---
_id: '49326'
abstract:
- lang: eng
  text: Many networked systems are governed by non-pairwise interactions between nodes.
    The resulting higher-order interaction structure can then be encoded by means
    of a hypernetwork. In this paper we consider dynamical systems on hypernetworks
    by defining a class of admissible maps for every such hypernetwork. We explain
    how to classify robust cluster synchronization patterns on hypernetworks by finding
    balanced partitions, and we generalize the concept of a graph fibration to the
    hypernetwork context. We also show that robust synchronization patterns are only
    fully determined by polynomial admissible maps of high order. This means that,
    unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order,
    i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork,
    for the degree of the polynomial admissible maps that determine robust synchronization
    patterns. We also demonstrate that this degree is optimal by investigating a class
    of examples. We conclude by demonstrating how this effect may cause remarkable
    synchrony breaking bifurcations that occur at high polynomial degree.
author:
- first_name: Sören
  full_name: von der Gracht, Sören
  id: '97359'
  last_name: von der Gracht
  orcid: 0000-0002-8054-2058
- first_name: Eddie
  full_name: Nijholt, Eddie
  last_name: Nijholt
- first_name: Bob
  full_name: Rink, Bob
  last_name: Rink
citation:
  ama: 'von der Gracht S, Nijholt E, Rink B. Hypernetworks: Cluster Synchronization
    Is a Higher-Order Effect. <i>SIAM Journal on Applied Mathematics</i>. 2023;83(6):2329-2353.
    doi:<a href="https://doi.org/10.1137/23m1561075">10.1137/23m1561075</a>'
  apa: 'von der Gracht, S., Nijholt, E., &#38; Rink, B. (2023). Hypernetworks: Cluster
    Synchronization Is a Higher-Order Effect. <i>SIAM Journal on Applied Mathematics</i>,
    <i>83</i>(6), 2329–2353. <a href="https://doi.org/10.1137/23m1561075">https://doi.org/10.1137/23m1561075</a>'
  bibtex: '@article{von der Gracht_Nijholt_Rink_2023, title={Hypernetworks: Cluster
    Synchronization Is a Higher-Order Effect}, volume={83}, DOI={<a href="https://doi.org/10.1137/23m1561075">10.1137/23m1561075</a>},
    number={6}, journal={SIAM Journal on Applied Mathematics}, publisher={Society
    for Industrial &#38; Applied Mathematics (SIAM)}, author={von der Gracht, Sören
    and Nijholt, Eddie and Rink, Bob}, year={2023}, pages={2329–2353} }'
  chicago: 'Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “Hypernetworks: Cluster
    Synchronization Is a Higher-Order Effect.” <i>SIAM Journal on Applied Mathematics</i>
    83, no. 6 (2023): 2329–53. <a href="https://doi.org/10.1137/23m1561075">https://doi.org/10.1137/23m1561075</a>.'
  ieee: 'S. von der Gracht, E. Nijholt, and B. Rink, “Hypernetworks: Cluster Synchronization
    Is a Higher-Order Effect,” <i>SIAM Journal on Applied Mathematics</i>, vol. 83,
    no. 6, pp. 2329–2353, 2023, doi: <a href="https://doi.org/10.1137/23m1561075">10.1137/23m1561075</a>.'
  mla: 'von der Gracht, Sören, et al. “Hypernetworks: Cluster Synchronization Is a
    Higher-Order Effect.” <i>SIAM Journal on Applied Mathematics</i>, vol. 83, no.
    6, Society for Industrial &#38; Applied Mathematics (SIAM), 2023, pp. 2329–53,
    doi:<a href="https://doi.org/10.1137/23m1561075">10.1137/23m1561075</a>.'
  short: S. von der Gracht, E. Nijholt, B. Rink, SIAM Journal on Applied Mathematics
    83 (2023) 2329–2353.
date_created: 2023-11-29T10:50:05Z
date_updated: 2023-11-29T10:52:23Z
department:
- _id: '101'
doi: 10.1137/23m1561075
external_id:
  arxiv:
  - '2302.08974'
intvolume: '        83'
issue: '6'
keyword:
- Applied Mathematics
language:
- iso: eng
page: 2329-2353
publication: SIAM Journal on Applied Mathematics
publication_identifier:
  issn:
  - 0036-1399
  - 1095-712X
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: 'Hypernetworks: Cluster Synchronization Is a Higher-Order Effect'
type: journal_article
user_id: '97359'
volume: 83
year: '2023'
...
