Food-sharing and its relevance for the evolution of human cooperation

Human beings are social animals that have built many forms to organize their life in community due to their remarkable ability to coordinate and jointly work to attain common goals. A fingerprint of these behaviors can be tracked in the myriad types of social networks human beings develop, such as family, friends, and communities of interest. Namely, it has been argued that a distinctive attribute of social networks is that they have communities or modules, that is, cohesive groups of agents more densely connected among them than with the rest of the network. This feature may help to explain other special attributes of social networks [1].

One of the key factors for the successful adaptation of the human species to almost every habitat on the planet along its evolutionary history [2], is the unique complexity of its patterns of food sharing, which extends well beyond infancy lactation to the whole life and across adults and families [3]. These patterns may have emerged as a response to a greater offspring dependency associated with the development of a larger brain requiring a nutrient-rich diet, more difficult feeding strategies, and foraging skills that may not develop until late in life, in contrast to other wild primates having more predictable diets [3–5]. The large human brains may have had a central role both to the formation of modular social networks enabling adaptive collective computation, to the optimization of foraging behaviors [6]. These behaviors ensuring the most elemental subsistence may have given the material basis for inter-generational accumulation of cultural innovations [7], and the whole range of social interactions carried out by hunter-gatherer bands, such as cooperative breeding [8, 9], establishing political alliances and reinforcing social bonds [10–13], labor sharing [14, 15] and costly signaling or show-off for mating purposes [16, 17]. Therefore, investigating the evolutionary formation of food-sharing networks may yield important insights for the study of social dynamics. With this in mind, we aim to theoretically contrast the hypothesis that the modular character of food-sharing networks, may be an optimal response for avoiding the risk of individual and collective starvation.

Characteristics and drivers of food-sharing networks

It has been reported that empirical food-sharing networks present the same traits of other general social networks such as degree assortativity [18], or positive correlation on neighbors degree, transitivity or global clustering [19], the fact that agent i is likely connected to k if i is connected to j and j to k. These two network characteristics may be an indication of communities [1]. Other such characteristics are reciprocity [20], which is a significant proportion of agent pairs with bidirectional connections, and a multilevel structure [21] of cohesive groups hierarchically organized. Among the few theories aiming to explain some of these network patterns, it has been argued that the multilevel structure would be related to the efficiency of information flow [22] which may in turn, facilitate the diffusion of cultural innovations [23]. We think that, though the efficiency information flow may be an important force shaping social networks, it does not fully serve as an explanation for the evolution of food-sharing networks. On the one hand, it does not have a clear connection to the usual motives for food-sharing discussed in the evolutionary anthropology literature [3]. On the other hand, the theory of cultural innovation may impose high cognitive requirements [24] which do not explain, for example, why these multilevel networks may be observed also in other social animals and non-mammals [25]. Another work [21], though it is not a theory of multilevel structure, gives suggestive evidence relating three nested network levels, the household, the cluster, and the camp, to distinct social and economic cooperative functions: intersexual provisioning of a couple and their dependent children in a household, kin provisioning, and risk reduction reciprocity, respectively.

The motives for food-sharing most supported by empirical field studies are kinship, need, reciprocity, and inter-household distance, where the latter is usually considered a proxy of the tolerated theft motive [20, 26–32]. The costly signaling hypothesis, which claims that food-sharing is made in order to gain status, has received minor support in recent works as a factor for food transfers, though status is an important driver of reproductive fitness [33–35]. Probably the most accepted causal interplay between these motives is the following. Reciprocal altruism, or contingency on past transfers, arises as an important, or the strongest, predictor in most studies, usually explained as a means of lowering the uncertainty of food generation [10, 20, 21, 26–32]. Kinship, or the preference for relatives, is another important driver of food transfers [10, 20, 21, 26, 27, 29, 31, 32], which appears to be as a bias, or partner-selection mechanism, interacting with reciprocity or need by increasing the associations to transfers with respect to non-kin [20, 26, 27, 31]. Tolerated theft, or sharing when the benefits of hoarding are outweighed by the cost of retaining the resource, is often a less important but significant motive of transfers [29, 36], though it may be empirically difficult to disentangle from the alternative explanations [10, 29, 36]. Some works have suggested that tolerated theft may be an evolutionary precursor of reciprocal altruism, given that the former is more common among other animals, and the latter has higher cognitive requirements that makes it almost nonexistent in non-human animals [4, 37]. Finally, the motive of need, or giving to those with a lower relative net energy production, often reflected in a positive correlation of transfers with receiving household size, is also an important force guiding food transfers [20, 21, 27, 31, 32].

The issue with need motive and a possible solution

The motive of need for food transfers has a difference with the other reviewed motives: it is not evolutionary adaptative, at least in the usual individual fitness models [38]. These models assert that cooperation requires to be assortative to be stable, that is, directed to other cooperative individuals [39]. This prediction has only partial support in real-world food sharing, where it is common that families or foragers in relative need like the elderly, tend to receive more resources [21, 31, 32] or that free-riders, those who take shares, but do not reciprocate, are not excluded from sharing [40]. A possible theoretical remedy that has been suggested [32] is to consider a diminishing returns value function of the transfers, which may produce a long-term equilibrium in transfers. In a previous publication [41], it was argued that assuming reciprocal altruism, sharing is an optimal individual strategy for values following a diminishing returns function of the shares, rather than just the raw quantity of food.

