Structural power and the evolution of collective fairness in social networks

From work contracts and group buying platforms to political coalitions and international climate and economical summits, often individuals assemble in groups that must collectively reach decisions that may favor each part unequally. Here we quantify to which extent our network ties promote the evolution of collective fairness in group interactions, modeled by means of Multiplayer Ultimatum Games (MUG). We show that a single topological feature of social networks—which we call structural power—has a profound impact on the tendency of individuals to take decisions that favor each part equally. Increased fair outcomes are attained whenever structural power is high, such that the networks that tie individuals allow them to meet the same partners in different groups, thus providing the opportunity to strongly influence each other. On the other hand, the absence of such close peer-influence relationships dismisses any positive effect created by the network. Interestingly, we show that increasing the structural power of a network leads to the appearance of well-defined modules—as found in human social networks that often exhibit community structure—providing an interaction environment that maximizes collective fairness.


Introduction
The human predisposition to be fair shapes decision-making and drives the outcome of social interactions [1][2][3][4][5][6]. The influence of fairness is often strong enough to overcome rationality and selfishness, posing a challenge to mathematical models that aim to incorporate the complexity of human interaction and thus justify fair behavior. Factors such as the cultural setting [7], community size, engagement in large-scale institutions [8], or even the socio-economic segment of the individuals [9], may provide clues regarding the propensity to be fair. Concerns about fairness may even lead individuals to decide, collectively, to give up some of their wealth to punish unfair behavior of others [10]. For instance, in the collective bargaining of work contracts, recognized in international human rights conventions, one has groups of individuals with different interests, where the fairness level of the outcome is ultimately shaped by the collective decision of employees and employer(s). Another less formal example is found in the Chinese concept of tuangou, where a group of people approaches a seller, offering to buy a network). In line with previous studies [25][26][27], each neighborhood defines a group, whereas the fitness F i of an individual i of degree k is determined by the payoffs resulting from the game instances occurring in k+1 groups: one centered on her neighborhood plus k others centered on each of her k neighbors. In other words, each node with degree k defines a group with size N = k+1, including that node (focal) and the neighbors. Fig 1 provides pictorial representations of this group formation process. In homogeneous populations, every individual participates in the same number of groups (and MUG instances), all with the same size. Often, however, individuals face different numbers of collective dilemmas (depending, e.g., on their social position) that may also have different sizes. Such a dimension of social diversity is introduced here (Fig 4) by considering heterogeneous networks [30].
Social success drives the evolution of strategies in the population, that is, we implement strategy revision by social learning [26,[31][32][33][34][35], assuming that the behavior of individuals that perform better (i.e. achieve higher fitness) will spread faster in the population as they will be imitated with higher probability (see Methods for details).
We assume that individuals do not have direct access to the set of rules that define the behavior of others-instead, they perceive their actions, and therefore, errors of perception may be relevant. Consequently, whenever a pair (p,q) is copied, the final value will be perturbed by a random shift uniformly drawn from the interval [-ε,ε], reflecting the myopic nature of the imitation process. This process occurs along the social ties defined by the underling network [25]. We represent all the groups where a focal individual, positioned in the blue node, participates. In both (a) and (b), the focal individual has a connectivity of 3 (blue links) thereby playing in 4 different groups: one centered on herself (represented by a grey circle) and the 3 others centered on her (numbered) neighbors (represented by yellow ellipses). For instance, the groups represented by the ellipses iv contain all neighbors of individuals with number 4 (including the focal individual). The motifs presented in (a) and (b) differ in the overlap of the groups where the blue nodes take part, and consequently, the influence that those neighbors exert and are subject to. In (a), none of the 1 st neighbors of the focal individual are 1 st neighbors of each other; thus, the focal individual only meets each of her/his neighbors in two groups. In (b), all 1 st neighbors of the focal individual are directly connected, which means that individuals 2, 3 and 4 take part in all the groups where the focal individual also takes part. Thus, they influence eachother more in (b) than in (a). The structural power (SP, defined in Methods and based on the prevalence of one individual in the interaction groups of another), provides a quantitative measure of the influence capacity of any node onto another (see Methods where we show that the influence extends to second neighbors). When applied to the entire network, the SP is thus higher in (b) than in (a).

