Adaptivity of Social Grooming Strategies caused by Social Closeness

Human beings tend to cooperate with close friends, therefore they have to construct strong social relationships to recieve cooperation from others. Therefore they should have acquired their strategies of social relationship construction through an evolutionary process. The behavior of social relationship construction is know as"social grooming."In this paper, we show that there are four classes including a human-like strategy in evolutionary dynamics of social grooming strategies based on an evolutionary game simulation. Social relationship strengths (as measured by frequency of social grooming) often show a much skewed distribution (a power law distribution). It may be due to time costs constraints on social grooming, because the costs are too large to ignore for having many strong social relationships. Evolution of humans' strategies of construction of social relationships may explain the origin of human intelligence based on a social brain hypothesis. We constructed an individual-based model to explore the evolutionary dynamics of social grooming strategies. The model is based on behavior to win over others by strengthening social relationships with cooperators. The results of evolutionary simulations show the four classes of evolutionary dynamics. The results depend on total resources and the ratio of each cooperator's resource to the number of cooperators. One of the four classes is similar to a human strategy, i.e. the strategies based on the Yule--Simon process of power law.


Introduction
Cooperation is common among humans and it is fundamental to our society [1,2]. The amoun of cooperation by a cooperator is limited because they have to pay costs (e.g., money, time, opportunities, food, etc.) [3,4]. Therefore cooperators choose their partners to interact with inorder to receive cooperation [5][6][7].
Actually, people tend to cooperate with close friends. In a survey based on Facebook, people were only helped by about four close friends despite having about 150 social relationships [8]. An experimental study using the donation game showed that participants tend to cooperate with closer friends [9]. Another study using the public good game showed friend groups are more cooperative with each other than with other groups [10]. Additionally, in a data analysis study using the data set of a social network game, people's frequent communication increases their cooperative behavior [11,12].
Thus, it is important that humans have stronger social relationships in greater numbers with cooperators than others. The behavior for construction of social relationships is called "social grooming". It is widely observed in primates including humans [11][12][13][14][15][16][17][18]. In doing so, they face cognitive constraints [19] (e.g. memory and processing capacity) and time constraints (i.e. time costs) in constructing and maintaining social relationships. These time constraints are not negligible, as people spend a fifth of their day in social grooming [20] for maintaining social relationships [21,22]. Therefore, the mean strength of existing social relationships has a negative correlation with the number of social relationships [23,24].
The skewed distributions of social relationships are generated by a strategy where individuals select social grooming partners in proportion to the strength of their social relationships [18,33]; known as Yule-Simon process [39][40][41]. Individuals should pay time costs to win the competitions with others by the Social Groomers! Cooperative Groomees! Social Groom! Cooperate! Figure 1: Concept of our model. Social groomers interact with cooperative groomees depending on their social grooming strategies. Cooperative groomees cooperate with social groomers who are top R c on the strengths of social relationships. Groomer strategies evolve based on their fitness which is the amount of cooperation from groomees. strength of their social relationships with cooperators, assuming that having strong social relationships with cooperators is to increase cooperation from them.
Humans' societies using these strategies are much larger than non-human primates. Based on the social brain hypothesis, human intelligence has evolved to adapt to large societies. Therefore, evolution of human strategies of social relationship construction may explain the origin of human intelligence.
In this paper, we aim to show adaptivity of the social grooming strategy in order to explore the evolution of human social intelligence based on the social brain hypothesis. Especially, we focus on how environments evolve a humanlike social grooming strategy. For this purpose, we simulated the evolution of the strategy to receive cooperation from others. In the simulation, strategies depend on the strength of social relationships.