We propose a distinct intuition guiding the abstraction of our model. We propose that the argument for using diminishing returns comes from the fact that the essential problem a hunter-gatherer community is solving when sharing food, is to satisfy the need given by hunger. This is the intuition of the law of diminishing marginal utility given by Mill [42], stating that if there is enough of a good to satisfy a basic level of need, then each additional increment of the good satisfies the need less than the previous increment. The basic social dilemma for the food-sharing network then comes from the conflict between the goal of satisfying the own need and the need of other people. If the satisfaction of the hunger need is defined as the goal of minimizing the risk of starvation, or the time without eating, then the own need may be embedded in the tolerated theft or reciprocal altruism motives, while the hunger of other people may be represented by the need motive. We do not explicitly incorporate kinship in our model in order to keep it simple. It has been suggested that many kinship configurations may be a product of sharing patterns [12, 43, 44], which suggests that the basic social dilemma posed is enough to recover the basic dynamics of food-sharing networks, such as cohesive groups given by nuclear families with a few interconnections that comprise a cluster of akin families [21].

Our proposed model for food-sharing networks, its assumptions, and contributions

In this work, we propose a normative model [45] for food-sharing networks, formulated as a network optimization. That is, the model does not aim at fitting real networks, but rather derives idealized results from a set of simplified assumptions. The optimizing goals correspond to the minimization of starvation risk, either at the individual level or as a group. These goals are named in the model as Reduction of Variability (RV) and Welfare (WEF). The assumptions taken by the model are the following.

• An individual who has food eats only the amount required to survive, and shares all the remaining food. This is inspired by the empirical finding that giving a high proportion of food is a common practice [41].
• The structural imbalance between production and consumption leading to adult sharing to provide infants and juveniles [3], is implemented by a special set of network nodes called hunters, which are those who inject food into the network. The number of hunters in a network of N agents corresponds to the parameter nh.
• Since large and unpredictable food items such as meat or honey are more widely shared [4, 40, 41, 46], the model poses that at each time step, there is an amount F of food able to feed several individuals that is produced with certain probability ph. These quantities, F and ph, are parameters of the model.
• Any additional motives for sharing, like kinship, which would introduce, for example, preferential sharing partners, are abstracted. Therefore, given a food-sharing network, and an agent that receives food, this agent chooses randomly one of their neighbors to share. Thus, the transfers of food in the model correspond to recurring exchanges among stable connections, that may well reflect the food sharing in a small community formed by a few akin families. This assumption is also consistent to the choice taken for sampling the population size parameter used in the simulation. Technologies for storing food are not considered in this model. These assumptions are taken to keep the model simple.

The model aims to recover the constraints for optimal network organization derived from material transfers of food under the stated assumptions. This emphasis in the material dimension of food-sharing, does not look for neglecting that this behavior is a complex sociocultural phenomenon [47]. Rather, the model shows that evolutionary models of social dynamics may benefit from a resource distribution perspective and the explicit inclusion of the group level of analysis, given that the typical small-world structure of social networks given by the presence of communities, is recovered under the simultaneous optimization of the two goals, that is, the individual but also the group survival. Specifically, the contributions of our work are the following.

First, networks optimizing both criteria may exhibit a community structure of cohesive groups under stringent conditions of food supply. These networks are resemblant to the food-sharing networks observed by Dyble et al (2016) [21], where there are cohesive groups corresponding to households provisioned by an adult couple, and a set of households forms a cluster with a small number of inter-household connections. This model result suggests that this organization of nuclear families may have been evolutionarily functional for resource distribution optimality. We have additionally obtained that optimal welfare networks, where each hunter is connected to one big, homogeneous and dense group of non-hunters, resemble social networks that promote more egalitarian income distribution in a lab gift game reported by Chiang (2015) [48]. This model result is consistent with the view that social networks structured originally according to ecological considerations, may have created the environment in which prosocial tendencies and equity response elicitation evolved [49, 50]. Finally, distinct distributions of reciprocity are obtained for each optimization regime, which may be consistent with the empirical finding [51] that often the sharing occurs in waves: first among hunters, and then hunters with their families. This model result gives a broader picture of the usual notion that reciprocity is driven by the minimization of food production uncertainty [28].

Paper structure

The rest of the paper is structured as follows. In the next section, “Materials and methods”, all the modeling assumptions and analysis techniques are described at a high level. Several details are set aside for the “Supporting information” section at the end of the article. After the methods section, there is a section of results where a description is given regarding the structural behaviors observed in the three types of optimal networks analyzed, WEF, RV, and simultaneous optimization of both criteria. The interpretation of these results is addressed next in the Discussion Section. We end the article with the conclusions.

Food-sharing protocol and assumptions

We propose a network optimization model in which each agent is a node in a loop-free directed network D = (V, E). There is a special set H ⊆ V of agents in D that we call hunters. Agents in H are, intuitively, responsible for injecting food in the network by hunting preys. We assume that there is a real number ph ∈ [0,1], a parameter of the model, that determines the probability that an agent in H hunts at each time step. The prey obtained in every hunt is enough to feed F agents (including the hunter). The value F is a fixed positive integer that is also a parameter of the model. The food resulting from a hunt is shared from the hunters to the rest of the network, only through network edges e ∈ E, and is used to feed the agents as we next explain. These assumptions implement the following two empirical facts: food items more widely shared are those unpredictable and coming in large packages [4, 40, 41, 46], and the fact that all agents require eating for survival, but only some of them can produce food [3].