Results and discussion
We start by simulating MUG on regular rings (regular) [36], and in homogeneous random networks (horand) [37] (see Methods for information regarding the construction and characterization of both networks, together with details of the simulation procedures). As Despite the fact that both classes of networks exhibit the same Degree Distribution (DD), they have quite different Clustering Coefficients (CC) and also Average Path Lengths (APL) [36,37]. The regular ring networks warrant a high CC which, in turn, ensures that individuals appear repeatedly in the interaction groups of others. The prevalence of a given individual in the interaction groups of another may be understood as a power relation [15,38,39], that is, as a measure of the influence that an individual A has in the goals (here, fitness) of another individual B. This influence is enhanced by the fraction of interaction groups of B in which A appears (see Methods). To further characterize this property, we define an explicit quantity, that we call the Structural Power (SP). At the individual level, the structural power of an individual A over another individual B is given by the fraction of all groups in which B participates that also include A. This quantity, conveniently normalized between 0 and 1, is further extended to define the (average) SP of a node in a network, as well as the (average) SP of an entire network. Full details are provided in Methods. It is important to point out, however, that SP and CC convey different properties of a network: For instance, whereas CC only accounts for the triangular motifs present in a network, the computation of SP also reflects existing square motifs. To isolate the effect of SP from CC-and also from APL and DD-we calculate the average proposals <p> and average acceptance threshold <q> emerging when MUG is played in a class of networks in which CC always remains close to 0, but SP is not negligible (see Fig 3 and Methods). , when MUG is played on unstructured populations (well-mixed), on regular rings (regular) or on random networks with homogeneous degree distribution (homogeneous random, horand, generated by swapping the edges initially forming a ring [37,40,66]). M has a positive effect on the average values of <p> [22]. Notwithstanding, this effect is much more pronounced in the case of regular networks, where we also witness a similar increase in the average values of <q>. Other parameters: average degree <k> = 6 (meaning that groups have a constant size of N = 7); population size, Z = 1000; mutation rate, μ = 0.001; imitation error, ε = 0.05 and selection strength, β = 10 (see Methods for definitions of all these parameters). In particular, we interpolate between two low CC networks: i) A triangle-free regular ring (which can also be interpreted as a regular bipartite graph, with links connecting odd-numbered nodes to even-numbered nodes, exhibiting a high SP, Fig 3c) and ii) a homogeneous random graph (low SP , Fig 3d), obtained by randomly rewiring the links of the triangle-free network (see Methods). The interpolation is implemented by means of a parameter r defining the fraction of links to be randomly rewired. The procedure keeps the DD unchanged, as pairs of links are swapped during the rewire process [37,40]. As we depict in Fig 3, irrespectively of CC, APL and DD, the dependence of both <p> and <q> is fully correlated with that of SP and not with any of the other quantities.
To further analyze the impact of SP on the levels of fairness, we design networks with different SP by optimizing the link structure of a random network until a desired pre-defined SP is achieved (see Methods). Fig 4 shows the average proposal (<p>), and acceptance threshold (<q>) we obtain, now as a function of the network SP. Clearly, high values of SP lead to higher values of <p> and <q>, in which case individuals adopt fairer strategies. Fig 5 illustrates the structural effects induced by maximizing the SP of a network, while keeping the average degree <k> constant. Additionally, we concentrate our analysis on sparse structures (<k> << Z), as it is often the case in social networks [41,42]. When maximizing the SP under these constraints, one witnesses the emergence of highly modular sub-structures, with the concomitant appearance of different communities [43]. In fact, each node acquires high SP by repeatedly appearing in the interaction groups of individuals belonging to the same community, which leads, as a consequence, to a distinguishing characteristic of modular networks: high average SP.
The prevalence of fairness in small communities where members share a large number of connections fits well with the empirical studies that argue for the trust and confidence provided by this kind of community structures [44][45][46][47]: All the individuals of the community monitor the interactions occurring within links, referred as strong ties or embedded links. In fact, it is worth noting how easier is to render punishment effective (where punishment here means having a high acceptance threshold, q, thus rejecting low proposals and preventing unfair Proposers to earn payoff) when networks exhibit high SP and, therefore, many local communities: In such cases one punisher may be enough to derail all the low proposals of one unfair neighbor. On the other hand, in structures with low SP, the fact that each unfair individual plays in several groups with different opponents requires different punishers to be present in those groups, in order for unfairness to be effectively penalized. This way, communities  provide the needed lever to trigger fair strategies. Naturally, explicit forms of sanctioning could be devised, such that its impact, together with its relation with particular network topologies, can be extended to other social dilemmas and interaction contexts [10,17,[48][49][50][51]. In particular, we conjecture that networks with high SP should play an increased role when working as interacting structures for multiplayer games with thresholds [26,27,52,53].
Additionally, networks with high SP, besides providing the right context for preventing unfair proposals, can also confer a relative advantage to individuals prone to reject low offers and make fair proposals. Having one individual with high SP and high acceptance threshold often implies that the only proposal accepted in the neighborhood is precisely her/his own. Naturally, this can only happen if that individual is able to take part in a large fraction of his/ her peers, which, again, translates into a high SP.
Here we investigate fairness based on group decisions by means of an evolutionary game theoretical model employing MUG, played along the links of complex social networks. Our results show that the SP of a network constitutes a key observable indicating the feasibility that fairness emerges in the population, in both regular (Figs 2 and 3) and heterogeneous networks (Fig 4) and in situations where resorting to well-established quantities such as CC can be elusive (Fig 3).
Finally, this work can be related with the important concept of governance by means of polycentric sanctioning institutions [52,54]. To this end, let us assume that every Responder conceals a potential punisher and each group where MUG is played constitutes a center of decision. This perspective re-positions the present model into an interestingly polycentric perspective, as now Responders with high SP spawn many overlapping interaction groups which, in turn, can be related to the problem of interdependence between groups. As mentioned by V. Ostrom "Polycentric connotes many centers of decision making that are formally independent of each other. Whether they actually function independently, or instead constitute an interdependent system of relations, is an empirical question in particular cases" [54,55]. We find that the question of the interdependence of groups taking part in collective decisions, here quantified by means of the SP, may be central in promoting seemingly paradoxical human features such as fairness.

