Pub Date : 2025-01-01Epub Date: 2025-09-06DOI: 10.1007/s41468-025-00213-z
David Beers, Jacob Leygonie
Consider the space of continuous functions on a geometric tree X whose persistent homology gives rise to a finite generic barcode D. We show that there are exactly as many path connected components in this space as there are merge trees whose barcode is D. We find that each component is homotopy equivalent to a configuration space on X with specialised constraints encoded by the merge tree. For barcodes D with either one or two intervals, our method also allows us to compute the homotopy type of this space of functions.
{"title":"The fiber of persistent homology for trees.","authors":"David Beers, Jacob Leygonie","doi":"10.1007/s41468-025-00213-z","DOIUrl":"10.1007/s41468-025-00213-z","url":null,"abstract":"<p><p>Consider the space of continuous functions on a geometric tree <i>X</i> whose persistent homology gives rise to a finite generic barcode <i>D</i>. We show that there are exactly as many path connected components in this space as there are merge trees whose barcode is <i>D</i>. We find that each component is homotopy equivalent to a configuration space on <i>X</i> with specialised constraints encoded by the merge tree. For barcodes <i>D</i> with either one or two intervals, our method also allows us to compute the homotopy type of this space of functions.</p>","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"9 3","pages":"22"},"PeriodicalIF":0.0,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12414068/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145024820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish tight bi-Lipschitz bounds certifying quasi-universality (universality up to a constant factor) for various distances between Reeb graphs: the interleaving distance, the functional distortion distance, and the functional contortion distance. The definition of the latter distance is a novel contribution, and for the special case of contour trees we also prove strict universality of this distance. Furthermore, we prove that for the special case of merge trees the functional contortion distance coincides with the interleaving distance, yielding universality of all four distances in this case.
{"title":"Tight quasi-universality of Reeb graph distances.","authors":"Ulrich Bauer, Håvard Bakke Bjerkevik, Benedikt Fluhr","doi":"10.1007/s41468-025-00203-1","DOIUrl":"10.1007/s41468-025-00203-1","url":null,"abstract":"<p><p>We establish tight bi-Lipschitz bounds certifying quasi-universality (universality up to a constant factor) for various distances between Reeb graphs: the interleaving distance, the functional distortion distance, and the functional contortion distance. The definition of the latter distance is a novel contribution, and for the special case of contour trees we also prove strict universality of this distance. Furthermore, we prove that for the special case of merge trees the functional contortion distance coincides with the interleaving distance, yielding universality of all four distances in this case.</p>","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"9 1","pages":"7"},"PeriodicalIF":0.0,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11821802/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143434453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01Epub Date: 2025-11-25DOI: 10.1007/s41468-025-00218-8
Barbara Giunti, Guillaume Houry, Michael Kerber, Matthias Söls
We study the algorithmic complexity of computing the persistence barcode of a randomly generated filtration. We provide a general technique to bound the expected complexity of reducing the boundary matrix in terms of the density of its reduced form. We apply this technique finding upper bounds for the average fill-in (number of non-zero entries) of the boundary matrix on Čech, Vietoris-Rips and Erdős-Rényi filtrations after matrix reduction, thus obtaining bounds on the expected complexity of the barcode computation. Our method is based on previous results on the expected Betti numbers of the corresponding complexes. Our fill-in bounds for Čech and Vietoris-Rips complexes are asymptotically tight up to a logarithmic factor. In particular, both our fill-in and computation bounds are better than the worst-case estimates. We also provide an Erdős-Rényi filtration realizing the worst-case fill-in and computation.
{"title":"Expected Complexity of Barcode Reduction.","authors":"Barbara Giunti, Guillaume Houry, Michael Kerber, Matthias Söls","doi":"10.1007/s41468-025-00218-8","DOIUrl":"10.1007/s41468-025-00218-8","url":null,"abstract":"<p><p>We study the algorithmic complexity of computing the persistence barcode of a randomly generated filtration. We provide a general technique to bound the expected complexity of reducing the boundary matrix in terms of the density of its reduced form. We apply this technique finding upper bounds for the average fill-in (number of non-zero entries) of the boundary matrix on Čech, Vietoris-Rips and Erdős-Rényi filtrations after matrix reduction, thus obtaining bounds on the expected complexity of the barcode computation. Our method is based on previous results on the expected Betti numbers of the corresponding complexes. Our fill-in bounds for Čech and Vietoris-Rips complexes are asymptotically tight up to a logarithmic factor. In particular, both our fill-in and computation bounds are better than the worst-case estimates. We also provide an Erdős-Rényi filtration realizing the worst-case fill-in and computation.</p>","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"9 4","pages":"29"},"PeriodicalIF":0.0,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12647303/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145643700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-15DOI: 10.1007/s41468-024-00187-4
Ulrike Tillmann, Gunnar Carlsson
{"title":"Proceedings of ATMCS10","authors":"Ulrike Tillmann, Gunnar Carlsson","doi":"10.1007/s41468-024-00187-4","DOIUrl":"https://doi.org/10.1007/s41468-024-00187-4","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"10 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141646285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-01DOI: 10.1007/s41468-024-00183-8
Peter Bubenik, Nikola Milićević
{"title":"Homotopy, homology, and persistent homology using closure spaces","authors":"Peter Bubenik, Nikola Milićević","doi":"10.1007/s41468-024-00183-8","DOIUrl":"https://doi.org/10.1007/s41468-024-00183-8","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"4 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141698290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-07DOI: 10.1007/s41468-024-00172-x
Changlin Wan, Muhan Zhang, Pengtao Dang, Weiyu Hao, Sha Cao, Pan Li, Chi Zhang
{"title":"Ambiguities in neural-network-based hyperedge prediction","authors":"Changlin Wan, Muhan Zhang, Pengtao Dang, Weiyu Hao, Sha Cao, Pan Li, Chi Zhang","doi":"10.1007/s41468-024-00172-x","DOIUrl":"https://doi.org/10.1007/s41468-024-00172-x","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"129 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141002946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-03DOI: 10.1007/s41468-024-00173-w
R. Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, M. Saghafian
{"title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","authors":"R. Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, M. Saghafian","doi":"10.1007/s41468-024-00173-w","DOIUrl":"https://doi.org/10.1007/s41468-024-00173-w","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"44 188","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141016743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-24DOI: 10.1007/s41468-024-00163-y
Yasuaki Hiraoka, Tomoyuki Shirai
{"title":"Torsion-weighted spanning acycle entropy in cubical lattices and Mahler measures","authors":"Yasuaki Hiraoka, Tomoyuki Shirai","doi":"10.1007/s41468-024-00163-y","DOIUrl":"https://doi.org/10.1007/s41468-024-00163-y","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"13 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140435018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-10DOI: 10.1007/s41468-023-00162-5
Luis Alberto
{"title":"Pseudospheres: combinatorics, topology and distributed systems","authors":"Luis Alberto","doi":"10.1007/s41468-023-00162-5","DOIUrl":"https://doi.org/10.1007/s41468-023-00162-5","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":" 1146","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139786963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-10DOI: 10.1007/s41468-023-00162-5
Luis Alberto
{"title":"Pseudospheres: combinatorics, topology and distributed systems","authors":"Luis Alberto","doi":"10.1007/s41468-023-00162-5","DOIUrl":"https://doi.org/10.1007/s41468-023-00162-5","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"29 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139846834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号