We assume that all the exchanges of food derived from one successful hunt, occur at the same time step where the prey is caught. When an agent v ∈ V receives f units of food from a neighbor, it consumes a single unit of food. After this, agent v chooses uniformly at random one of its neighbors, say v′, and sends all the remaining food (f − 1 units) to v′. This assumption is inspired by the empirical finding that giving a high proportion of food is a common practice [41]. Notice that this sharing from v to v′ is possible only if v initially received f > 1 units of food. Additional food received in the same time step, cannot be mixed with other food packages, nor stored for later usage. For simplicity, we assume that a hunt by a hunter is equivalent to receiving F units of food, though in this case the food is not received from a neighbor but from the environment. Thus, after feeding itself, a hunter sends F − 1 units of food to a random neighbor. These assumptions are taken to keep the model simple and to abstract the possible motives for agents’ actions.

In our model, time is discrete, and at each time step, every hunter may or may not hunt with the same probability ph. Due to previous assumptions, those agents at distance greater than (F − 1) to every hunter, simply cannot receive food. Thus, every node v ∈ V in the network at a distance at most (F − 1) to some hunter, may or may not eat depending on whether v receives food, either by hunting or by sharing. In order to model the effect of feeding an agent, we assume that agents have a life span of , and that there is a critical time window k < n, in which, a run of k consecutive time steps without receiving food turns out to be seriously detrimental for every agent, resulting in its death. Only non-dead agents can hunt, eat, send or receive food. Therefore, the agents will try to eat as frequently as possible, to avoid spending too many rounds without eating.

To sum up, the parameters of our model are the probability of hunting in a time step (ph), the units of food produced by a prey (F), the number of time steps of life span (n), and the maximum number of time steps an agent can survive without eating (k). Moreover, a network in our model is defined by D = (V, E) plus a set H ⊆ V of hunters. We note that, even when fixing the parameters of the model, different networks behave in distinct ways depending on their connectivity (and the presence of hunters).

Probability of eating

In this section, we give some intuition about how the probabilities of eating behave according to the previously described protocol of food sharing. The interested reader may look at S1 Appendix of S1 File in order to see the analytical derivation of these probabilities. For now, let us see an example. Fig 1 shows a network with 2 hunters, 0 and 4. Assume that ph = 0.2 and obtain the probabilities of eating according to (4) (S1 Appendix of S1 File) for distinct values of F, displayed in Table 1. For F = 2, 5 is surely fed by 4 with the probability of hunting ph, while 1 and 3 get a distributed chance of receiving food. In the case of F = 3, now 6 is also fed with probability 0.2, and there are two 2–walks from 0: {(0, 1), (1, 2)} and {(0, 3), (3, 1)}. Since 1 is fed by the 2 equiprobable walks, its probability of eating is 2 * 0.1. When F = 4, 1 gets food by another walk, raising this probability to 0.36 = 0.2 + 0.2 − (0.2)² thanks to Inclusion-Exclusion. Finally, notice in F = 5 that, though node 3 is reached twice by one walk, this does not increase the probability of eating since this probability does not measure the quantity of food, but rather the distinct events of food supply.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. A network with 7 nodes and 2 (yellow) hunters.

https://doi.org/10.1371/journal.pone.0272733.g001

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Probabilities of eating, distinct F values.

https://doi.org/10.1371/journal.pone.0272733.t001

Optimizing criteria

We have said our model may have two possible goals, welfare, the egalitarian group access to food, and avoidance of individual starvation. Since our model implements food exchanges as a probabilistic protocol, the access to food resources is expressed as the probability of eating. Our model is framed as the minimization of an objective function, thus, we aim to penalize the dispersion of this probability among agents. Next, we formalize the two goals as minimization objectives.

Definition 0.1 (Welfare). Given fixed values for variables ph, F, n, k and nh = |H|, Welfare (WEF) criterion for minimization is the standard deviation of probabilities of eating over network nodes.

Now, the minimization of individual starvation risk is formalized as follows. We can think of the time steps in our model, and the possibility of eating at each time as a sequence of Bernouilli trials, where a success is given by the event of not receiving food, otherwise we have a failure. That is, at each time step, node v obtains a success with probability 1 − pe(v), where pe(v) is computed as explained in the previous sections. Since network nodes attempt to avoid k consecutive steps without food, or a success run of length k, in order to stay alive, a possible formulation of reduction of starvation risk can be expressed as the minimization of the number G_n,k of success runs of length greater than or equal to k in n trials. The distribution and moments of this random variable can be computed exactly with a Markov chain embedding technique [52], which allows defining this criterion as the minimization of the sum of the expected value and standard deviation of G_n,k.

The random variable G_n,k defined above is expressed as an integer number of runs. One can normalize this variable G_n,k by the total maximum number of runs of length k that may be observed in n trials. In this way, this criterion could be interpreted as the probability of dying within the life span -or observing a k–length (or greater) run with no food in n trials- as a function of the probability of receiving food at each trial. Since the number of runs of length at least k within n trials is maximized when all the runs have the minimum length k, the normalizing denominator may be estimated as the floor of , where the “+1” appears to count the spaces necessary to split a new run from the previous one. With these ingredients, we formally define the second minimization criterion.

Definition 0.2 (Reduction of variability function). Given fixed values for variables n, k, and 1 − pe(v) for the probability of a success, let G_n,k be the random variable given by the number of success runs of length greater than or equal to k in n trials. We will refer to the expected value and the standard deviation of this variable as and σ(G_n,k(pe(v))). The Reduction of Variability of food intake (RV) function of the probability of eating, is defined as .

