Magnitude: a bibliography

 

This bibliography is maintained by Tom Leinster and Mark Meckes.

This is a list of all the publications on magnitude of which we are aware. Here we mean "magnitude" in the specific sense of these papers.

The criterion for inclusion is that the paper is specifically about magnitude, or magnitude plays a major role in the work, rather than it merely including a citation.

We have included papers on magnitude of categories, which is usually called Euler characteristic. However, we have not included papers on the diversity measures that are closely related to magnitude, except for those in which magnitude is discussed explicitly.

Within each topic, papers are ordered by arXiv date. We'll add journal publication details as we learn them, and would be happy to hear of any we've missed.

You may also be interested in the list of n-Category Café blog posts on magnitude, diversity, and closely related ideas.

Magnitude (Euler characteristic) of ordinary categories:

  1. Tom Leinster.
    The Euler characteristic of a category.
    arXiv:math.CT/0610260, 2006; Documenta Mathematica 13 (2008), 21–49.
  2. Clemens Berger and Tom Leinster.
    The Euler characteristic of a category as the sum of a divergent series.
    arXiv:0707.0835, 2007; Homology, Homotopy and Applications 10 (2008), 41–51.
  3. Martin Wedel Jacobsen and Jesper Møller.
    Euler characteristics and Möbius algebras of p-subgroup categories.
    arXiv:1007.1890, 2010; Journal of Pure and Applied Algebra 216 (2012), 2665–2696.
  4. Kazunori Noguchi.
    The Euler characteristic of acyclic categories.
    arXiv:1004.2547, 2010; Kyushu Journal of Mathematics 65 (2011), 85–99.
  5. Tom Leinster.
    Notions of Möbius inversion.
    arXiv:1201.0413, 2012; Bulletin of the Belgian Mathematical Society—Simon Stevin 19 (2012), 911–935.
  6. Kazunori Noguchi.
    Euler characteristics of categories and barycentric subdivision.
    arXiv:1104.3630, 2011; Münster Journal of Mathematics 6 (2013), 85–116.
  7. Kazunori Noguchi.
    The zeta function of a finite category.
    arXiv:1203.6133, 2012; Documenta Mathematica 18 (2013), 1243–1274.
  8. Kazunori Noguchi.
    The zeta function of a finite category which has Möbius inversion.
    arXiv:1205.4380, 2012.
  9. Kazunori Noguchi.
    The zeta function of a finite category and the series Euler characteristic.
    arXiv:1207.6750, 2012.
  10. Jesper Møller.
    Euler characteristics of centralizer subcategories.
    arXiv:1502.01317, 2015.
  11. Kohei Tanaka.
    Čech complexes for covers of small categories.
    arXiv:1508.03688, 2015; Homology, Homotopy and Applications 19 (2017), 281–291.
  12. Kohei Tanaka.
    Discrete Euler integration over functions on finite categories.
    arXiv:1508.06391, 2015; Topology and its Applications 204 (2016), 185–197.
  13. Mustafa Akkaya and Özgün Ünlü.
    The Euler characteristic of finite categories.
    arXiv:2301.08966, 2023.
  14. Stephanie Chen and Juan Pablo Vigneaux.
    Categorical magnitude and entropy.
    arXiv:2303.00879, 2023.
  15. Stephanie Chen and Juan Pablo Vigneaux.
    A formula for the categorical magnitude in terms of the Moore–Penrose pseudoinverse.
    arXiv:2303.12176, 2023; Bulletin of the Belgian Mathematical Society---Simon Stevin 30 (2023), 341–353.

Magnitude of enriched categories (see also the paper "The magnitude of metric spaces" below):

  1. Kazunori Noguchi and Kohei Tanaka.
    The Euler characteristic of an enriched category.
    arXiv:1405.3356, 2014; Theory and Applications of Categories 31 (2016), 1–30.
  2. Yasuhiko Asao.
    Magnitude and magnitude homology of filtered set enriched categories.
    arXiv:2303.05677, 2023.
  3. Steve Huntsman.
    Magnitude of arithmetic scalar and matrix categories.
    arXiv:2304.08334, 2023.
  4. Juan Pablo Vigneaux.
    A combinatorial approach to categorical Möbius inversion and pseudoinversion.
    arXiv:2407.14647, 2024.

