The social brain: scale-invariant layering of Erdős–Rényi networks in small-scale human societies

Abstract

The cognitive ability to form social links that can bind individuals together into large cooperative groups for safety and resource sharing was a key development in human evolutionary and social history. The ‘social brain hypothesis’ argues that the size of these social groups is based on a neurologically constrained capacity for maintaining long-term stable relationships. No model to date has been able to combine a specific socio-cognitive mechanism with the discrete scale invariance observed in ethnographic studies. We show that these properties result in nested layers of self-organizing Erdős–Rényi networks formed by each individual's ability to maintain only a small number of social links. Each set of links plays a specific role in the formation of different social groups. The scale invariance in our model is distinct from previous ‘scale-free networks’ studied using much larger social groups; here, the scale invariance is in the relationship between group sizes, rather than in the link degree distribution. We also compare our model with a dominance-based hierarchy and conclude that humans were probably egalitarian in hunter–gatherer-like societies, maintaining an average maximum of four or five social links connecting all members in a largest social network of around 132 people.

1. Introduction

A key distinguishing feature of human society is the size and adaptive complexity of our social dynamics and the individual cognitive ability to develop and maintain stable interactions. This socio-cognitive coupling has played an important role in human evolutionary and social history. The ‘social brain hypothesis’ argues that the maximum group size is limited by some socio-cognitive mechanism for which each group member has a limited capacity and thereby limits the typical group size. Dunbar [1,2] was the first to observe that there is a correlation between an allometric ratio C_r of two neural volumes and the social group size N for primates across a large range of genera. The ratio measured was

For example, for the pygmy marmoset C_r = 1.43 and N = 6.0, whereas for the chimpanzee C_r = 3.22 and N = 53.5. By extrapolating to humans (C_r = 4.10), Dunbar [3] estimated the maximum average stable group size to be N = 147.8 (95% confidence interval = [100.2,231.1]). The social brain hypothesis [4] then posits that the primary evolutionary drive in small-scale or hunter–gatherer societies was the expansion of the brain's neocortical executive function to manage the complex interactions of our social relationships.

This hypothesis places an upper limit on the human capacity to manage social interactions, but it does not tell us what socio-cognitive process mediates this limit or what specifically is limited in capacity. Partial explanations have come from functional magnetic resonance imaging studies that have localized this function to the orbital prefrontal cortex [5,6]: the larger the neural volume in this region, the larger an individual's social network tends to be. However, this is a key area of the brain's executive function and it is involved in many different tasks, such as planning [7], reward estimation [8,9] and encoding of novel information [10,11], and it is integrated into many other decision-making processes [12,13]. Other cognitive constraints are also well documented: the number of objects we can rapidly and confidently identify without explicitly counting (subitizing) is limited to about four items [14,15] and we have a limited capacity to hold independent ‘chunks’ of information in our working memory, often quoted as: ‘7 ± 2’ [16]. In a similar vein, what is currently missing in our understanding of social group sizes is a quantifiable mechanism that connects our limited ability in a specific task and the discrete layers of our social networks.

The hypothesis we test is that in order to connect together individuals to form a social network, each person needs to maintain a discrete number of stable social links, each of a different grade, or level of social closeness, that we can only form a small number of these and that they are explicitly the mechanism through which different social layers form. Such social links correspond to particular social group sizes (figure 1a): intimate friends form support cliques, more distant friends form sympathy groups and so on [17]. These groups are in turn mediated by different cognitive processes [18]: support clique size is mediated by an ability to understand another person's perspective and sympathy group size is mediated by memory. As there are in general many possible mechanisms that could generate the observed group sizes, we begin by distilling three properties (desiderata) of small groups that have been observed and that will characterize a class of possible mechanisms:

• (i) Scale-invariant branching factor. Either the number of individuals [19] or the frequency of group sizes [20] at each social layer, or scale, is a fixed multiple of the corresponding quantity at the next level.

• (ii) Log-linearity [20,21]. The log of the size of each social layer i is a linear function of the layer index i.

• (iii) Social binding [22]. Each layer must have more than half of all individuals linked together in order to act as a nucleus for social cohesion.

Figure 1.

Two different social network models. (a) Random links form between sub-group members. As the average number of links per person increases in discrete steps, the network size also increases in predictable, discrete, steps. (b) A structured hierarchy similar to modern military, bureaucratic and corporate structures in which each layer is ‘managed’ by a coordinator. (Online version in colour.)