Definition 0.3 (Individual starvation risk criterion). Given fixed values for variables ph, F, n, k and nh = |H|, the Individual starvation risk criterion for minimization is the average over network nodes of the RV function, evaluated at the respective probability of eating of each node. For brevity, and without causing ambiguity (since the function is for nodes, and the criterion for networks), we refer also to this criterion as RV.

We have taken the average to approximate the individual agent risk since this function is not a robust statistic of central tendency [53], and may be extremely affected by the presence of outliers. That is, the mean of RV over network nodes may be minimized by a few number of agents with low RV, while a great number still experiment high starvation risk. Now, in Fig 2 we can observe distinct instances of the RV function, for different values of k and n = 1000. These RV curves have a maximum value corresponding to some value pe· ∈ (0, 0.5), and the larger the length k of the run, the smaller the probability of eating pe· at which this maximum is obtained, which makes intuitive sense. In the following, we will continue working with the values k = 10 and n = 1000, since from inspection of this plot, this function displays all relevant behaviors within a non-degenerate probability domain. Therefore, the resting free variables, with which we will work are F, ph and nh = |H|.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Reduction of variability function of the probability of eating.

The X-axis is the probability of eating or receiving food, as defined in the last sections. The vertical axis is the Reduction of variability (RV) function of this probability. Each curve corresponds to a distinct instance of the variables n, the number of life span time steps, and k, the maximum number of time steps an agent can survive without eating.

https://doi.org/10.1371/journal.pone.0272733.g002

Model solution

Pipeline of analysis

The outcome of the evolutionary algorithms described previously is a set of networks, for every sampled setting of parameters F, ph, and nh = |H|. In the S2 Appendix of S1 File, the structure of the simulation carried out is explained, which comprises the choice of sampled parameters. Now, to obtain a high-level description of the set of different optima networks, we processed the data according to the following steps. For each network, a set of features is computed to build a table of networks and features, for either single criteria or multiobjective case. See Tables 1, 2 in the S3 Appendix of S1 File to consult the sizes of these datasets. These networks are then clustered by their feature similarity, where each group depicts a set of qualitatively similar networks. Finally, a decision tree to discriminate distinct groups is computed, which permits an objective split of groups in terms of the feature values. Next, we review in more detail each of these steps.

Welfare optima

Summary of distinct types of WEF optima.

It is instructive to start by inspecting the following tree of WEF optima in Fig 3, belonging to the accuracy-stability-Pareto optimal set, which has an average accuracy of 0.872 to discriminate the 7 clustering classes. In these trees, the loss corresponds to the Gini coefficient of samples at that node, the predicted class is the class with a maximum number of samples, and proportion is the relative proportion of node samples with respect to the total in the root. This total and the other statistics present in the tree are computed on the training set. The accuracy, on the other hand, is computed on the test set, which is the complement of the training set with respect to the whole WEF dataset of size 18686 (see Table 1 in S3 Appendix of S1 File).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. An efficient tree (accuracy = 0.872) to discriminate the clustering labels of Welfare optima.

Statistics in the tree are computed on the training set, while average accuracy is computed in the test set. See S3 Appendix of S1 File for the sizes of the datasets used, and Paragraph Description of clusters by classification trees for the general procedure of tree construction. See the first paragraph from Section Welfare optima for an explanation of the variables displayed in tree nodes.

https://doi.org/10.1371/journal.pone.0272733.g003

The first remark is the presence of the condition F * nh ≤ 13.5 at root, which turns out to be very stable since in each of the 9 distinct trees found at the Pareto efficient tree set, the structure formed by internal nodes of IDs 0, 1, 3 is present. The threshold 13.5 is an artifact from averaging the two most proximal F * nh values, 12 and 15, and since F * nh is a product of two integer variables, the condition F * nh ≤ 13.5 can be equivalently replaced by F * nh ≤ 12 = N. A possible topological marker of model variable F * nh, which reflects the relative food abundance, is the mean in-degree of hunters, a variable with which there is a significant Spearman correlation of 0.81. Another possible marker is the number of strongly connected components (SCC), variable with which the correlation, in this case, is −0.574. These relationships are due to the fact that optimal WEF networks tend to form a densely connected group of non-hunters that is fed by hunters, except if there is enough food, in whose case there is only 1 strongly connected component where hunters are also fed back. An important exception to this trend in SCC is shown by networks in node 16, where there are only networks with ph = 0.6 and a mean of 3.796 SCC. That is, when there is enough food with a high probability of hunting, the network is segmented into distinct SCC’s that provide an egalitarian probability of eating. These structures are shown in the representative networks on Fig 4.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Most central networks of nodes 2 (left), 13 (center) and 16 (right) from tree in Fig 3.

Hunters are filled in yellow. Each network specifies the values of model parameters used as input, and for which the respective network is a local minimum.

https://doi.org/10.1371/journal.pone.0272733.g004

General statistics and modularity on WEF optima.

WEF optima are networks of high clustering, mean out-degree, moderate reciprocity, and low assortativity and modularity, as shown in Table 2, which speaks of one large community with low segmentation. The WEF optimal networks with the largest modularity values, obtain this structure from the special feature of minimizing the standard deviation of in-degree. An example of these networks is in Fig 5 from node 8, having a modularity of 0.305 for the partition {{0, 2, 6, 7, 9, 10}, {1, 3, 4, 5, 8, 11}}.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. A WEF optima minimizing in-degree variability.

