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Transitivity and structural balance in marmot social networks

  • Tina W. WeyEmail author
  • Ferenc Jordán
  • Daniel T. Blumstein
Original Article

Abstract

Social relationships are composed of both positive (affiliative) and negative (agonistic) interactions, representing opposing effects. Social network theory predicts that positive relationships should be transitive; thus, the friend of a friend is more likely to be a friend. Further, when considering both positive and negative relationships jointly, structural balance theory predicts that certain configurations of positive and negative relationships in a triad are inherently less stable (unbalanced) and should tend to be eliminated. However, structural balance has been rarely examined in nonhuman social systems. We tested for transitivity and structural balance in social networks of socially flexible yellow-bellied marmots (Marmota flaviventer) and asked if group size, network density, or group composition affected the degree of structural balance. We found a consistent pattern of significant transitivity in positive interactions, some transitivity in negative interactions, and some evidence of structural balance. In particular, a “weak” definition of structural balance is probably more common than “strong” structural balance, which used a stricter definition of balance. Network size limited the ability to detect these social processes, and smaller networks were less likely to show significant transitivity or structural balance. The proportion of adult females in a group affected the level of transitivity but did not affect the degree of structural balance. Our study suggests that there are intriguing similarities in social processes across diverse animal societies and that studying triads and network motifs may help identify basic social mechanisms linking local to global structure.

Significance statement

Social network theory predicts that basic social mechanisms should lead to similar structural properties across different societies. For example, positive relationships should be transitive (a friend of a friend is a friend), and certain combinations of positive and negative relationships represent conflict and should be unstable over time (e.g., a friend of a friend being an enemy is an unstable state). This latter theory, called structural balance, has rarely been examined in nonhuman societies; hence, we tested for transitivity and structural balance in groups of free-living yellow-bellied marmots. Positive interactions were generally transitive, but evidence for structural balance was inconsistent. Furthermore, group composition could affect network transitivity, and small network size (associated with few interactions) limits ability to detect significant patterns. Our results suggest that transitivity is fundamental in structuring positive relationships, while some forms of structural balance are present but not widespread.

Keywords

Social networks Structural balance Network transitivity Triad analysis Yellow-bellied marmots 

Notes

Acknowledgments

We thank Adriana Maldonado-Chaparro and Julien Martin for database curation and all the marmoteers over the years for data collection. This work was also assisted through participation in the Animal Social Networks Investigative Workshop at the National Institute for Mathematical and Biological Synthesis, sponsored by the National Science Foundation (award no. DBI-1300426), with additional support from the University of Tennessee, Knoxville. We thank Josh Firth and an anonymous reviewer for extremely constructive feedback on an earlier version of the manuscript.

Funding

This work was supported by a U.S. Department of Education Graduate Assistance in Areas of National Need Fellowship, a National Science Foundation GK-12 Fellowship, a UCLA Chancellor’s Prize, and a Rocky Mountain Biological Laboratory (RMBL) Snyder Graduate Research Fellowship (to TWW); by the National Geographic Society (grant no. 8140-06), the UCLA Faculty Senate and Division of Life Sciences, and a RMBL research fellowship (to DTB); by the National Science Foundation (IDBR-0754247, DEB-1119660, and 1557130 to DTB; DBI 0242960 and 0731346 to the RMBL); and by the National Research, Development, and Innovation Office (NKFIH grant OTKA K 11607 to FJ).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

All applicable international, national, and/or institutional guidelines for the care and use of animals were followed. All procedures performed in studies involving animals were in accordance with the ethical standards of the institution or practice at which the studies were conducted. Marmots are studied under protocols approved by UCLA (2001-191, renewed annually) and the Colorado Division of Wildlife (TR914, renewed annually).

Supplementary material

265_2019_2699_MOESM1_ESM.xlsx (23 kb)
ESM 1 Table S1 Network properties and triad summaries in marmot social networks. “AF” = adult females. “Structural Balance (Method 1)” assigns links with both + and - interactions as - (tending towards conflict). “Structural Balance (Method 2)” omits links with both + and – interactions (ambiguous relationships not included). The “P-value” is the probability of obtaining a transitivity score from 1000 permutations of the observed network (i.e., maintaining network size and density) that is as large as or larger than the value from the observed network. “P-value1” and “P-value 2” are the probabilities of obtaining a level of structural balance as large as or larger than the observed level, again from 1000 permutations, where structural balance is defined as either the number of balanced triads or the proportion of balanced triads, respectively. “Link Diff.” is the difference in number of links from Method 1 and Method 2 (i.e., the number of node pairs that had both + and - interactions). “Prop. Diff.” is the Link Diff. / Links from Method 1. (XLSX 23 kb)
265_2019_2699_MOESM2_ESM.r (3 kb)
ESM 2 R functions to calculate strong and weak structural balance (R 2 kb)

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Family Medicine and Emergency MedicineUniversity of SherbrookeSherbrookeCanada
  2. 2.Department of Ecology and Evolutionary BiologyUniversity of California at Los AngelesLos AngelesUSA
  3. 3.MTA Centre for Ecological ResearchHungarian Academy of SciencesBudapestHungary
  4. 4.The Microsoft Research-University of Trento Centre for Computational and Systems BiologyRoveretoItaly
  5. 5.Rocky Mountain Biological LaboratoryCrested ButteUSA

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