Skip to main content

Modeling the Emergence of Modular Leadership Hierarchy During the Collective Motion of Herds Made of Harems


Gregarious animals need to make collective decisions in order to keep their cohesiveness. Several species of them live in multilevel societies, and form herds composed of smaller communities. We present a model for the development of a leadership hierarchy in a herd consisting of loosely connected sub-groups (e.g. harems) by combining self organization and social dynamics. It starts from unfamiliar individuals without relationships and reproduces the emergence of a hierarchical and modular leadership network that promotes an effective spreading of the decisions from more capable individuals to the others, and thus gives rise to a beneficial collective decision. Our results stemming from the model are in a good agreement with our observations of a Przewalski horse herd (Hortobágy, Hungary). We find that the harem-leader to harem-member ratio observed in Przewalski horses corresponds to an optimal network in this approach regarding common success, and that the observed and modeled harem size distributions are close to a lognormal.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Abegglen, J.J.: On Socialization in Hamadryas Baboons: A Field Study. Bucknell University Press, Lewisburg (1984)

    Google Scholar 

  2. 2.

    Adamcsek, B., Palla, G., Farkas, I.J., Derényi, I., Vicsek, T.: CFinder: locating cliques and overlapping modules in biological networks. Bioinformatics 22, 1021–1023 (2006)

    Article  Google Scholar 

  3. 3.

    Ákos, Zs., Beck, R., Nagy, M., Vicsek, T., & Kubinyi, E.: Leadership and path characteristics during walks are linked to dominance order and individual traits in dogs. PLoS Comput. Biol. 10, e1003446 (2014)

  4. 4.

    Baird, R.: The killer whale: Foraging specializations and group hunting. In: Mann, J., Connor, R.C., Tyack, P., Whitehead, H. (eds.) Cetacean Societies, pp. 127–153. University of Chicago Press, Chicago (2000)

    Google Scholar 

  5. 5.

    Bourjade, M., Thierry, B., Maumy, M., Petit, O.: Decision-making in Przewalski horses (Equus ferus przewalskii) is driven by the ecological contexts of collective movements. Ethology 115, 321–330 (2009)

    Article  Google Scholar 

  6. 6.

    Boyd, L., Houpt, K.A.: Przewalski\(^{\prime }\)s Horses. State University of New York Press, New York (1994)

    Google Scholar 

  7. 7.

    Dunbar, R.I.M., Dunbar, E.P.: Social dynamics of gelada baboons. Contrib. Primatol. 6, 1–157 (1975)

    ADS  Google Scholar 

  8. 8.

    Eades, P., Lin, X., Smyth, W.F.: A fast and effective heuristic for the feedback arc set problem. Inf. Process. Lett. 47(6), 319–323 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Feh, C., Munkhtuya, B., Enkhbold, S., Sukhbaatar, T.: Ecology and social structure of the Gobi khulan Equus hemionus subsp in the Gobi B National Park. Mongolia Biol. Cons. 101, 51–61 (2001)

    Article  Google Scholar 

  10. 10.

    Fischhoff, I.R., Sundaresan, S.R., Cordingley, J., Larkin, H.M., Sellier, M.-J., Rubenstein, D.I.: Social relationships and reproductive state influence leadership roles in movements of plains zebra, Equus burchellii. Anim. Behav. 73, 825–831 (2007)

    Article  Google Scholar 

  11. 11.

    Guimera, R., Nunes Amaral, L.A.: Functional cartography of complex metabolic networks. Nature 433, 895–900 (2005)

  12. 12.

    Grueter, C.C., Matsuda, I., Zhang, P., Zinner, D.: Multilevel societies in primates and other mammals: introduction to the special issue. Int. J. Primatol. 33, 993–1001 (2012)

  13. 13.

    Grueter, C.C., Chapais, B., Zinner, D.: Evolution of multilevel societies in nonhuman primates and humans. Int. J. Primatol. 33, 1002–1037 (2012)

    Article  Google Scholar 

  14. 14.

    Houpt, K.A., Fraser, A.F.: Przewalski horses. Appl. Anim. Behav. Sci. 21, 1–3 (1988)

    Article  Google Scholar 

  15. 15.

    Jacobs, A., Sueur, C., Deneubourg, J.-L., Petit, O.: Social network influences decision making during collective movements in brown lemurs. Int. J. Primatol. 32, 721–736 (2011)

    Article  Google Scholar 

  16. 16.

    King, A.J., Douglas, C.M.S., Huchard, E., Isaac, N.J.B., Cowlishaw, G.: Dominance and affiliation mediate despotism in a social primate. Curr. Biol. 18, 1833–1838 (2008)

    Article  Google Scholar 

  17. 17.

    King, A.J., Sueur, C., Huchard, E., Cowlishaw, G.: A rule-of-thumb based on social affiliation explains collective movements in desert baboons. Anim. Behav. 82, 1337–1345 (2011)

    Article  Google Scholar 

  18. 18.

    Kummer, H.: Social Organisation of Hamdryas Baboons. A Field Study. University of Chicago Press, Chicago (1968)

    Google Scholar 

  19. 19.