Obtained under variables: F = 4, nh = 2, ph = 0.6. Hunters filled in yellow.

https://doi.org/10.1371/journal.pone.0272733.g005

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. WEF optima general statistics.

https://doi.org/10.1371/journal.pone.0272733.t002

Reduction of variability optima

Relation between RV and WEF costs.

We start this section by analyzing the structure of the relation between costs RV and WEF, and derive from it a profile of the distinct networks obtained as RV local optima. We have mentioned in S2 Appendix of S1 File the intuition regarding the expected behavior of RV optima from RV function. That is, for values of ph smaller than pe⋆, the probability producing the RV maximum, there is an every man for himself regime, or free-riding situation where non-hunters just do not receive food and are isolated nodes. Now, for ph > pe⋆, there is progressive cooperation, an increasing trend of feeding non-hunters, particularly when RV reaches its range of flat values. On the other hand, depending on the values of model variables, hunters often share food in order to decrease the RV mean. We can visualize this pattern in Fig 6, where we graph the mean correlation values between RV and WEF for randomly sampled networks and distinct ph values.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Correlation of costs in function of the probability of hunting.

Each curve represents the mean of Spearman correlation between RV and WEF for a specific configuration of F and nh, and a range of 25 equidistant ph values. Each point is the mean of 100 correlation points, each one computed for a uniform random sample of 100 networks. Each mean point is surrounded by the percentile confidence interval using the standard error of the mean with error α = 0.05, that is, assuming the sample mean is normally distributed.

https://doi.org/10.1371/journal.pone.0272733.g006

In this graph, it can be seen that these mean correlation values start approximately at −1 for low ph values, which corresponds to the free-riding regime, and in the extent ph increases the correlation curves tend to stabilize around values near to perfect correlation. Observe that most curves become increasing functions only for values of ph greater than pe⋆ = 0.0899. The fact that greater values of the product F · nh have a faster convergence to cooperation may be explained, if one assumes a uniform feeding probability, from Equation (7) (S6 Appendix of S1 File), where a greater F · nh implies greater mean probability of eating, which makes these probabilities closer to the flat RV region.

Distribution of arc types and representative RV optimal networks.

This pattern of costs behavior is reflected also in the distribution of distinct types of arcs. If we separate arcs in three groups depending on whether they connect, without arc direction: two hunters, a hunter and a non-hunter, and two non-hunters, we obtain the following distribution of the number of arcs depicted in Fig 7. It can be seen in the figure that, in the two smallest ph values, non-hunters are isolated in RV optima, which is part of the so-called every-man-for-himself phase. If ph increases, the relative numbers of arcs gradually converge to the same ordering displayed by WEF optima, which occurs for lower values of ph if F · nh > 12. This pattern is consistent to what is displayed in Fig 6.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Distributions of arc type for distinct optima and food supply condition.

The three types of arc graphed (which do not consider arc direction) are: connecting two hunters (intra_h), connecting two non-hunters (intra_nonh), and connecting a hunter with a non-hunter (inter_h_nonh). Each number of arc type average is computed for every sampled ph value, and surrounded by a percentile bootstrap confidence interval of error α = 0.05 and 1000 bootstrap samples of 70% of dataset. See the paragraph Summary of distinct types of WEF optima from Section Welfare optima, for the result justifying the use of condition F · nh ≤ N to aggregate datasets.

https://doi.org/10.1371/journal.pone.0272733.g007

The distinctive profiles of the number of arcs from these graphs may be used to identify representative networks obtained as RV optima. In Fig 8, a decision tree that discriminates the classes of RV optima with an accuracy of 0.907 is displayed, together with representative networks from some of its leaves in Fig 9. In the left part of the tree, there are the networks with a lesser number of strongly connected components, which are networks with greater values of F · nh. The networks from Fig 9(a)–9(c), show the prototypical behaviors shown in Fig 7(c), which are, respectively, the absence of connectivity for non-hunters if ph is low, a network with WEF-like topology of minimization of in-degree variability if ph ≥ 0.3, and a transition between the two regimes similar to the last one, but where hunters have more intra-connectivity rather than inter-connectivity.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. An efficient tree (accuracy = 0.907) to discriminate the clustering labels of RV optima.

Statistics in the tree are computed on the training set, while average accuracy is computed in the test set. See S3 Appendix of S1 File for the sizes of the datasets used, and Paragraph Description of clusters by classification trees for the general procedure of tree construction. See the first paragraph from Section Welfare optima for an explanation of the variables displayed in tree nodes.

https://doi.org/10.1371/journal.pone.0272733.g008

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. Example networks from distinct DT leaves of the tree in Fig 8.

From left to right, and from top to bottom: (a) Leaf 3 (most central), (b) Leaf 4, (c) Leaf 4, (d) Leaf 15, (e) Leaf 17, (f) Leaf 7 (most central), (g) Leaf 19 (most central) and (h) Leaf 17. Each subcaption displays values of the condition under which these optima were obtained. Hunters filled in yellow.

https://doi.org/10.1371/journal.pone.0272733.g009

Now, some of the patterns from Fig 7 (a) are displayed in Fig 9(d)–9(f). In the first of them, there is a network from Leaf 15, where networks have arcs mainly among hunters, and may present some few connections to non-hunters. The networks of this kind are among those in RV optima having the greatest proportions of reciprocated arc pairs, usually only between hunters. This is used as a strategy to lower the high RV values in this range of ph values (ph = 0.15). Then, in Fig 9(e), there is more connectivity to non-hunters, but it is still comparable to that between hunters. Finally, Fig 9(f) displays a network with the cooperative regime on ph = 0.6 where intra-non-hunters connectivity surpasses the number of connections to hunters, and these display a minimal intra-connectivity. The case when ph = 0.6 and there is just one hunter, as in Fig 9(g), is special. These networks present a one-way walk with no forks, since a fork lowers the probability of eating with respect to the no-fork situation, increasing the RV cost. If there are isolated nodes, as is observed in most optima obtained from an evolutionary optimization, there is a big difference in the probability of eating between connected and isolated nodes, which increases WEF cost. This affects the Spearman correlation between ph and WEF in the whole dataset, which is 0.313. If we restrict to those data where nh > 3, this correlation lowers to −0.319, and to −0.73 if nh > 6, consistent with Fig 6.