The two interpretations of scale-invariant branching in point (i) are related. If we denote the number of individuals at level i by G_i, scale invariance can be defined by G_i = BG_i−1 for some constant B [19]. Alternatively, if we define f(G_i) as the frequency of groups of size G_i within a larger society, then scale invariance can be defined as f(G_i−1) = Cf(G_i) for some constant C; this is the Horton–Strahler branching ratio [20]. If there is a power-law relationship between size and frequency, then one type of scale-invariant network implies the other: and B^γ = C. This distinction is important for the two models we consider.

2. Models of social structure

In previous work, it has been argued that there was a major transition in human social evolution from small-scale non-centralized egalitarian groups to large-scale centralized and hierarchical societies [23,24]. These more recent non-egalitarian societies were integrated by symbolic markers and chains of command which allowed for a reduction in the cognitive load of maintaining large networks of interactions. The two models that we test address this distinction between dominance and egalitarianism, where cognitive ability plays different social roles.

In our first model, we explicitly connect the social layer index i with the number of social links that an individual is able to form. Formally, we propose that the average number of links per person in a social network is identical with i + δ for some 0 < δ < 1, where we will use and to denote averages and set cardinality, respectively. The first model is the dynamic egalitarianism model:

2.1

where is the number of people within a group at layer i, is the branching factor between successive layers, i is the social layer index [20,21], i.e. support clique: i = 1, sympathy group: i = 2 and so on (figure 1a), and is a parameter that allows model (2.1) to fit a non-integer average number of links for each social layer:

For this model, we define a sub-group S_i,j at social layer i in the following way. In a small-scale society, there is a group V containing all individuals and links of differing strengths exist between members of V. For each i, an identifying index j is given by a function that labels every member as belonging to a single group at layer i, such that contains all people in group j. For example, V_2,j, are the three-coloured sets shown in figure 1a, whereas V_1,j, are the 10 sets of nodes contained in the smallest circles. Then, we define a labelling function mapping every link in E to a social layer i, We define S_i,j within layer i as

and

The links within a sub-group S_i,j include all stronger social links formed at layers m < i, in addition to the links at layer i, but exclude weaker social links for layers m > i. Figure 1a, illustrates the notion of sub-group layers: the encircled group of red nodes for i = 2 includes the sub-groups of the dyadic relationships at i = 0, the support cliques at i = 1, the sympathy groups at i = 2, but not the higher order members at i = 3.

Our proposal means that the discrete step from to is mediated by adding, on average and across a large society containing many instances of such social groups, an extra link per person such that the number of links per person increases from to Taking the averages across j, scale invariance in group sizes is maintained on average:

The alternative model is the hierarchical dominance model:

2.2

in which there are ‘hierarchs’ (coordinators in figure 1b) per group on average. Note that equation (2.2) does not satisfy the scale invariance in the number of individuals , but it is scale invariant in the frequency of groups of a given size. To verify this, note that there are groups of size subordinated by direct connections to each i-level hierarch. Formally, yielding scale invariance in the number of groups of a given size: with the branching factor between successive layers given by A simple example illustrates this argument (figure 1b: ): there are three groups of size at i = 2, i.e. and within each of these groups their hierarch subordinates groups at level i = 1, resulting in nine groups of size formally,

The difference between these two forms of scale invariance is especially pronounced for smaller group sizes. It is also evident by construction (figure 1b) that for a structured hierarchy and so the number of social links does not increase with group size and, therefore, the average cognitive demand on each individual does not increase. Consequently, the hierarchical structure does not have the log-linearity property. However, this model plays an important counterpoint in social anthropology in that it represents a dominance-based social hierarchy. Additionally, for much larger social groups, the scale invariance in the number of individuals is an asymptotic property of this model as well. Given equation (2.2), for large i. This also follows from the observation that where and

3. Data

Our primary data come from Zhou et al. [19], who used a technique developed for analysing discrete, scale-invariant systems to extract underlying parameters from empirical data. It was applied to several dozen earlier ethnographic studies and the authors reported point estimates of group sizes of and a branching factor of There has been one other large-scale summary analysis of ethnographic data reporting scale-invariant properties in human social networks: in [20] hunter–gatherer societies were studied and a mean branching factor of B = 3.6 was reported. We also analysed the data in this study and the results are consistent with the results shown below; for details see the electronic supplementry material.

