*Ecological Archives* E096-062-A1

E. Vander Wal, M. Festa-Bianchet, D. Réale, D. W. Coltman, F. Pelletier. 2015. Sex-based differences in the adaptive value of social behavior contrasted against morphology and environment. *Ecology* 96:631–641. http://dx.doi.org/10.1890/14-1320.1

Appendix A. Supplementary methods pertaining to the choice and measurement of centrality metrics.

SUPPLEMENTARY METHODS

Simple measures of centrality, for example, degree (the number of neighbors an animal is connected to via edges) may not capture the variation in pairwise associations shared between animals in dense networks. The equivalent measure for weighted networks, such as ours, is graph strength: the sum of all weighted edges an individual shares with its neighbors (Csardi and Nepusz 2006). Graph strength only accounts for direct associations among individuals. Here, however, we focus on a more complex measure, eigenvector centrality (*EC*), to measure of centrality. EC accounts not only for direct associations between nodes but also for indirect associations and affinities. EC calculates the prominence of an individual in a weighted network where the centrality of each individual is proportional to the sum of the centralities of individuals to which it is connected (i.e., observed in the same group). See Bonacich (1987) for details and equations used in the R package igraph (Csardi and Nepusz 2006).

Eigenvector centrality is highly effective at approximating centrality in a network (Costenbader and Valente 2003, Maiya and Berger-Wolf 2010). In general, individuals with high eigenvector centralities are connected to many other individuals that are, in turn, connected to many individuals (Csardi and Nepusz 2006). The largest values are obtained by individuals in high-density substructures (Csardi and Nepusz 2006) or highly connected to their neighbors.

Graph strength and eigenvector centrality are highly correlated. Using a general linear mixed model which controlled for year (as graph strength can vary according to population size) the centrality measures are highly correlated. The correlation is significant in both sexes (*p* < 0.001) and explains considerable variation: *R*² = 0.97 and 0.80 for females and males respectively. Despite this we replicate all results from the body of the manuscript, based on eigenvector centrality, using graph strength (Tables B1 and B2).

Literature cited

Bonacich, P. 1987. Power and centrality: a family of measures. American Journal of Sociology 92:1170–1182.

Costenbader, E., and T. W. Valente. 2003. The stability of centrality measures when networks are sampled. Social Networks 25:283–307.

Csardi, G., and T. Nepusz. 2006. The igraph software package for complex network research. R.

Maiya, A. S., and T. Y. Berger-Wolf. 2010. Online sampling of high centrality individuals in social networks. Pages 91–98 Advances in Knowledge Discovery and Data Mining. Springer.