Modularity on RV optima.

We close this section by remarking on the behavior of modularity on this type of optima. In general, RV optima are networks of low modularity, see Table 3 for its general statistics, but in fact, high modularity may be observed. This high modularity is a by-product of survival or network arrangements for enhancement of probabilities of eating, as can be pointed from its correlation with F · nh which is −0.359. However, the relationship is stronger for higher ph values, since restricting to ph > 0.2 gives a correlation of −0.474. This higher modularity is usually more related to a segmentation in the network than to the presence of cohesive groups of nodes. An example of this is the same network from Fig 9(g), which has a modularity of 0.208 for the partition {{0, 1, 2, 3, 5, 7, 8, 11}, {4, 6, 9, 10}}. Another example is that of Fig 9(h) having a modularity of 0.32 for partition {{0, 6, 8, 9}, {1, 10, 11}, {2, 3, 4, 5, 7}}.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Modularity general statistics on RV optima.

https://doi.org/10.1371/journal.pone.0272733.t003

Multi-objective optima

Community structure on PF optimal networks.

One of the main results is displayed in Fig 10, where there are two measures of community structure, the already mentioned network modularity, and a new measure called average intra-partition clustering. This measure corresponds to the average of local clustering over the subnetworks induced by each of the modules of the partition maximizing modularity found by the Monte Carlo algorithm. Thus, intra-partition clustering is a measure of the cohesiveness of the partition maximizing modularity. A first remark to note is that, though maximum values of this measure are obtained by WEF optima, this pattern is more a product of the high connectivity across the whole network found in this type of optima, than to the presence of clear cohesive modules. This observation is supported by the low values of modularity exhibited by WEF optima, see Fig 10(c), and is coherent with the hypothesis that the partitioning found in this optima obeys more to the random nature of the Monte Carlo algorithm.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 10. Distributions of different measures of community structure for each type of network optima.

From left to right: Average intra-partition clustering (left), percentile 90 of average intra-partition clustering. (center), and mean modularity (right), for each type of network optima. Average intra-partition clustering (AIC) is the mean of local clustering computed in each subnetwork induced by the modules of the partition maximizing modularity. These statistics are computed on each of the 500 bootstrap samples with replacement of size 70% from dataset, used to compute percentile confidence intervals with error α = 0.05 for each statistic.

https://doi.org/10.1371/journal.pone.0272733.g010

Now, if one inspects the mean values of average intra-partition clustering (AIC) of PF optima, one notes that they are not too distinct from those of RV (Fig 10(a)). As an example, for the observed PF networks with ph ≤ 0.15, the mean is 0.148 and the median is 0.056, and one can find typical networks such as those in Fig 11(a)–11(c), with AIC values of 0.083, 0.313 and 0.023, respectively. However, if one looks at the highest values of this measure in Fig 10(b), it is noted PF gets higher values than RV, for ph ≤ 0.15. This pattern is due to networks such as that of Fig 12(a), where there are two modules of connected nodes around distinct hunters, which is reflected in a value of 0.454 for intra-partition clustering. On the other hand, if community structure is measured with modularity index, it can be seen in Fig 10(c) that PF mean increases with ph, and it is higher than that of single optima networks. This behavior is influenced mainly by networks that indeed can exhibit relatively large modularity, but with weakly connected modules, similar to the architecture of some RV optima. Fig 12(b) displays an example of this type of PF optima.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 11. Representative PF networks of regime ph ≤ 0.15.

Model variables under which these Pareto optimal networks were obtained are displayed under each network. Hunters are filled in yellow. To obtain representative networks, these were chosen as the most central from leaves of efficient trees with a relative proportion higher than 8%. To see the trees, consult the S8 Appendix of S1 File.

https://doi.org/10.1371/journal.pone.0272733.g011

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 12. Representative PF networks with some social network pattern.

Model variables under which these Pareto optimal networks were obtained are displayed under each network. Hunters are filled in yellow. Average intra-partition clustering (AIC) is the mean of local clustering computed in each subnetwork induced by the modules of the partition maximizing modularity. From left to right: network (a) has a modularity of 0.392 and AIC of 0.454, network (b) has a modularity of 0.221 and AIC of 0, while network (c) has a modularity of 0 and AIC of 0.029. Additionally, this last network has 7 reciprocated arc pairs between hunters and non-hunters.

https://doi.org/10.1371/journal.pone.0272733.g012

Reciprocity on PF optimal networks.

The behavior of the other social network pattern we study, reciprocity, can be appreciated in Fig 13. There it can be seen that, particularly, the presence of reciprocated arcs between a hunter and non-hunters is very salient on PF networks compared to the other optima. This pattern is observed in networks such as that of Fig 12(c). On the other hand, the presence of reciprocity among non-hunters in Fig 13(b), is associated with networks similar to that of Fig 11(b), which for example has 5 reciprocated arcs among non-hunters. The intuition for the presence of reciprocated arcs between a hunter and non-hunters, as well as other patterns found on PF optima, is given by the notion of dominance, which defines the Pareto optimal set, and will be explained next through the example of Fig 14.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 13. Distributions of different types of reciprocated arc pairs for each type of network optima.