Magnitude of metric spaces:

  1. Tom Leinster and Simon Willerton.
    On the asymptotic magnitude of subsets of Euclidean space.
    arXiv:0908.1582, 2009; Geometriae Dedicata 164 (2013), 287–310.
  2. Tom Leinster.
    A maximum entropy theorem with applications to the measurement of biodiversity.
    arXiv:0910.0906, 2009.
  3. Simon Willerton.
    Heuristic and computer calculations for the magnitude of metric spaces.
    arXiv:0910.5500, 2009.
  4. Simon Willerton.
    On the magnitude of spheres, surfaces and other homogeneous spaces.
    arXiv:1005.4041, 2010; Geometriae Dedicata 168 (2014), 291–310.
  5. Tom Leinster.
    The magnitude of metric spaces.
    arXiv:1012.5857, 2010; Documenta Mathematica 18 (2013), 857–905.
  6. Mark Meckes.
    Positive definite metric spaces.
    arXiv:1012.5863, 2010; Positivity 17 (2013), 733–757.
  7. Simon Willerton.
    Spread: a measure of the size of metric spaces.
    arXiv:1209.2300, 2012; International Journal of Computational Geometry and Applications 25 (2015), 207–225.
  8. Mark Meckes.
    Magnitude, diversity, capacities, and dimensions of metric spaces.
    arXiv:1308.5407, 2013; Potential Analysis 42 (2015), 549–572.
  9. Juan Antonio Barceló and Anthony Carbery.
    On the magnitudes of compact sets in Euclidean spaces.
    arXiv:1507.02502, 2015; American Journal of Mathematics 140 (2018), 449–494.
  10. Tom Leinster and Mark Meckes.
    Maximizing diversity in biology and beyond.
    arXiv:1512.06314, 2015; Entropy 18 (2016), article 88.
  11. Tom Leinster and Mark Meckes.
    The magnitude of a metric space: from category theory to geometric measure theory.
    arXiv:1606.00095, 2016; in Nicola Gigli (ed.), Measure Theory in Non-Smooth Spaces, de Gruyter Open, 2017.
  12. Heiko Gimperlein and Magnus Goffeng.
    On the magnitude function of domains in Euclidean space.
    arXiv:1706.06839, 2017; American Journal of Mathematics 143 (2021), 939–967.
  13. Simon Willerton.
    The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials.
    arXiv:1708.03227, 2017; Discrete Analysis 2020, number 5.
  14. Simon Willerton.
    On the magnitude of odd balls via potential functions.
    arXiv:1804.02174, 2018.
  15. Mark Meckes.
    On the magnitude and intrinsic volumes of a convex body in Euclidean space.
    arXiv:1904.08923, 2019; Mathematika 66 (2020), 343–355.
  16. Glenn Fung, Eric Bunch and Dan Dickinson.
    Approximating the convex hull via metric space magnitude.
    arXiv:1908.02692, 2019.
  17. Tom Leinster and Emily Roff.
    The maximum entropy of a metric space.
    arXiv:1908.11184, 2019; Quarterly Journal of Mathematics 72 (2021), 1271–1309.
  18. Eric Bunch, Daniel Dickinson, Jeffery Kline and Glenn Fung.
    Practical applications of metric space magnitude and weighting vectors.
    arXiv:2006.14063, 2020.
  19. Tom Leinster.
    Entropy and Diversity: The Axiomatic Approach.
    arXiv:2012.02113, 2020; Cambridge University Press, 2021.
  20. Eric Bunch, Jeffery Kline, Daniel Dickinson, Suhaas Bhat and Glenn Fung.
    Weighting vectors for machine learning: numerical harmonic analysis applied to boundary detection.
    arXiv:2106.00827, 2021.
  21. Heiko Gimperlein and Magnus Goffeng.
    The Willmore energy and the magnitude of Euclidean domains.
    arXiv:2109.10097, 2021; Proceedings of the American Mathematical Society 151 (2023), 897–906.
  22. Michael Adamer, Edward De Brouwer, Leslie O'Bray and Bastian Rieck.
    The magnitude vector of images.
    arXiv:2110.15188, 2021; Journal of Applied and Computational Topology, 2024.
  23. Tom Leinster and Mark Meckes.
    Spaces of extremal magnitude.
    arXiv:2112.12889, 2021; Proceedings of the American Mathematical Society 151 (2023), 3967–3973.
  24. Nikoletta Louca.
