Topological graph persistence

1. D. Alberici, P. Contucci, E. Mingione, and M. Molari. Aggregation models on hypergraphs. Annals of Physics, 376:412–424, 2017.10.1016/j.aop.2016.12.001Search in Google Scholar

2. M. G. Bergomi, M. Ferri, and A. Tavaglione. Steady and ranging sets in graph persistence. arXiv preprint arXiv:2009.06897, 2020.Search in Google Scholar

3. M. G. Bergomi, M. Ferri, P. Vertechi, and L. Zu . Beyond topological persistence: Starting from networks. arXiv preprint arXiv:1901.08051, 2019.Search in Google Scholar

4. M. G. Bergomi and P. Vertechi. Rank-based persistence. Theory and Applications of Categories, 35(9):228–260, 2020.Search in Google Scholar

5. A. Bondy and U. Murty. Graph Theory. Graduate Texts in Mathematics. Springer London, 2011.Search in Google Scholar

6. R. Boppana and M. M. Halldórsson. Approximating maximum independent sets by excluding subgraphs. BIT Numerical Mathematics, 32(2):180–196, 1992.10.1007/BF01994876Search in Google Scholar

7. G. Carlsson. Topology and data. Bull. Amer. Math. Soc., 46(2):255–308, 2009.10.1090/S0273-0979-09-01249-XSearch in Google Scholar

8. F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Y. Oudot. Proximity of persistence modules and their diagrams. In SCG ’09: Proceedings of the 25th annual symposium on Computational geometry, pages 237–246, New York, NY, USA, 2009. ACM.10.1145/1542362.1542407Search in Google Scholar

9. S. Chowdhury and F. Mémoli. Persistent homology of asymmetric networks: An approach based on dowker filtrations. arXiv preprint arXiv:1608.05432, 2016.Search in Google Scholar

10. S. Chowdhury and F. Mémoli. Persistent path homology of directed networks. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1152–1169. SIAM, 2018.10.1137/1.9781611975031.75Search in Google Scholar

11. D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discr.Comput. Geom., 37(1):103–120, 2007.10.1007/s00454-006-1276-5Search in Google Scholar

12. D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Extending persistence using Poincaré and Lefschetz duality. Foundations of Computational Mathematics, 9(1):79–103, 2009.10.1007/s10208-008-9027-zSearch in Google Scholar

13. M. d’Amico, P. Frosini, and C. Landi. Using matching distance in size theory: A survey. Int. J. Imag. Syst. Tech., 16(5):154–161, 2006.Search in Google Scholar

14. M. d’Amico, P. Frosini, and C. Landi. Natural pseudo-distance and optimal matching between reduced size functions. Acta Applicandae Mathematicae, 109(2):527–554, 2010.10.1007/s10440-008-9332-1Search in Google Scholar

15. P. Dłotko, K. Hess, R. Levi, M. Nolte, M. Reimann, M. Scolamiero, K. Turner, E. Muller, and H. Markram. Topological analysis of the connectome of digital reconstructions of neural microcircuits. arXiv preprint arXiv:1601.01580, 2016.Search in Google Scholar

16. C. H. Dowker. Homology groups of relations. Annals of mathematics, pages 84–95, 1952.10.2307/1969768Search in Google Scholar

17. H. Edelsbrunner and J. Harer. Persistent homology—a survey. In Surveys on discrete and computational geometry, volume 453 of Contemp. Math., pages 257–282. Amer. Math. Soc., Providence, RI, 2008.10.1090/conm/453/08802Search in Google Scholar

18. H. Edelsbrunner and J. Harer. Computational Topology: An Introduction. American Mathematical Society, 2009.10.1090/mbk/069Search in Google Scholar

19. A.-H. Esfahanian. Connectivity algorithms. In Topics in structural graph theory, pages 268–281. Cambridge University Press, 2013.Search in Google Scholar

20. P. Frosini and C. Landi. Size theory as a topological tool for computer vision. Pattern Recognition and Image Analysis, 9(4):596–603, 1999.Search in Google Scholar

21. C. Giusti, R. Ghrist, and D. S. Bassett. Two’s company, three (or more) is a simplex. Journal of computational neuroscience, 41(1):1–14, 2016.10.1007/s10827-016-0608-6Search in Google Scholar

22. C. Giusti, E. Pastalkova, C. Curto, and V. Itskov. Clique topology reveals intrinsic geometric structure in neural correlations. Proceedings of the National Academy of Sciences, 112(44):13455–13460, 2015.10.1073/pnas.1506407112Search in Google Scholar

23. A. Hagberg, P. Swart, and D. S Chult. Exploring network structure, dynamics, and function using networkx. Technical report, Los Alamos National Lab.(LANL), Los Alamos, NM (United States), 2008.Search in Google Scholar

24. A. Hatcher. Algebraic Topology. Algebraic Topology. Cambridge University Press, 2002.Search in Google Scholar

25. D. Horak, S. Maletić, and M. Rajković. Persistent homology of complex networks. Journal of Statistical Mechanics: Theory and Experiment, 2009(03):P03034, 2009.10.1088/1742-5468/2009/03/P03034Search in Google Scholar

26. W. Huang and A. Ribeiro. Persistent homology lower bounds on high-order network distances. IEEE Transactions on Signal Processing, 65(2):319–334, 2017.10.1109/TSP.2016.2620963Search in Google Scholar

27. R. Jayaraman, G. Raja, A. K. Bashir, C. S. Hussain, A. Hassan, and M. A. Alqarni. Interference mitigation based on radio aware channel assignment for wireless mesh networks. Wireless Personal Communications, 101(3):1539–1557, 2018.10.1007/s11277-018-5776-4Search in Google Scholar