From left to right: Mean number of reciprocated pairs between a hunter and a non-hunter (left), mean number of reciprocated pairs between non-hunters (center), and mean number of reciprocated pairs between only hunters (right). These means are computed on each of the 500 bootstrap samples with replacement of size 70% from dataset, which are used to compute percentile bootstrap confidence intervals with error α = 0.05 for each statistic.

https://doi.org/10.1371/journal.pone.0272733.g013

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 14. Three networks with increasing number of reciprocated arcs between hunter and non-hunters.

We assume there is 1 hunter, node 0, ph = 0.08 and F = 4. Network D₁ gets costs RV = 0.261 and WEF = 0.024, D₂ obtains RV = 0.315, WEF = 0.02 and D₃, RV = 0.332, WEF = 0.018.

https://doi.org/10.1371/journal.pone.0272733.g014

Probability of eating.

As a final observation, we look at the behavior of the probability of eating in a network. The mean probability (over network nodes) is on Fig 15(a), and it turns out that PF obtains lower means over dataset samples. This is due to the high prevalence of networks with one hunter and reciprocated arcs between hunter and non-hunters, which usually get low mean probabilities, due to low food availability. If now we look at the median probability on Fig 15(b), it is observed that PF offers a greater probability than RV on low ph values. This is because in PF there are networks similar to WEF minima that deliver better probabilities of eating that do not leave isolated nodes as with the RV optima. This is confirmed by Fig 15(c), which shows the greatest differences between hunters and non-hunters are in the RV optima, and by the remarkable fact in Fig 15(b) that WEF optima shows the greatest efficiency in the probability of eating in the whole range of ph.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 15. Display of three variables associated with the probability of eating for each type of network optima.

From left to right: mean probability of eating, median probability of eating, and difference in means of the probability of eating between hunters and non-hunters. In each graph, the mean of each statistic is drawn surrounded by percentile bootstrap confidence intervals with error α = 0.05, computed on each of the 500 bootstrap samples with replacement of size 70% from the dataset.

https://doi.org/10.1371/journal.pone.0272733.g015

Condition for community network structure

Communities given by cohesive network modules appear in food sharing networks optimized to provide egalitarian access to food resources jointly with reduced individual risk. This result is obtained under conditions of uncertainty and scarcity in the food supply (see Figs 10(b) and 12), which suggests their adaptive value for this context. The simultaneous optimization of the two goals is necessary since RV optima may present community structure but with rather sparse connectivity (see Fig 9), while WEF may present communities under the special case of degree dispersion minimization (see Fig 5). These PF networks are resemblant to the food-sharing networks observed by Dyble et al (2016) [21], where there are cohesive groups corresponding to households provisioned by an adult couple, and a set of households forms a cluster with a small number of inter-household connections. In our model, the cohesive groups correspond to consumers, or non-hunters, grouped around one or more hunters. This result suggests that not only the camp level, which groups several clusters, is functionally associated with risk reduction reciprocity as argued by Dyble et al (2016), but also the hierarchically lower levels of households and clusters may have been related evolutionarily to some resource distribution optimality that includes the egalitarian access to resources. This idea is consistent with some studies suggesting that many kinship configurations may be a product of sharing patterns [12, 43, 44].

The claim that the two minimization criteria are required for the emergence of cohesive communities is reinforced by the condition of uncertain food generation, not only because it is under this regime where communities are obtained (Fig 12), but also because when food supply is safer, the two criteria become indistinct to the dense optimal WEF networks with low modularity (Fig 9(f)). The fact that sharing is directed mainly towards large and uncertain packages of food is acknowledged by several empirical studies [4, 40, 41, 46]. The necessity of the confluence of both the individual and group interests has also been pointed out in the simulation agent-based study by Briz i Godino et al (2014) [72] which models the cooperative public calls through smoke signals of a hunter-fisher-gatherer society conditional on the exceptional accumulation of fish. The model shows that a low weight on social capital relative to individual consumption of meat in the agents’ fitness, is enough to obtain most society cooperating as the most likely outcome, but this result may be reversed when this weight is absent. This social capital component of reproductive fitness seems to be a plausible mechanism according to Chaudhary et al (2016) [73], where it is reported that relational wealth, or the cooperative relationships an individual has, provides advantages in buffering food risk, body mass index, fertility, and is partially heritable.

Finally, the result that a stringent resource availability would be necessary for the evolution of cooperation is supported by another agent-based model named Cooperation under resource pressure (CURP) by Pereda et al (2017) [74]. This work shows that under moderate survival stress, populations of agents self-organize in an indirect reciprocity system consisting in sharing the part of the resource that is not strictly necessary for survival, achieving to collectively lowering the chances of starving. Additionally, a high-stress regime turns into unstable behavior, where the population constantly searches for survival strategies, while low-stress does not exert selection and strategies remain almost constant and randomly drifting. Since the variables in our model may be understood as a form of resource-pressure, and even a parallel may be established between the two models’ variables (prob-resource to ph, and min-energy to F), the results of our model roughly resemble those of CURP, taking into account that CURP is not a social networks model, and that ours is not an agent-based evolutionary model. This aclaration allows us to understand the fact that our model does not produce a clear distinction between moderate and high-stress scenarios in the sense CURP does. Our model does not incorporate a notion of dynamics like evolutionary models, but rather gives a fixed optimal organization under certain conditions. On the other hand, CURP does not make a distinction between hunters and non-hunters, and all agents can produce food.