Game, payoff and fitness
Following the conventional notation of UG [17], the total amount initially given to the Proposer playing MUG is equal to 1. In a group of N individuals, the proposal made is p ∊ [0,1] and each of the N-1 Responders has an acceptance threshold q ∊ [0,1]. Once the proposal is made, each Responder will individually state his acceptance (if q p) or rejection (if q>p). Overall, the group acceptance depends upon a minimum fraction of individual acceptances, M. This can be summarized in a variable a i , assuming the value 1 if the proposal by individual i is accepted, and 0 otherwise [22]: where Θ(x) is the Heaviside function, assuming the value 0 when x<0 and 1 otherwise. The payoff P i earned by an individual i in a group of N individuals, will be given by adding the result of acting once as the Proposer-P P = (1 − p i )a i -and N-1 times as a Responder- where p k is the offer of individual k and a k refers to the proposal of Structural power and the evolution of collective fairness in social networks individual k. It is worth noting that the maximum payoff of an individual i is obtained when p i is the smallest possible and all other p k (the offers of opponents) are maximized. Therefore, there is a high pressure to free-ride, that is, offering less and expecting that others will contribute. Furthermore, dividing the game in two stages and reasoning in a backward fashion, the conclusions regarding the sub-game perfect equilibrium of this game anticipate the use of the smallest possible p i and q i , irrespectively of N and M [56], mimicking the conclusions for the traditional 2-person UG [57]. The fitness is given by the accumulated payoff earned after playing in all possible groups.