Model
Consider two type individuals which are social groomers and cooperative groomees (Fig. 1). Groomers construct their social relationships with groomees depending on their social grooming strategies in a "grooming stage." Cooperative groomees cooperate with groomers depending on social relationship strengths in a "cooperation stage." Groomer strategies evolve based on their fitness which is the amount of cooperation from groomees to them. Groomees' cooperation strategy are static.
In a grooming stage, groomer i repeats to interact with cooperative groomees R g times depending on their social grooming strategy (s i , q i ). q i is a ratio that i constructs a new social relationship with strange groomee j, and s i is a parameter of a probabilistic function p(d ij ; s i ) which determines social grooming partner j with depending on d ij (d ij > 0). The function is following. where where w ij shows strength of social relationships, i.e., the number of social grooming from i to j. M is the number of groomees. b(x; α, β) is a normalized beta distribution is a beta function. Figure 3 shows examples of p(d ij ; s i ).
In a cooperation stage, groomee j cooperate groomers who are top R c on {w 1j , w 2j , . . . , w N j }. R c M shows all resources in the environment (R c , M ), i.e. total amount of cooperation.
Next generations are generated by the Roulette Selection depending on the groomers' fitness. In an each generation, s mutates using Gaussian distribution (µ = 0, σ = 0.2) and q mutates using Gaussian distribution (µ = 0, σ = 0.05), where µ is a mean of the distribution and σ is a standard deviation of the distribution, where q ∈ [0, 1] (if q is out of range by mutation, then it is set the nearest value in [0, 1]). Groomers' s and q in an initial generation are set by Gaussian distribution (µ = 0, σ = 5.0) and by uniform distribution [0, 1].
We conducted evolutionary simulation 30 times on each R c and M by using this model (R c ∈ {5, 10, . . . , 50}, M ∈ {5, 10, . . . 200}). The number of groomers N is 100, the number of social grooming actions R g in each grooming stage is 300 (we also use R g = 100 whose experiments are shown in Supplementary Information (SI)), and the number of generation T is 200. The behavior of class 2 and 3 were roughly like human strategies as described hereinbelow.

Results
As the results of evolutionary simulations, we found four classes on trends of evolution (Fig. 3). These classes are explained by total resources R c M and the ratios of each cooperator's resource to the amount of cooperators R c /M (see also SI Fig. A-2).
Groomers evolved to class 1 when R c M was small. Their s evolved to large and their q evolved to small. This strategy are concentrated investment to strong social relationships (e.g. s = 4 in Fig. 3). Groomers tended to evolve to class 4, where s < 0, when R c M was large. This strategy invests widely to many weak social relationships (e.g. s = −4 in Fig. 3). These classes' s did not converge, i.e. they do not have characteristic values.
On the other hand, s converged to 0 < s < 2 on classes 2 and 3. They evolved in intermediate range between class 1 and 4, and R c /M determined whether groomers evolved to class 2 or 3. Groomers evolved to class 2 when R c /M was large, where q evolved to large. They evolved to class 3 when R c /M was small, where q evolved to small. s of class 2 tend to be larger s than of class 3. The both strategies are diversified investment (e.g. s = 1 and s = 0.5 in Fig. 3), i.e. groomers invest intensively to strong social relationships while they also invest widely to weak social relationships. Additionally, M where groomers evolved to class 2 and 3 is larger when R g is large (see SI Fig. A-2).
Next, we show how the four classes emerged in the evolution of this model and how groomers constructed social structures in each class. For the former, figure 4 shows evolutionary pressures of each combination of s and q, and typical orbits of evolution. For the latter, figure 5 shows strategies of social grooming (a-d) and social structures of each class, i.e. distributions of w (e-h).
As above, class 1 evolved in environments with small R c M . Groomers are   [42]. That is, we calculated the mean difference of s and q of next generations of a population in which individuals' s and q obeyed Gaussian distribution (µ = s, σ = 0.2) (if the distribution generated a value in out of range q then the value was the nearest value in the range) on each cell (s, q). These orbits were drawn based on the average selection pressures and noises which is a normal distribution with µ = 0 an σ = 0.01. Incidentally, there are not cells of (ds, dq) = (0, 0). Evolutionary dynamics in R g = 100 showed a similar trend to this (see SI Fig. A-3).
in intense competition for receiving cooperation from groomees in the environments. Therefore they evolved to concentrated investment to a few poor groomee, i.e. large s and small q ((R c , M ) = (5, 5) on Fig. 4 and Fig. 5a). As the results, they only had very strong social relationships in environments with small R c M (Fig. 5e). As above, class 4 evolved in environments with large R c M . Groomers easily get cooperation from groomees in the environments. Thus, they constructed many weak social relationships with many rich cooperators ((R c , M ) = (50, 200) on Fig. 4 and Fig. 5d and h) As above, classes 2 and 3 evolved in between class 1 environments and class 4 environments. Their s converged to (0, 2), i.e. groomers with these strategies invest intensively to strong social relationships while they also invest widely to weak social relationships ((R c , M ) = (15,45) and (5, 200) on Fig. 4). Their social grooming probability is proportion with each strength of social relationship ( Fig.  5b and c), i.e. their construction processes of social relationships were similar to Yule-Simon process. As the results, their social structures were similar to power law distributions ( Fig. 5f and g).
The main difference between classes 2 and 3 are q which depended on R c /M . When R c /M was small, groomers had to confine the number of social relationships with groomees for having strong social relationships, because they competed intensively in each social relationship (i.e. small R c ).Therefore, they evolved to small q with small R c /M (class 3; (R c , M ) = (5, 200) on Fig. 4). On the other hand, when R c /M was large, they did not have to confine the number of social relationships with groomees, because their competition was not intensive in each social relationship (i.e. large R c ) and the maximum number of their social relationships are small (i.e. small M ). Therefore they evolved to large q with large R c /M (class 2; (R c , M ) = (15, 45) on Fig. 4). Interestingly, these trends of evolution shows non-continuous transition (see SI Fig. A-5).