There are two caveats for these data. The original social brain hypothesis and the work that followed sets the cognitive limit for human social networks at around 150. As the nearest data point from Zhou et al. [19] is 132.5, we set this as the upper limit given our focus on cognitive limitations, although a complete analysis including all data points is available in the electronic supplementry material. We also note that the previous studies assumed a smallest ‘social’ unit of the individual N₀ = 1. The gap between N₀ and N₁ = 4.6 appears to exclude long-term pair bonding, e.g. married couples, but an average group size at this layer was not reported in the ethnographic studies we have used. This warrants a closer analysis of our models at this smallest group level. Our results suggest a very small group lying between the individual and the support clique N₁; for details see the electronic supplementry material. Therefore, we assume that there is a point N₀ = y, which can be extrapolated from the model so the primary dataset is for

We emphasize this point by noting that the estimate of δ for the two models was quite different across all data (see the electronic supplementry material). For the hierarchical model (figure 1b), values were typically near 0, whereas for the self-organizing model (figure 1a), the values were between 0.18 and 0.36. This suggests two things: the first is that there is only one organizer (hierarch) for each group in the hierarchical model and the second is that the self-organizing model has a non-zero δ, implying a smallest group size larger than the ego (i.e. N₀ ≠ 1 for i = 0). This can be explained by monadic and dyadic relationships, such as long-term human pair bonding [25,26]. The non-zero δ value strongly suggests that there is a small group at this layer, and as such the self-organizing model provides a prediction of the size of this group.

4. Results

The two models are evaluated using the distance between the corresponding G_i(B, δ) and each element of the empirical data N_i and then numerically estimating the {B*, δ*} parameter pair that minimizes this distance. To do so, the following error functional was used:

4.1

where Φ is the minimized average percentage error per data point. Using equations (2.1) and (4.1) and the empirical N_i, with , and the estimates of are , where is extrapolated. An error of less than 1% is low relative to the other model analysed next and the quality of these estimates can be assessed by finding exact solutions for by allowing it to vary for each i in the minimization; we find: and Φ = 1.42 × 10⁶%. This suggests that closely estimates the best fit found by excluding any error induced by setting to be a fixed constant and that the estimate is consistent with the with no large outliers. For model (2.2): with and the modelled As model (2.1) is a better fit than model (2.2), the notation that follows is simplified:

5. Network size

If larger social networks are formed by aggregating smaller units, then a naive assumption would be that the branching ratio of B ≈ 3 implies that for every individual at level i there are two members at i + 1 that individuals at level i connect with. Owing to pre-existing relationships at level i, this would lead to significant redundancy in the number of links necessary to bind together the three smaller groups. A more efficient approach is to randomly (uniformly) form a link between any two individuals that were not already linked at level i until the average number of links in the larger network has increased sufficiently to bind the smaller groups together into a larger whole. We show next that increasing the links per person by an average of a single link achieves this, validating the relationship

A minimal network model describing social group sizes given by model (2.1) and the desiderata is the Erdős–Rényi (ER) class of random networks [27]. In such a network, the probability that any two individuals are connected in a group of size N with a given average number of links per person is The and N are then associated with equation (2.1): and N ≡ G_i. Substituting G_i = 3.065^i+0.362 into gives the plot of shown in figure 2 (red curve). If is interpreted as a continuous curve for i ≥ 1 – 0.362, then it interpolates between the N_i found using equation (4.1). Each empirical data point in N_i intersects with (unfilled circles for ) and lies halfway between both the nearest network sizes and (horizontal spread) and the nearest link probabilities and (vertical spread), given by the ER model for discrete links It can also be seen that the G_i satisfy the second property of log-linearity and that the expected percentage of connected individuals (shown in the background colour of figure 2 from 0 to 1) is greater than half for all estimates of N_i, thereby satisfying the third property.

Figure 2.

Red curve: continuous parametric interpretation of Open circles: discrete values of The five curves for are the diagonal lines plotted from left () to right (). The vertical blue lines indicate the empirical data points N_i. The background shade indicates the number of nodes that are connected to a single large cluster: 0 = none, 1 = all. An N_i × P_ER = 147 × 2001-sized grid was used; for each point in this grid an ER network was simulated 100 times and the average percentage connectivity reported. (Online version in colour.)