Networks
An underlying network of contacts defines the groups in which individuals play. One node (focal) and its direct neighbors define a group. An individual placed in a node with connectivity k will play in k+1 different groups. In Fig 1 we provide intuitive representations for this group formation process (where the structural power SP is defined next). We use four classes of networks, namely, i) regular rings [36], ii) regular triangle-free rings, iii) homogeneous random networks [37] and iv) networks with pre-defined average SP. Regular rings, with degree k, are traditionally constructed by i) creating a numbered list of nodes and ii) connecting each node to the k nearest neighbours in that list [36]. Similarly, we generate regular triangle-free rings (with degree k) by connecting one node (source) with the closest k nodes, yet only those at an odd distance (in the list) to the source (in the language of graph theory, this corresponds to define a (k,k)-biregular graph using the odd-numbered and even-numbered nodes as disjoints sets). This allows preventing the occurrence of triangles (i.e., adjacent nodes of a given node that are, themselves, connected) which would contribute to increase CC. In Fig 3, we interpolate between a regular triangle-free ring and a homogeneous random graph following the algorithm proposed in [37]. We introduce a parameter r which gives the fraction of edges to be randomly rewired: for r = 0 we have a regular triangle-free ring, whereas for r = 1 all edges are randomly rewired and we obtain a homogeneous random graph. We adopt a rewiring mechanism which does not change the degree distribution [37,40]. The algorithm resumes to repeat the following two-step circular procedure until a fraction r of all edges are successfully rewired: 1) choose-randomly and independently-two different edges which have not been used yet in step 2, and 2) swap the ends of the two edges if no duplicate connections arise. In Fig 4, to generate networks with pre-defined average SP, we apply an optimization algorithm to a random network. The random networks are generated by rewiring all the edges of regular ring [36]. Let us now assume that we want to build a network with average SP equal to sp max . We re-organize the link structure of the initial network using a stochastic multi-step process such that, in each step, an edge of network is rewired at random (with no repeated edges allowed). The move is accepted if two criteria are met: 1) the resulting network remains connected and 2) the average SP of the resulting network (sp t ) increases (compared to the previous value) or passes the following stochastic criterion: a move in which SP decreases is accepted with probability λ(sp max -sp t ), where λ controls the probability of accepting an erroneous move. That means that the probability of accepting a rewire that decreases SP is lower as we get close to the desired SP. This is an optimization feature similar in spirit to the wellknown simulated annealing [58]. We used λ = 0.001.

Structural Power (SP)
The population structure provides the definition of the different groups of interaction, which may overlap to variable extent [59]. Considering the usual group formation that we address (in which one node defines, together with his/her direct neighbors, a group), individuals may appear repeatedly in the interaction groups of others. As said, this repetition may provide increased SP to some individuals over others.
We define the SP of A over B as SP A;B ¼ jIðAÞ\IðBÞj jIðBÞj , where I(X) represents the groups in which individual X appears and |I(X)| represents the number of groups in I(X). One may note that, using the Kronecker δ A,B to identify edges between A and B (e.g, 1 if an edge connects nodes A and B and 0 otherwise), and denoting by o A,B (overlap) the number of common neighbors of A and B and by k X the number of neighbors of X, then the SP of A over B is given by Intuitively, if one individual is a direct neighbor of other (δ A,B = 1), they will meet in at least two groups, where each one will be the focal in each group. They will meet again if they have a common neighbor i (δ A,i × δ i,B = 1), and thus whenever A and B are direct neighbors, o A,B counts the number of triangular motifs involving both A and B. If B has connectivity k B , then this node participates in k B +1 groups, providing the proper normalization to SP A,B . Importantly, even if A and B are not direct neighbors, SP A,B will not be zero, in general (e.g., square motifs may lead to o A,B 6 ¼0).
The average SP of one node is defined as is the set of individuals reached by individual A, either directly or through a common neighbor, and |R (A)| is the size of this set. Finally, the average SP of one network is the average SP taken over all of its nodes. As an example, in Table 1 we show the average structural power (SP) of several social networks [60] including a sample of Facebook [61], an email communication network (Enron email network, in which nodes are email addresses and edges represent at least one email sent between addresses [62,63]), and several collaboration networks inferred from the co-authorship of papers on arXiv [64], in topics such as General Relativity (GrQc), High Energy Physics Phenomenology (HepPh), High Energy Physics Theory (HepTh), Astrophysics (AstroPH) or Condensed Matter (CondMat). Interestingly, all the abovementioned networks show a global SP significantly higher that the one obtained from a random network [36,65] with the same size (Z) and average degree (<k>) (see SP rand 1). A similar result is obtained if, instead, we compare the SP of empirical networks with randomized networks of the same Z, <k>, and degree distributions (SP rand 2). Following refs. [37,40,66], SP rand 2 was computed as the SP of the network that results from swapping random pairs of edges for 10Z<k> times.

