Discrete hierarchical organization of social group sizes

Abstract

The 'social brain hypothesis' for the evolution of large brains in primates has led to evidence for the coevolution of neocortical size and social group sizes, suggesting that there is a cognitive constraint on group size that depends, in some way, on the volume of neural material available for processing and synthesizing information on social relationships. More recently, work on both human and non-human primates has suggested that social groups are often hierarchically structured. We combine data on human grouping patterns in a comprehensive and systematic study. Using fractal analysis, we identify, with high statistical confidence, a discrete hierarchy of group sizes with a preferred scaling ratio close to three: rather than a single or a continuous spectrum of group sizes, humans spontaneously form groups of preferred sizes organized in a geometrical series approximating 3-5, 9-15, 30-45, etc. Such discrete scale invariance could be related to that identified in signatures of herding behaviour in financial markets and might reflect a hierarchical processing of social nearness by human brains.

Figure 1

Presentation of our dataset of 61 group sizes. The ordinate is an arbitrary ordering of data sources and the abscissa gives the group sizes reported in each source. The symbols refer to the classification used in each of the studies: circles (support cliques), triangles (sympathy groups), diamonds (bands), stars (cognitive groups), and squares (small and large tribes). This classification is not used in our systematic analysis summarized in the other figures, to avoid any bias.

Figure 2

Probability density function f(s) of size s estimated with a Gaussian kernel estimator in the variable ln(s) with a bandwidth h=0.14. Varying h by 100% does not change f(s) significantly.

Figure 3

Typical (H,q)-derivative $D_q^{H}f(s)$ of the probability density f(s) as a function of size s with H=0.5 and q=0.8.

Figure 4

Normalized Lomb periodograms P_N(ω) as a function of angular log-frequency ω of the (H,q)-derivative $D_q^{H}f(s)$ for different pairs of (H,q) with 0.5≤H≤0.9 and 0.65≤q≤0.95.

Figure 5

Average Lomb periodogram P_N(ω) of the (H,q)-derivative $D_q^{H}f(s)$ with respect to the number of receivers of the residual contact frequency for each individual in the Christmas card experiment, as a function of the angular log-frequency ω of the (H,q)-derivative, over the 42 individuals and different pairs of (H,q) with −1≤H≤1 and 0.80≤q≤0.95.