---
_id: '53340'
author:
- first_name: Kevin J.
  full_name: Painter, Kevin J.
  last_name: Painter
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Painter KJ, Winkler M. Phenotype Switching in Chemotaxis Aggregation Models
    Controls the Spontaneous Emergence of Large Densities. <i>SIAM Journal on Applied
    Mathematics</i>. 2023;83(5):2096-2117. doi:<a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>
  apa: Painter, K. J., &#38; Winkler, M. (2023). Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities. <i>SIAM
    Journal on Applied Mathematics</i>, <i>83</i>(5), 2096–2117. <a href="https://doi.org/10.1137/22m1539393">https://doi.org/10.1137/22m1539393</a>
  bibtex: '@article{Painter_Winkler_2023, title={Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities}, volume={83},
    DOI={<a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>}, number={5},
    journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial
    &#38; Applied Mathematics (SIAM)}, author={Painter, Kevin J. and Winkler, Michael},
    year={2023}, pages={2096–2117} }'
  chicago: 'Painter, Kevin J., and Michael Winkler. “Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities.” <i>SIAM
    Journal on Applied Mathematics</i> 83, no. 5 (2023): 2096–2117. <a href="https://doi.org/10.1137/22m1539393">https://doi.org/10.1137/22m1539393</a>.'
  ieee: 'K. J. Painter and M. Winkler, “Phenotype Switching in Chemotaxis Aggregation
    Models Controls the Spontaneous Emergence of Large Densities,” <i>SIAM Journal
    on Applied Mathematics</i>, vol. 83, no. 5, pp. 2096–2117, 2023, doi: <a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>.'
  mla: Painter, Kevin J., and Michael Winkler. “Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities.” <i>SIAM
    Journal on Applied Mathematics</i>, vol. 83, no. 5, Society for Industrial &#38;
    Applied Mathematics (SIAM), 2023, pp. 2096–117, doi:<a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>.
  short: K.J. Painter, M. Winkler, SIAM Journal on Applied Mathematics 83 (2023) 2096–2117.
date_created: 2024-04-07T12:52:03Z
date_updated: 2024-04-07T12:52:06Z
doi: 10.1137/22m1539393
intvolume: '        83'
issue: '5'
keyword:
- Applied Mathematics
language:
- iso: eng
page: 2096-2117
publication: SIAM Journal on Applied Mathematics
publication_identifier:
  issn:
  - 0036-1399
  - 1095-712X
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: Phenotype Switching in Chemotaxis Aggregation Models Controls the Spontaneous
  Emergence of Large Densities
type: journal_article
user_id: '31496'
volume: 83
year: '2023'
...
---
_id: '63277'
author:
- first_name: Kevin J.
  full_name: Painter, Kevin J.
  last_name: Painter
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Painter KJ, Winkler M. Phenotype Switching in Chemotaxis Aggregation Models
    Controls the Spontaneous Emergence of Large Densities. <i>SIAM Journal on Applied
    Mathematics</i>. 2023;83(5):2096-2117. doi:<a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>
  apa: Painter, K. J., &#38; Winkler, M. (2023). Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities. <i>SIAM
    Journal on Applied Mathematics</i>, <i>83</i>(5), 2096–2117. <a href="https://doi.org/10.1137/22m1539393">https://doi.org/10.1137/22m1539393</a>
  bibtex: '@article{Painter_Winkler_2023, title={Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities}, volume={83},
    DOI={<a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>}, number={5},
    journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial
    &#38; Applied Mathematics (SIAM)}, author={Painter, Kevin J. and Winkler, Michael},
    year={2023}, pages={2096–2117} }'
  chicago: 'Painter, Kevin J., and Michael Winkler. “Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities.” <i>SIAM
    Journal on Applied Mathematics</i> 83, no. 5 (2023): 2096–2117. <a href="https://doi.org/10.1137/22m1539393">https://doi.org/10.1137/22m1539393</a>.'
  ieee: 'K. J. Painter and M. Winkler, “Phenotype Switching in Chemotaxis Aggregation
    Models Controls the Spontaneous Emergence of Large Densities,” <i>SIAM Journal
    on Applied Mathematics</i>, vol. 83, no. 5, pp. 2096–2117, 2023, doi: <a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>.'