Evolutionary dynamics in structured populations
Instead of revising their strategies through rational reasoning, humans often resort to the experiences and successes of others to select their next move, as, in fact, has been shown to be the case in the context of public donations [67][68][69]. Such an interacting dynamical process, grounded on peer-influence and imitation, creates a behavioral ecosystem in which strategies and behaviors evolve in time, whereas the returns of each individual depend on the actual frequency of each strategy present in its neighborhood. Fitness is said to be context-dependent.
Here we adopt such social learning dynamics [17, 23, 25-27, 31-35, 70, 71], which is also well suited to be used in the framework of evolutionary game theory. The baseline assumption is that individuals performing better when playing MUG (i.e. those achieving higher accumulated payoffs) will be imitated more often and thus their strategies will spread in the population. Social success drives the adoption of strategies in the population. Imitation occurs by copying behavior through the social ties, statically defined by the underlying network.

Simulations
Numerical results were obtained for structured populations of size Z = 1000. Simulations take place for 50000 generations, considering that, in each generation, all the individuals have the opportunity to revise their strategy through imitation. At every (discrete and asynchronous) time step, two individuals A and B (neighbors) are randomly selected from the population and their individual fitness is computed as the accumulated payoff in all possible groups, provided by the underlying structure; subsequently, A copies the strategy of B with a probability χ that is a monotonic increasing function of the fitness difference f B -f A , following the pairwise comparison update rule [72]-w ¼ ð1 þ e À bðf B À f A Þ Þ À 1 . The parameter β conveniently specifies the selection pressure (β = 0 represents neutral drift and β!1 represents a purely deterministic imitation dynamics). Additionally, imitation is myopic: The copied p and q values will suffer a perturbation due to errors in perception, such that the new parameters will be given by p' = p + ξ p (ε) and q' = q + ξ q (ε), where ξ p (ε) and ξ q (ε) are uniformly distributed random variables drawn from the interval [-ε,ε]. This feature not only i) models a slight blur in perception but also ii) helps to avoid the random extinction of strategies, and iii) ensures a complete exploration of the strategy spectrum, given that the pairwise comparison does not introduce new strategies in the population [73]. To guarantee that p' and q' are not lower than 0 or higher than 1, we implement reflecting boundaries at 0 and 1, e.g., if p'>1 then p' is set to 2-p' [73][74][75]. Furthermore, with probability μ, imitation will not occur and the individual will adopt random values of p and q, proceeding through a random exploration of behaviors. We use μ = 1/Z throughout this work. The effect of varying this parameter is similar to the one verified when changing ε: an overall increase of randomness leads to higher chances of fairer offers [22,76,77]. For each combination of parameters, the simulations were repeated 100 times (using 10 different networks from each class studied), whereas each simulation starts from a population where individuals are assigned random values of p and q drawn uniformly from an evenly discretized strategy space in the interval [0,1] containing 101 values. The average values of p and q obtained, denoted by <p> and <q>, are both a time and ensemble average, taken over all the runs and considering the last 25% of generations, disregarding the initial transient period.

Author Contributions
Conceptualization: FPS JMP AP FCS.