    A pseudodifferential approach to the fractional Laplacian and magnitude.
    PhD thesis, Heriot-Watt University and the University of Edinburgh, 2021.
  25. Steve Huntsman.
    Diversity enhancement via magnitude.
    arXiv:2201.10037, 2022.
  26. Steve Huntsman.
    Parallel black-box optimization of expensive high-dimensional multimodal functions via magnitude.
    arXiv:2201.11677, 2022.
  27. Heiko Gimperlein, Magnus Goffeng and Nikoletta Louca.
    Semiclassical analysis of a nonlocal boundary value problem related to magnitude.
    arXiv:2201.11357, 2022; Journal d'Analyse Mathématique 153 (2024), 401–487.
  28. Heiko Gimperlein, Magnus Goffeng and Nikoletta Louca.
    The magnitude and spectral geometry.
    arXiv:2201.11363, 2022; American Journal of Mathematics, to appear.
  29. Miguel O'Malley, Sara Kalisnik and Nina Otter.
    Alpha magnitude.
    arXiv:2205.09521, 2022.
  30. Mark Meckes.
    Magnitude and Holmes–Thompson intrinsic volumes of convex bodies.
    arXiv:2206.02600, 2022; Canadian Mathematical Bulletin 66 (2023), 854–867.
  31. Steve Huntsman.
    Quality-diversity in dissimilarity spaces.
    arXiv:2211.12337, 2022.
  32. Miguel O'Malley.
    Magnitude, alpha magnitude, and applications.
    PhD thesis, Wesleyan University, 2023.
  33. Rayna Andreeva, Katharina Limbeck, Bastian Rieck and Rik Sarkar.
    Metric space magnitude and generalisation in neural networks.
    arXiv:2305.05611, 2023; in Proceedings of the 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), number 221, p.242–253, 2023.
  34. Yasuhiko Asao.
    Classification of metric fibrations.
    arXiv:2307.04387, 2023.
  35. Kiyonori Gomi.
    A direct proof for the positive definiteness of four point metric spaces.
    arXiv:2310.20690, 2023.
  36. Katharina Limbeck, Rayna Andreeva, Rik Sarkar and Bastian Rieck.
    Metric space magnitude for evaluating the diversity of latent representations.
    arXiv:2311.16054, 2023.
  37. Emily Roff and Masahiko Yoshinaga.
    The small-scale limit of magnitude and the one-point property.
    arXiv:2312.14497, 2023.
  38. Jun O'Hara.
    Magnitude function identifies generic finite metric spaces.
    arXiv:2401.00786, 2024.
  39. Nick Kunz.
    Fractal dimensions in theory and practice.
    Master's thesis, ETH Zürich, 2024.
  40. Rayna Andreeva, Benjamin Dupuis, Rik Sarkar, Tolga Birdal and Umut Şimşekli.
    Topological generalization bounds for discrete-time stochastic optimization algorithms
    arXiv:2407.08723, 2024.
  41. Hiroki Kodama and Jun O'Hara.
    Identification of circular spaces by magnitude and discrete Riesz energy.
    arXiv:2408.06091, 2024.
  42. Rayna Andreeva, James Ward, Primoz Skraba, Jie Gao and Rik Sarkar.
    Approximating metric magnitude of point sets.
    arXiv:2409.04411, 2024.
  43. Heiko Gimperlein and Magnus Goffeng.
    Riesz energies and the magnitude of manifolds.
    arXiv:2409.19969, 2024. Proceedings of the American Mathematical Society, to appear.
  44. Ksenia Pereverdieva, André Deutz, Tessa Ezendam, Thomas Bäck, Hèrm Hofmeyer and Michael T. M. Emmerich.
    Comparative analysis indicators for multiobjective diversity optimization.
    arXiv:2410.18900, 2024.
  45. Simon Willerton.
    Metric-like spaces as enriched categories: three vignettes.
    arXiv:2501.00416, 2024.
  46. Yemeen Ayub.
    Towards metric measure space learning.
    PhD thesis, George Mason University, 2024.
  47. Tai-Danae Bradley and Juan Pablo Vigneaux.
    The magnitude of categories of texts enriched by language models.
    arXiv:2501.06662, 2025.
  48. Hirokazu Katsumasa, Emily Roff and Masahiko Yoshinaga.
    Is magnitude 'generically continuous' for finite metric spaces?
    arxiv:2501.08745, 2025.
  49. Jun O'Hara.
    Identification of generic finite metric spaces by eigenvalues.
    arXiv:2502.08980, 2025.