Discussion
We analyzed evolutionary dynamics of social grooming strategies and social structures. As the results, we showed that the evolutionary dynamics depends on total resources (i.e. R c M ) and the ratios of each cooperator's resource to the number of cooperators (i.e. R c /M ). In a poor small group, individuals' strategies evolved to concentrated investment to strong social relationships. In a rich large group, their strategies evolved to wide investment to many weak social relationships. When medium between the both, their strategies evolved the Yule-Simon process strategy, which invest intensively to strong social relationships while they also invest widely to weak social relationships. As the results of these strategies, power law distributions of social relationship strengths were generated.
There are two class strategies based on the Yule-Simon process. The one evolved in relative rich and small groups. Individuals with this strategy constructed social relationships with all group members, and they reinforce their  Fig. A-4). In the formers, we drew w when the number of samples was more than 20. The figures of class 2 and 3 of the latters (f, g) are shown by using double logarithmic plot. On the social structure of class 1 (e), many weak social relationships were caused by mutation noises of q.
relationships in proportion to strength of social relationships. The another one evolved in relative poor and large groups. Individuals with this strategy constructed social relationships with parts of group members, and they reinforce their relationships. In humans' primitive group, all humans know each other, because a size of their community is about 150 [43] and humans have about 150 social relationships [16,21,[44][45][46]. Hence, humans' social grooming strategy may evolved in the former group. Non-human primates (baboon [36][37][38]) also may have similar strategies, because they also constructs skew social structures while their group size are different with humans. It may caused by a positive correlation between group sizes and amount of social grooming in primates [47,48]. Our experiments showed increasing amount of social grooming R g increased group sizes M in which social grooming strategies evolved to the Yule-Simon process strategy (see SI Fig. A-2). As the results, same social grooming strategies are stable in different group size.
If a social grooming strategy based on the Yule-Simon process is universal in primates including humans and group sizes depend on external factors (e.g., predators, foods, and so on), then the strategies may be evolutionary stable by automatically controlling their amount of social grooming. This relationship between group sizes and strategies may be clear by comparison among humankind, non-human primates, and other social animals.   Average selection pressures (ds, dq) on each (s, q) and and typical orbits from (s, q) = (0, 0.5) of evolution on typical parameters for four classes. Evolutionary dynamics in R g = 300 showed a similar trend to this (see Fig. 4).