28. J. Jonsson. Simplicial complexes of graphs, volume 3. Springer, 2008.10.1007/978-3-540-75859-4Search in Google Scholar

29. C. Landi and P. Frosini. New pseudodistances for the size function space. In R. A. Melter, A. Y. Wu, and L. J. Latecki, editors, Proceedings SPIE, Vision Geometry VI, volume 3168, pages 52–60, 1997.10.1117/12.279674Search in Google Scholar

30. M. Lesnick. The theory of the interleaving distance on multidimensional persistence modules. Foundations of Computational Mathematics, pages 1–38, 2015.Search in Google Scholar

31. L.-D. Lord, P. Expert, H. M. Fernandes, G. Petri, T. J. Van Hartevelt, F. Vaccarino, G. Deco, F. Turkheimer, and M. L. Kringelbach. Insights into brain architectures from the homological sca olds of functional connectivity networks. Frontiers in Systems Neuroscience, 10, 2016.10.3389/fnsys.2016.00085Search in Google Scholar

32. L. Lovász. Kneser’s conjecture, chromatic number, and homotopy. Journal of Combinatorial Theory, Series A, 25(3):319–324, 1978.10.1016/0097-3165(78)90022-5Search in Google Scholar

33. S. Maletić, M. Rajković, and D. Vasiljević. Simplicial complexes of networks and their statistical properties. In International Conference on Computational Science, pages 568–575. Springer, 2008.10.1007/978-3-540-69387-1_65Search in Google Scholar

34. S. Maletić, Y. Zhao, and M. Rajković. Persistent topological features of dynamical systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(5):053105, 2016.10.1063/1.4949472Search in Google Scholar

35. P. Masulli and A. E. Villa. The topology of the directed clique complex as a network invariant. SpringerPlus, 5(1):388, 2016.10.1186/s40064-016-2022-ySearch in Google Scholar

36. E. I. Moser, E. Krop , and M.-B. Moser. Place cells, grid cells, and the brain’s spatial representation system. Annu. Rev. Neurosci., 31:69–89, 2008.10.1146/annurev.neuro.31.061307.090723Search in Google Scholar

37. S. Pal, T. J. Moore, R. Ramanathan, and A. Swami. Comparative topological signatures of growing collaboration networks. In Workshop on Complex Networks CompleNet, pages 201–209. Springer, 2017.10.1007/978-3-319-54241-6_18Search in Google Scholar

38. G. Petri, M. Scolamiero, I. Donato, and F. Vaccarino. Topological strata of weighted complex networks. PloS one, 8(6):e66506, 2013.10.1371/journal.pone.0066506Search in Google Scholar

39. T. Pino, S. Choudhury, and F. Al-Turjman. Dominating set algorithms for wireless sensor networks survivability. IEEE Access, 6:17527–17532, 2018.10.1109/ACCESS.2018.2819083Search in Google Scholar

40. E. Prisner. Convergence of iterated clique graphs. Discrete Mathematics, 103(2):199–207, 1992.10.1016/0012-365X(92)90270-PSearch in Google Scholar

41. F. P. Ramsey. On a problem of formal logic. Proceedings of the London Mathematical Society, s2-30(1):264–286, 1930.10.1112/plms/s2-30.1.264Search in Google Scholar

42. M. W. Reimann, M. Nolte, M. Scolamiero, K. Turner, R. Perin, G. Chindemi, P. Dłotko, R. Levi, K. Hess, and H. Markram. Cliques of neurons bound into cavities provide a missing link between structure and function. Frontiers in Computational Neuroscience, 11:48, 2017.10.3389/fncom.2017.00048546743428659782Search in Google Scholar

43. C. Shannon. The zero error capacity of a noisy channel. IRE Transactions on Information Theory, 2(3):8–19, 1956.10.1109/TIT.1956.1056798Search in Google Scholar

44. G. Singh, F. Mémoli, and G. E. Carlsson. Topological methods for the analysis of high dimensional data sets and 3d object recognition. In SPBG, pages 91–100, 2007.Search in Google Scholar

45. A. Sizemore, C. Giusti, and D. S. Bassett. Classification of weighted networks through mesoscale homological features. Journal of Complex Networks, 5(2):245–273, 2017.10.1093/comnet/cnw013Search in Google Scholar

46. A. E. Sizemore, C. Giusti, A. Kahn, J. M. Vettel, R. F. Betzel, and D. S. Bassett. Cliques and cavities in the human connectome. Journal of Computational Neuroscience, 44(1):115–145, Feb 2018.10.1007/s10827-017-0672-6576985529143250Search in Google Scholar

47. E. H. Spanier. Algebraic topology, volume 55. Springer Science & Business Media, 1994.Search in Google Scholar

48. R. Tarjan. Depth-first search and linear graph algorithms. SIAM journal on computing, 1(2):146–160, 1972.10.1137/0201010Search in Google Scholar

49. E. Tomita, A. Tanaka, and H. Takahashi. The worst-case time complexity for generating all maximal cliques and computational experiments. Theoretical computer science, 363(1):28–42, 2006.10.1016/j.tcs.2006.06.015Search in Google Scholar

50. K. Turner. Rips filtrations for quasimetric spaces and asymmetric functions with stability results. Algebraic & Geometric Topology, 19(3):1135–1170, 2019.10.2140/agt.2019.19.1135Search in Google Scholar

51. A. Verri, C. Uras, P. Frosini, and M. Ferri. On the use of size functions for shape analysis. Biol. Cybern., 70:99–107, 1993.10.1007/BF00200823Search in Google Scholar

52. D. J. Watts and S. H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393(6684):440, 1998.10.1038/309189623998Search in Google Scholar