    Lusseau, D., Conradt, L.: The emergence of unshared consensus decisions in bottlenose dolphins. Behav. Ecol. Sociobiol. 63, 1067–1077 (2009)

    Article  Google Scholar 

  20. 20.

    Mones, E., Vicsek, L., Vicsek, T.: Hierarchy measure for complex networks. PLoS ONE 7(3), e33799 (2012)

    ADS  Article  Google Scholar 

  21. 21.

    Nagy, M., Ákos, Z., Biro, D., Vicsek, T.: Hierarchical group dynamics in pigeon flocks. Nature 464, 890–893 (2010)

    ADS  Article  Google Scholar 

  22. 22.

    Nagy, M., Vásárhelyi, G., Pettit, B., Roberts-Mariani, I., Vicsek, T., Biro, D.: Context-dependent hierarchies in pigeons. PNAS 110, 13049–13054 (2013)

    ADS  Article  Google Scholar 

  23. 23.

    Nepusz, T., Vicsek, T.: Hierarchical self-organization of non-cooperating individuals. PLoS ONE 8(12), e81449 (2013)

    ADS  Article  Google Scholar 

  24. 24.

    Palla, G., Derényi, I., Illés, F., Vicsek, T.: Uncovering the overlapping community structure of complex networks in nature and society. Nature 435, 814–818 (2005)

    ADS  Article  Google Scholar 

  25. 25.

    Petit, O., Gautrais, J., Leca, J.-B., Theraulaz, G., Deneubourg, J.-L.: Collective decision-making in white-faced capuchin monkeys. Proc. R. Soc. B. 276, 3495–3503 (2009)

    Article  Google Scholar 

  26. 26.

    Petit, O., Bon, R.: Decision-making process: the case of collective movements. Behav. Process 84, 635–647 (2010)

    Article  Google Scholar 

  27. 27.

    Ravasz, E., Somera, A.L., Mongru, D.A., Oltvai, Z.N., Barabási, A.-L.: Hierarchical organization of modularity in metabolic networks. Science 297, 1551–1555 (2002)

    ADS  Article  Google Scholar 

  28. 28.

    Rubenstein, D.I., Hack, M.: Natural and sexual selection and the evolution of multi-level societies: insights from zebras with comparisons to primates. In: Kappeler, P.M., van Schaik, C.P. (eds.) Sexual Selection in Primates: New and Comparative Perspectives, pp. 266–279. Cambridge University Press, New York (2004)

    Chapter  Google Scholar 

  29. 29.

    Sueur, C., King, A.J., Pelé, M., Petit, O.: Fast and accurate decisions as a result of scale-free network properties in two primate species. In: Proceedings of the European Conference on Complex Systems 2012, Springer Proceedings in Complexity, pp. 579–584 (2013)

  30. 30.

    Sueur, C., MacIntosh, A.J.J., Jacobs, A.T., Watanabe, K., Petit, O.: Predicting leadership using nutrient requirements and dominance rank of group members. Behav. Ecol. Sociobiol. 67, 457–470 (2013)

    Article  Google Scholar 

  31. 31.

    Whitehead, H., Antunes, R., Gero, S., Wong, S.N.P., Engelhaupt, D., Rendell, L.: Multilevel societies of female sperm whales (Physeter macrocephalus) in the Atlantic and Pacific: why are they so different? Int. J. Primatol. 33, 1142–1164 (2012)

    Article  Google Scholar 

  32. 32.

    Wittemyer, G., Douglas-Hamilton, I., Getz, W.M.: The socioecology of elephants: analysis of the processes creating multitiered social structures. Anim. Behav. 69, 1357–1371 (2005)

    Article  Google Scholar 

  33. 33.

    Wittemyer, G., Getz, W.M.: Hierarchical dominance structure and social organization in African elephants, Loxodonta africana. Anim. Behav. 73, 671–681 (2007)

    Article  Google Scholar 

  34. 34.

    Zafeiris, A., Vicsek, T.: Group performance is maximized by hierarchical competence distribution. Nat. Commun. 4, 3484 (2013)

    Article  Google Scholar 

Download references


We are grateful to Kristin Brabender and the Directorate of Hortobágy National Park for providing us their data as well as officially authorizing the carrying out of our observations and to Waltraut Zimmermann, Kölner Zoo for cooperation. We thank Gábor Vásárhelyi and Gergő Somorjai for helping our observations with flying robots. This research was partially supported by the EU ERC COLLMOT project (Grant No. 227878).

Author information



Corresponding author

Correspondence to Katalin Ozogány.

Electronic supplementary material

Below is the link to the electronic supplementary material.

ESM 1 (MOV 35647 kb)

ESM 2 (MOV 8426 kb)

ESM 3 (PDF 593 kb)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ozogány, K., Vicsek, T. Modeling the Emergence of Modular Leadership Hierarchy During the Collective Motion of Herds Made of Harems. J Stat Phys 158, 628–646 (2015).

Download citation


  • Collective animal behaviour
  • Leadership hierarchy
  • Multilevel societies
  • Collective decision making
  • Modular hierarchy