  mla: Painter, Kevin J., and Michael Winkler. “Phenotype Switching in Chemotaxis
    Aggregation Models Controls the Spontaneous Emergence of Large Densities.” <i>SIAM
    Journal on Applied Mathematics</i>, vol. 83, no. 5, Society for Industrial &#38;
    Applied Mathematics (SIAM), 2023, pp. 2096–117, doi:<a href="https://doi.org/10.1137/22m1539393">10.1137/22m1539393</a>.
  short: K.J. Painter, M. Winkler, SIAM Journal on Applied Mathematics 83 (2023) 2096–2117.
date_created: 2025-12-18T19:15:29Z
date_updated: 2025-12-18T20:11:36Z
doi: 10.1137/22m1539393
intvolume: '        83'
issue: '5'
language:
- iso: eng
page: 2096-2117
publication: SIAM Journal on Applied Mathematics
publication_identifier:
  issn:
  - 0036-1399
  - 1095-712X
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: Phenotype Switching in Chemotaxis Aggregation Models Controls the Spontaneous
  Emergence of Large Densities
type: journal_article
user_id: '31496'
volume: 83
year: '2023'
...
---
_id: '63349'
author:
- first_name: Nicola
  full_name: Bellomo, Nicola
  last_name: Bellomo
- first_name: Kevin J.
  full_name: Painter, Kevin J.
  last_name: Painter
- first_name: Youshan
  full_name: Tao, Youshan
  last_name: Tao
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Bellomo N, Painter KJ, Tao Y, Winkler M. Occurrence vs. Absence of Taxis-Driven
    Instabilities in a May--Nowak Model for Virus Infection. <i>SIAM Journal on Applied
    Mathematics</i>. 2019;79(5):1990-2010. doi:<a href="https://doi.org/10.1137/19m1250261">10.1137/19m1250261</a>
  apa: Bellomo, N., Painter, K. J., Tao, Y., &#38; Winkler, M. (2019). Occurrence
    vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection.
    <i>SIAM Journal on Applied Mathematics</i>, <i>79</i>(5), 1990–2010. <a href="https://doi.org/10.1137/19m1250261">https://doi.org/10.1137/19m1250261</a>
  bibtex: '@article{Bellomo_Painter_Tao_Winkler_2019, title={Occurrence vs. Absence
    of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection}, volume={79},
    DOI={<a href="https://doi.org/10.1137/19m1250261">10.1137/19m1250261</a>}, number={5},
    journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial
    &#38; Applied Mathematics (SIAM)}, author={Bellomo, Nicola and Painter, Kevin
    J. and Tao, Youshan and Winkler, Michael}, year={2019}, pages={1990–2010} }'
  chicago: 'Bellomo, Nicola, Kevin J. Painter, Youshan Tao, and Michael Winkler. “Occurrence
    vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection.”
    <i>SIAM Journal on Applied Mathematics</i> 79, no. 5 (2019): 1990–2010. <a href="https://doi.org/10.1137/19m1250261">https://doi.org/10.1137/19m1250261</a>.'
  ieee: 'N. Bellomo, K. J. Painter, Y. Tao, and M. Winkler, “Occurrence vs. Absence
    of Taxis-Driven Instabilities in a May--Nowak Model for Virus Infection,” <i>SIAM
    Journal on Applied Mathematics</i>, vol. 79, no. 5, pp. 1990–2010, 2019, doi:
    <a href="https://doi.org/10.1137/19m1250261">10.1137/19m1250261</a>.'
  mla: Bellomo, Nicola, et al. “Occurrence vs. Absence of Taxis-Driven Instabilities
    in a May--Nowak Model for Virus Infection.” <i>SIAM Journal on Applied Mathematics</i>,
    vol. 79, no. 5, Society for Industrial &#38; Applied Mathematics (SIAM), 2019,
    pp. 1990–2010, doi:<a href="https://doi.org/10.1137/19m1250261">10.1137/19m1250261</a>.
  short: N. Bellomo, K.J. Painter, Y. Tao, M. Winkler, SIAM Journal on Applied Mathematics
    79 (2019) 1990–2010.
date_created: 2025-12-19T10:47:32Z
date_updated: 2025-12-19T10:47:44Z
doi: 10.1137/19m1250261
intvolume: '        79'
issue: '5'
language:
- iso: eng
page: 1990-2010
publication: SIAM Journal on Applied Mathematics
publication_identifier:
  issn:
  - 0036-1399
  - 1095-712X
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: Occurrence vs. Absence of Taxis-Driven Instabilities in a May--Nowak Model
  for Virus Infection
type: journal_article
user_id: '31496'
volume: 79
year: '2019'
...