Magnitude of higher categories:

  1. Kohei Tanaka.
    The Euler characteristic of a bicategory and the product formula for fibered bicategories.
    arXiv:1410.0248, 2014.

Magnitude of additive and linear categories:

  1. Joseph Chuang, Alastair King and Tom Leinster.
    On the magnitude of a finite dimensional algebra.
    arXiv:1505.04281, 2015; Theory and Applications of Categories 31 (2016), 63–72.
  2. Dawei Shen and Yaru Wu.
    The magnitude for Nakayama algebras.
    arXiv:2303.06553, 2023.

Magnitude of graphs (excluding magnitude homology):

  1. Tom Leinster.
    The magnitude of a graph.
    arXiv:1401.4623, 2014; Mathematical Proceedings of the Cambridge Philosophical Society 166 (2019), 247–264.
  2. Steve Huntsman.
    Magnitude and topological entropy of digraphs.
    arXiv:2205.05178, 2022.

Magnitude (co)homology:

  1. Richard Hepworth and Simon Willerton.
    Categorifying the magnitude of a graph.
    arXiv:1505.04125, 2015; Homology, Homotopy and Applications 19(2) (2017), 31–60.
  2. Tom Leinster and Michael Shulman.
    Magnitude homology of enriched categories and metric spaces.
    arXiv:1711.00802, 2017; Algebraic & Geometric Topology 21 (2021), 2175–2221.
  3. Ryuki Kaneta and Masahiko Yoshinaga.
    Magnitude homology of metric spaces and order complexes.
    arXiv:1803.04247, 2018; Bulletin of the London Mathematical Society 53(3) (2021), 893–905
  4. Benoît Jubin.
    On the magnitude homology of metric spaces.
    arXiv:1803.05062, 2018.
  5. Nina Otter.
    Magnitude meets persistence: homology theories for filtered simplicial sets.
    arXiv:1807.01540, 2018; Homology, Homotopy and Applications 24 (2022), 365–387.
  6. Nina Otter.
    The homology of data.
    PhD thesis, University of Oxford, 2018.
  7. Richard Hepworth.
    Magnitude cohomology.
    arXiv:1807.06832, 2018; Mathematische Zeitschrift 301 (2022), 3617–3640.
  8. Kiyonori Gomi.
    Smoothness filtration of the magnitude complex.
    arXiv:1809.06593, 2018; Forum Mathematicum 32 (2020), 625–639.
  9. Yuzhou Gu.
    Graph magnitude homology via algebraic Morse theory.
    arXiv:1809.07240, 2018.
  10. Kiyonori Gomi.
    Magnitude homology of geodesic space.
    arXiv:1902.07044, 2019.
  11. Yasuhiko Asao.
    Magnitude homology of geodesic metric spaces with an upper curvature bound.
    arXiv:1903.11794, 2019; Algebraic & Geometric Topology 21 (2021), 647–664.
  12. Victor Summers.
    Torsion in the Khovanov homology of links and the magnitude homology of graphs.
    PhD thesis, North Carolina State University, 2019.
  13. Simon Cho.
    Quantales, persistence, and magnitude homology.
    arXiv:1910.02905, 2019.
  14. Dejan Govc and Richard Hepworth.
    Persistent magnitude.
    arXiv:1911.11016, 2019; Journal of Pure and Applied Algebra 225 (2021), 106517.
  15. Radmila Sazdanovic and Victor Summers.
    Torsion in the magnitude homology of graphs.
    arXiv:1912.13483, 2019; Journal of Homotopy and Related Structures 16(2) (2021), 275–296.
  16. Yasuhiko Asao and Kengo Izumihara.
    Geometric approach to graph magnitude homology.
    arXiv:2003.08058, 2020; Homology, Homotopy and Applications 23 (2021), 297–310.
  17. Rémi Bottinelli and Tom Kaiser.
    Magnitude homology, diagonality, medianness, Künneth and Mayer–Vietoris.
    arXiv:2003.09271, 2020; Homology, Homotopy and Applications 23 (2021), 121–140.
  18. Yasuhiko Asao, Yasuaki Hiraoka and Shu Kanazawa.
    Girth, magnitude homology, and phase transition of diagonality.
    arXiv:2101.09044, 2021; Proceedings of the Royal Society of Edinburgh Section A: Mathematics 154 (2024), 221–247.
  19. Yu Tajima and Masahiko Yoshinaga.
    Magnitude homology of graphs and discrete Morse theory on Asao–Izumihara complexes.
    arXiv:2110.02458, 2021; Homology, Homotopy and Applications 25 (2023), 331–343.
  20. Vadim Lebovici.
    Hybrid transforms of constructible functions.
    arXiv:2111.07829, 2021.
  21. Yasuhiko Asao.
    Magnitude homology and path homology.
    arXiv:2201.08047, 2022; Bulletin of the London Mathematical Society 55(1) (2023), 375–398.
  22. Steve Huntsman.