Network structures promoting egalitarian resource distribution

Welfare optimal networks (WEF) display high connectivity but low network segmentation, where each hunter is connected to one big, homogeneous and dense group of non-hunters, see Fig 4. On the other hand, networks minimizing individual risk (RV) range from cooperation exclusive among hunters, to networks more similar to welfare networks when the food supply is safer, see Fig 9. These networks are very similar to those tested in the lab experiment by Chiang (2015) [48], where it is found that network structures linking agents with discrepant income levels promote more egalitarian distributions by motivating the rich agents to share their incomes with the poor. Particularly, the network SF_negative from the experiment is very similar to the undirected version of WEF optima from our model. On the contrary, networks where similar income agents are linked evolve to more unequal income distributions, which is also consistent with the RV optima in stringent resource conditions where hunters are only connected among themselves. This result is consistent with the view that social networks structured originally according to ecological considerations, may have created the environment in which prosocial tendencies and equity response elicitation as those explored in the experiment, evolved along human history [49, 50].

Finally, the fact that many WEF optima minimize degree variance is consistent with results of a previous simulation of need-based transfers evolution model showing that degree variance is anti-correlated with survival rate [75]. The minimization of in-degree variance is another pathway to cohesive communities found in our model, see Fig 5. This scenario may be the idealized case resultant of some other more realistic constraint as, for example, an upper bound on the number of connections an agent may have. This last idea has been proposed by the social brain hypothesis [76], which states that there are cognitive constraints on the number of face-to-face social interactions a human may have, where these limits would be inversely related to the emotional closeness of the bond. Thus, for example, the number of close friends one could maintain is limited to five. The evolutionary perspective suggests, again, that these cognitive constraints on social relationships may have been related originally to some kind of resource distribution optimality.

Distinct patterns of reciprocity

We have obtained distinct distributions of reciprocity among hunters and non-hunters, in the three types of optimal networks we study, which gives a broader picture to the usual notion that reciprocity is driven by the minimization of food production uncertainty [28, 77]. We only got high reciprocity under the reduction of variability criterion (RV) for reciprocal exchanges among hunters, see Fig 13(c). On the other hand, Pareto optimal networks (PF) maximize mean reciprocity between hunters and non-hunters, and Welfare (WEF) maximizes mean reciprocity among non-hunters, see Fig 13. These distinct distributions may suggest that the relative importance of every criterion guiding food transfers depends critically upon the context, which in turn may be a result of population size and division of labor.

One possibility is that for large communities, where hunters are only a small fraction of the whole population, there are hunters that reciprocally exchange food to minimize risk, and after that, new exchanges take place within subcommunities fed by each of these hunters, that may be driven to a greater or lesser extent by welfare considerations. This is not far from what has been reported for some hunter-gatherer societies, where there are two systems or stages of sharing: in the first, sharing occurs among participants in the cooperative effort of food acquisition as a form of labor reward, and then, in the secondary distribution each individual that received shares redistributes his or her share to families that did not participate [51]. This is consistent with the claim that exchanges motivated by need or welfare, would be more common within a household [20, 78], and with more recent observations reporting that Hadza men consumed a substantial amount of food while out of camp foraging [79], and that returning to camp empty-handed is indicative that he failed to produce enough surplus to share. This may suggest a greater relative weight of individual starvation risk minimization over welfare motivation in the first stage of sharing.

Implications for evolutionary and economic models of social dynamics, and future work

Though our model is not evolutionary in its formulation, its results suggest that motives not usually modeled by these approaches, such as the egalitarian group access to critical resources for survival, may be an important driver for social network formation. Typical evolutionary models of food-sharing [38] rely on the assumption of maximization of a function of subjective preferences, usual in economic models of network formation [80], which highlights an exclusive role of individual choice and may impose high cognitive requirements. We claim that evolutionary approaches may benefit from a wider repertoire of assumptions including a resource distribution perspective, the modeling of survival needs, and the explicit inclusion of the group level of analysis.

Our model looks for capturing the transfers of finite resources subject to uncertain and scarce production, and their effect on optimal network organization. Therefore, its insights may potentially be applied to other situations imposing an analog structure of restraints. For example, since parents need to spend most of their time raising their offspring, maybe the behavior of cooperative breeding is dependent on the uncertain generation of scarce free time available for the care of non-descendant children. On the other hand, some works have argued [32, 41] about the usefulness of using a diminishing marginal returns function over the exchanged quantities of food, which may be a symptom of the underlying phenomenon of satisfaction of needs [42]. Our model is based on an original formulation of starvation risk relying upon the expected number of success runs of a certain length in a sequence of trials to implement the hunger need over time. Our model may contribute a formal framework to proceed in this discussion. Finally, the explicit inclusion of group level of analysis may be implemented through the multi-level selection model [81, 82] that incorporates as evolutionary units the individuals, as well as the groups they form. A mapping from individual fitness to starvation risk, and from group fitness to egalitarian group access to resources may be studied in future work. A model like this may serve to shed light on the complex interactions between developmental, biological, social, economic and cultural factors influencing social networks formation. More direct extensions of our model would be, for example, more complex protocols of food-sharing allowing for preferential exchanges according to agent closeness, correlations in production of food [77], or others. The modular character of the model would allow making changes at several levels.