On the prevalence of hierarchies in social networks

In this paper, we introduce two novel evolutionary processes for hierarchical networks referred to as dominance- and prestige-based evolution models, i.e., DBEM and PBEM, respectively. Our models are deterministic in nature which allows for closed-form derivation of equilibrium points for such type of networks, for the special case of complete networks. After deriving these equilibrium points, we are somewhat surprised in recovering the exponential and power-law strength distribution as the shared property of the resulting hierarchal networks. Additionally, we compute the network properties, Geodesic distance distribution and centrality closeness, for each model in closed form. Interestingly, these results demonstrate very different roles of hubs for each model, shedding the light on the evolutionary advantages of hierarchies in social networks: in short, hierarchies can lead to efficient sharing of resources and robustness to random failures. For the general case of any hierarchical network, we compare the estimations of tie intensities and node strengths using the proposed models to open-source real-world data. The prediction results are statistically compared using the Kolmogorov-Smirnov test with the original data.