Fair Topologies: Community Structures and Network Hubs Drive Emergence of Fairness Norms

Fairness has long been argued to govern human behavior in a wide range of social, economic, and organizational activities. The sense of fairness, although universal, varies across different societies. In this study, using a computational model, we test the hypothesis that the topology of social interaction can causally explain some of the cross-societal variations in fairness norms. We show that two network parameters, namely, community structure, as measured by the modularity index, and network hubiness, represented by the skewness of degree distribution, have the most significant impact on emergence of collective fair behavior. These two parameters can explain much of the variations in fairness norms across societies and can also be linked to hypotheses suggested by earlier empirical studies in social and organizational sciences. We devised a multi-layered model that combines local agent interactions with social learning, thus enables both strategic behavior as well as diffusion of successful strategies. By applying multivariate statistics on the results, we obtain the relation between network structural features and the collective fair behavior.

Selecting appropriate network structure is a key challenge in our study. On the one hand, we need enough network structures that cover a wide range of topology parameters.
On the other hand, the number of network structures is limited by the computational cost of simulation. In order to find the networks that generate required variations of structural features for this study, we used a wide range of deterministic (e.g., Tree, Lattice, Ring, and Full) and stochastic (e.g., Barabási-Albert (BA), Watts-Strogatz (WS), and Erdős-Rényi (ER) ) network formation models. Based on these network models, we initially generated a  Figure S1), 3D-Lattice (12-Lattice in Figure S1), Circular Lattice (17-Lattice in Figure S1), Ring (6-Ring in Figure S1), and a fully connected graph (26-Full in Figure S1). The higher values of power of prefential attachement in the BA model enhance the tendency to link to highly connected nodes and accelarate the rich-gets-richer process. This results in a hub-and-spoke structure where most nodes are connected to a few central nodes [1].
The consequence of higher power of preferential attachment for networks with fixed average degree is higher skewenewss of degree distribution since the connections are shifted from a majority of nodes to a few hubs. Variations in transitivity and average degree (degree) are generated by the ER model in which both degree and transitivity increase by increasing the probability of connection between two arbitrary nodes. Finally, variations of modularity are mainly generated by the WS model in which network modularity decreases with increased probability of rewiring. The effects of variation of community structures obtained from the WS model and those structures generated by adjusting modularity based on other methods, such as [2], on the results of evolutionary games simulation were validated in previous work [3].
To balance the trade-off between computational efficiency and sensitivity to structural variations, we did a sensitivity analysis of average strategies at equilibrium against network size for different network models' parameters. For each network model, we selected two (three) network parameters and then compared the value of strategies against network size. Figure S3 depicts the results of network size sensitivity analysis for four network models. We compared two BA networks with m = 3 and p pa ∈ {1, 1.8} (where m is the the number of edges that are added in each time step and p pa is the power of the preferential attachment), three types of lattice structures (i.e., 2D lattice, circular lattice and 3D lattice), two tree structures where the number of children of each vertex is proportional to the network size (i.e., %8 and %16 of the total number of nodes), and two WS structures with k = 4 and p rw ∈ {0, 0.4} (where k is the number of nearest neighbors in the ring topology and p rw is the probability of rewiring each edge).
As depicted in Figure S3, for small network size, the difference in average strategies is not significant for different values of the network models. We selected network structures with 100 nodes (with two exceptions: circular lattice and 3D lattice which have 125 (5 × 5 × 5) nodes, where the results are not sensitive to the deviation of 100 ∼ 125) that capture the effect of varying parameters across all network models that are analyzed. It is worth noting that the average strategies also change with network size for certain network models and parameters values as structural features are also affected by the network size for fixed network parameters. For example, in WS model the average offer value becomes fix for larger network size when p rw = 0, however, it decreases with the network size when p rw = 0.4.

Convergence
A common practice to calculate the value at the equilibrium is to average strategies over a certain number of generations after a transient stage with fixed and large number of generations [4,5,6,7]. This, however, has two main shortcomings. Firstly, it may not be computationally efficient to consider a fixed and large number of generations for the transient stage of all simulation runs as it can be different from one network structure and initialization to another. Secondly, one needs to ensure that the agents' strategies are in fact stabilized after the transient stage, otherwise the averaged strategies would be misleading.
In this study, we used the adaptive convergence criteria suggested in [3]. For each initialization, the simulation of the evolutionary process continues over a number of generations until no more more than one agent (or a certain percentage of the whole population) updates its strategy for a consecutive window of 100 generations. To enhance convergence, we considered a noise threshold of = 0.05 for strategies below which agents do not adopt a new strategy. All runs converged for these parameters. As a result, the number of generations before convergence becomes a strong function of the network structure as it is depicted in Figure S4.

Robustness of Results
In order to find the required number of initializations, we analyzed the difference between averaged strategies at the equilibrium versus the number of initializations for all network structures. For each number of initializations, we collected 10 samples of the average strategies at the equilibrium. Figure S5 and S6 show the results for each network structure, respectively, for offer value and acceptance threshold. The results indicate that for 1024 initializations that is selected for this study, variation of average strategy at the equilibrium is less than 0.01.
The results of our initial sensitivity analysis also shows that the average strategy at equilibrium does not change much with varying selection intensity (β), yet it affects the number of generations before convergence as its increased value expedite the strategy update.
For this study, we used β=0.1.

Data Analysis
Based on the simulation results, we created a 2 × 11 data table (Table S1). Each row in the data table represents one network structure and contains average offer value, acceptance threshold, number of generations before convergence, computation time, and the network structural features. We then use this data table (excluding number of generations and computation time) as the input to our Principal Component Analysis (PCA). Table S2 shows the loading of the simulation variables and network structural features on the Principal Components (PC) where average offer is highly correlated with PC1 and moderately correlated with PC2. Among the structural features, modularity is highly correlated with PC1, while degree is negatively correlated with PC1. Skewness is highly correlated with PC2. We selected the first two PCs that explain more than %80 of the total variations in the dataset according to the scree plot shown in Figure S7(a). The biplot [8] in Figure S7 Table S2: The correlations between the initial variables and the independent principal components. Significant correlations are in bold. Average offer is highly correlated with PC1 and moderately correlated with PC2. Modularity is highly correlated with PC1, while Degree is negatively correlated with PC1. Skewness is highly correlated with PC2.        Figure S7: Principal Component Analysis results summary (a) The scree plot displays the eigenvalues associated with each principal component in descending. The first two components explain more than 80% of variations in the whole dataset. (b) The biplot the correlation between initial variables based on the first two principal components. The angles between the variables represent the level correlation.