    Discrete topological methods for cybersecurity, network science, and machine learning.
    hal-03688966, 2022.
  23. Emily Roff.
    The size and shape of things: magnitude, diversity, homology.
    PhD thesis, University of Edinburgh, 2022.
  24. Emily Roff.
    Magnitude, homology, and the Whitney twist.
    arXiv:2211.02520, 2022; Homology, Homotopy and Applications, to appear.
  25. Luigi Caputi and Carlo Collari.
    On finite generation in magnitude (co)homology and its torsion.
    arXiv:2302.06525, 2023; Bulletin of the London Mathematical Society 56 (2024), 3434–3451.
  26. Yu Tajima and Masahiko Yoshinaga.
    Causal order complex and magnitude homotopy type of metric spaces.
    arXiv:2302.09752, 2023; International Mathematics Research Notices 2024 (2024), 3176–3222.
  27. Shaobo Di, Sergei O. Ivanov, Lek Mukoseev and Mengmeng Zhang.
    On the path homology of Cayley digraphs and covering digraphs.
    arXiv:2305.15683, 2023.
  28. Wangying Bi, Jingyan Li and Jie Wu.
    The magnitude homology of a hypergraph.
    arXiv:2306.01534, 2023.
  29. Emily Roff.
    Iterated magnitude homology.
    arXiv:2309.00577, 2023.
  30. Richard Hepworth and Emily Roff.
    The reachability homology of a directed graph.
    arXiv:2312.01378, 2023.
  31. Sergei O. Ivanov.
    Nested homotopy models of finite metric spaces and their spectral homology.
    arXiv:2312.11878, 2023.
  32. Eudes Robert.
    Persistence, magnitude and blurred homology.
    Master's thesis, ETH Zürich, 2023.
  33. Luigi Caputi, Carlo Collari and Eric Ramos.
    The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology.
    arXiv:2401.01248, 2024.
  34. Yasuhiko Asao and Sergei O. Ivanov.
    Magnitude homology is a derived functor.
    arXiv:2402.14466, 2024.
  35. Chad Giusti and Giuliamaria Menara.
    Eulerian magnitude homology: subgraph structure and random graphs.
    arXiv:2403.09248, 2024.
  36. Yu Tajima.
    Discrete Morse theory on magnitude homotopy types of finite graphs.
    PhD thesis, Hokkaido University, 2024.
  37. Richard Hepworth and Emily Roff.
    Bigraded path homology and the magnitude-path spectral sequence.
    arXiv:2404.06689, 2024.
  38. Sergei O. Ivanov and Lev Mukoseev.
    On diagonal digraphs, Koszul algebras and triangulations of homology spheres.
    arXiv:2405.04748, 2024.
  39. Adrián Doña Mateo and Tom Leinster.
    Magnitude homology equivalence of Euclidean sets.
    arXiv:2406.11172, 2024; Algebraic and Geometric Topology, to appear.
  40. Yasuhiko Asao and Shun Wakatsuki.
    Minimal projective resolution and magnitude homology of geodetic metric spaces.
    arXiv:2408.12147, 2024.
  41. Yasuhiko Asao, Yu Tajima and Masahiko Yoshinaga.
    Magnitude homology and homotopy type of metric fibrations.
    arXiv:2409.03278, 2024.
  42. Giuliamaria Menara.
    On torsion in eulerian magnitude homology of Erdős-Rényi random graphs.
    arXiv:2409.03472, 2024.
  43. Ellen Gasparovic, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang and Lori Ziegelmeier.
    A survey of simplicial, relative, and chain complex homology theories for hypergraphs.
    arXiv:2409.18310, 2024.
  44. Giuliamaria Menara and Luca Manzoni.
    Computing eulerian magnitude homology.
    arXiv:2410.10376, 2024.
  45. Sergei O. Ivanov and Xiaomeng Xu.
    On lp-Vietoris-Rips complexes.
    arXiv:2411.01857, 2024.
  46. Daisuke Kishimoto and Yichen Tong.
    The fundamental group and the magnitude-path spectral sequence of a directed graph.
    arXiv:2411.02838, 2024.
  47. Matthew Burfitt and Tyrone Cutler.
    Inductive construction of path homology chains.
    arXiv:2411.09501, 2024.
  48. Luigi Caputi and Giuliamaria Menara.
    Eulerican magnitude homology: diagonality, injective words, and regular path homology.
    arXiv:2503.06722, 2025.
  49. Patrick Martin II and Radmila Sazdanovic.
    Torsion in magnitude homology theories.
    arXiv:2503.11976, 2025.
  50. Giuliamaria Menara.
    Combinatorial aspects of discrete structures — graphs and Latin squares.
    PhD thesis, University of Trieste, 2025.

 
This page was last changed on 7 April 2025.