “Persistence homology of networks: methods and applications,” Appl Netw Sci, vol. 4, no. 1, p. 61, Dec. 2019, doi: 10.1007/s41109-019-0179-3.
G. F. Monkam, M. J. De Lucia, and N. D. Bastian, “Preprocessing Network Traffic using Topological Data Analysis for Data Poisoning Detection,” in 2023 IEEE Conference on Dependable and Secure Computing (DSC), IEEE, Nov. 2023, pp. 1–8. doi: 10.1109/DSC61021.2023.10354143.
B. M. Balachandran and S. Prasad, “Challenges and Benefits of Deploying Big Data Analytics in the Cloud for Business Intelligence,” Procedia Comput Sci, vol. 112, pp. 1112–1122, 2017, doi: 10.1016/j.procs.2017.08.138.
A. Omorede, J. F. Prados-Castillo, and A. C. Casas-Jurado, “Researching entrepreneurship using big data: implementation, benefits, and challenges,” Int Entrep Manag J, vol. 21, no. 1, p. 85, Dec. 2025, doi: 10.1007/s11365-025-01100-w.
X. Meng, Y. Pei, and H. Takagi, “Evolutionary Multi - Modal Optimization Using Persistence-Based Clustering in Riemannian Manifolds,” in 2024 IEEE Congress on Evolutionary Computation (CEC), IEEE, Jun. 2024, pp. 1–8. doi: 10.1109/CEC60901.2024.10612013.
F. Chazal, D. Cohen-Steiner, and Q. Mérigot, “Geometric Inference for Probability Measures,” Found Comput Math, vol. 11, no. 6, pp. 733–751, Dec. 2011, doi: 10.1007/s10208-011-9098-0.
D. Barnes, L. Polanco, and J. A. Perea, “A Comparative Study of Machine Learning Methods for Persistence Diagrams,” Front Artif Intell, vol. 4, Jul. 2021, doi: 10.3389/frai.2021.681174.
J.-D. Boissonnat, F. Chazal, and B. Michel, “Topological Data Analysis,” 2022, pp. 247–269. doi: 10.1007/978-3-030-96173-2_9.
G. Rote and G. Vegter, “Computational Topology: An Introduction,” in Effective Computational Geometry for Curves and Surfaces, Springer Berlin Heidelberg, pp. 277–312. doi: 10.1007/978-3-540-33259-6_7.
P. Niyogi, S. Smale, and S. Weinberger, “A Topological View of Unsupervised Learning from Noisy Data,” SIAM J Comput, vol. 40, no. 3, pp. 646–663, Jan. 2011, doi: 10.1137/090762932.
V. Patrangenaru, P. Bubenik, R. L. Paige, and D. Osborne, “Challenges in Topological Object Data Analysis,” Sankhya A, vol. 81, no. 1, pp. 244–271, Feb. 2019, doi: 10.1007/s13171-018-0137-7.
K. Xia, Z. Li, and L. Mu, “Multiscale Persistent Functions for Biomolecular Structure Characterization,” Bull Math Biol, vol. 80, no. 1, pp. 1–31, Jan. 2018, doi: 10.1007/s11538-017-0362-6.
A. Adcock, E. Carlsson, and G. Carlsson, “The Ring of Algebraic Functions on Persistence Bar Codes,” Apr. 2013. http://arxiv.org/abs/1304.0530
I. Chevyrev, V. Nanda, and H. Oberhauser, “Persistence Paths and Signature Features in Topological Data Analysis,” IEEE Trans Pattern Anal Mach Intell, vol. 42, no. 1, pp. 192–202, Jan. 2020, doi: 10.1109/TPAMI.2018.2885516.
B. Okelo and A. Onyango, “Persistent Homology and Artificial Intelligence Analysis of COVID-19 in Topological Spaces,” DS J Digit Sci Technol, vol. 2, no. 3, pp. 1–8, Sep. 2023, doi: 10.59232/DST-V2I3P101.
L. M. Seversky, S. Davis, and M. Berger, “On Time-Series Topological Data Analysis: New Data and Opportunities,” in 2016 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), IEEE, Jun. 2016, pp. 1014–1022. doi: 10.1109/CVPRW.2016.131.
D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, “Stability of Persistence Diagrams,” Discrete Comput Geom, vol. 37, no. 1, pp. 103–120, Jan. 2007, doi: 10.1007/s00454-006-1276-5.
R. Kindelan, J. Frías, M. Cerda, and N. Hitschfeld, “A topological data analysis based classifier,” Adv Data Anal Classif, vol. 18, no. 2, pp. 493–538, Jun. 2024, doi: 10.1007/s11634-023-00548-4.
R. P. Singh, N. O. Malott, B. Sauerwein, N. Mcgrogan, and P. A. Wilsey, “Generating High Dimensional Test Data for Topological Data Analysis,” 2024, pp. 18–37. doi: 10.1007/978-981-97-0316-6_2.
E. Purvine et al., “Experimental Observations of the Topology of Convolutional Neural Network Activations,” Proc AAAI Conf Artif Intell, vol. 37, no. 8, pp. 9470–9479, Jun. 2023, doi: 10.1609/aaai.v37i8.26134.
M. Hajij, B. Assiri, and P. Rosen, “Parallel Mapper,” May 2020. http://arxiv.org/abs/1712.03660
A. F. Zobaa and T. J. Bihl, Eds., Big Data Analytics in Future Power Systems. Boca Raton : Taylor & Francis, a CRC title, part of the Taylor &: CRC Press, 2018. doi: 10.1201/9781315105499.
L. Zhang and G. B. White, “Analysis of Payload Based Application level Network Anomaly Detection,” in 2007 40th Annual Hawaii International Conference on System Sciences (HICSS’07), IEEE, Jan. 2007, pp. 99–99. doi: 10.1109/HICSS.2007.75.
B. Kuskonmaz, R. Wisniewski, and C. Kallesøe, “Topological Data Analysis-Based Replay Attack Detection for Water Networks,” IFAC-PapersOnLine, vol. 58, no. 4, pp. 91–96, 2024, doi: 10.1016/j.ifacol.2024.07.199.
R. Winding, T. Wright, and M. Chapple, “System Anomaly Detection: Mining Firewall Logs,” in 2006 Securecomm and Workshops, IEEE, Aug. 2006, pp. 1–5. doi: 10.1109/SECCOMW.2006.359572.
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