6. Discussion

This article contributes several significant findings to the literature. The first is that we provide a sound mathematical framework that connects cognitive limits with macroscopic social structures using layered ER networks. Kudo and Dunbar [22] have suggested that social link formation is related to social group sizes but this is the first study to show how such discrete links between individuals structurally define the aggregate layering of society. We are also able to describe a specific cognitive process—that of forming multiple one-to-one social bonds—that is quantifiable from large datasets and then to show that typically humans are able to manage four or five such connections. Specifically, in extending from a smaller to a larger social group, this expansion can be identified with the addition of (on average) a single one-to-one link for every individual such that the majority of individuals form a unique core of connected individuals around which the group can then socially cohere. Some of the possible neurobiological or cognitive mechanisms have been reported in the literature [18,28]: pair bonds are mediated by reward mechanisms and oxytocin, support cliques are mediated by perspective taking and sympathy groups are mediated by memory.

We have also contrasted two quite different models of human social organization. The top-down hierarchical branching model is common in larger businesses and military organizations where technology and institutional infrastructure often play an important role in determining the type and number of connections that can form between individuals. In contrast, the ER network model is driven by individuals deciding who to connect with and how strongly in a bottom-up process that induces a complex layering of group sizes. The statistical fit of the ER model is notable: it is considerably better in its error than the top-down model and it predicts a small and intuitively reasonable group size of N₀ = 1.5 that does not appear in the data; see the electronic supplementary material for the full statistical analysis. This smallest social group of 1.5 members is most likely to be composed of singles with no closest connections and pair-bonded couples with only one connection between them.

This work relates directly to the small-scale societies of human foraging populations that are believed to have been largely egalitarian [29]: they lived in small autonomous groups of between a few dozen and a few hundred individuals. In such a setting, complex social interactions mitigate the risk of strict authoritarian social hierarchies emerging [30] that can lead to an unequal distribution of a group's resources. During the Late Pleistocene, all humans lived in such hunter–gatherer societies, this mode of life having emerged at least hundreds of thousands of years earlier and played an important role in human social and cultural development [29,31,32]. Our model of such a social structure is illustrated in figure 1a, in which there is no explicit individual that plays an authoritarian role of managing conflicts that might arise between individuals but instead individuals manage their own relationships. This self-organization at each social level is in contrast to the social hierarchies observed in chimpanzee societies that can have up to four distinct hierarchical ranks of individuals (alpha, high, middle and low); this structure is common for all great apes [33] as well as recent humans living in highly stratified cultures [30]. This hierarchical model is illustrated in figure 1b.

This dynamic egalitarianism in hunter–gatherer cultures is distinct from that of both our nearest genetic relatives as well as more recent, structured human societies. As our societies have grown beyond these small groups, the cognitive demands may have quickly exceeded our individual capacity to manage the larger number of relationships necessary to connect together larger groups. Consequently, the need arises for hierarchical arrangements that do not increase the cognitive demands as group sizes grow. This is the advantage of the dominance hierarchy model and suggests why it might be more effective at connecting larger social groups. It is interesting to note that modern social platforms such as Twitter have not shifted this fundamental limit on group size [34], suggesting that the underlying socio-cognitive mechanism has not changed with technology use.

It is important to note that neither of the social structures considered here takes into account the specific role an individual and their connections plays within a group. At the simplest level, a hunter–gatherer society has both hunters and gatherers and neither of these roles is explicitly distinguished here. Individual roles may also include record keepers and other specialists important for the survival of the group. A more sophisticated example is the role of shared memory within a social group, i.e. the transactive theory of memory [35,36], where memories are dynamically and collectively stored with specific group members. While the data used in this study preclude the possible modelling of these roles in a structured network, it is known that such relationship properties played a key role in the social evolution of humans.

Finally, we also note that, in general, scale invariance indicates a subtle and complex structuring of individual interactions at all levels of a system. In our model, scale invariance is driven by average individual capacities that result in macroscopic structures emerging across multiple discrete levels of a society. It is significant that any specific instance of a group within our layered ER network is not itself scale invariant. Instead, the discrete scale invariance [37] observed here lies in the relationship between successive social layers and as such is quite distinct from the ‘scale-free’ networks observed in studies of much larger numbers of individuals and their social links [38].

Authors' contributions

M.S.H. developed the research question and the software scripts. M.S.H. and M.P. carried out the data analysis, developed the theoretical framework and wrote the article. Both authors gave the final approval for publication.

Competing interests

The authors declare no competing interests.

Funding

We received no funding for this study.