Emergence of power laws in hierarchical dynamics on multi-level graphs

Power law distributions are widely recognized in complex systems as indicative of underlying complexity in interaction networks and critical macroscopic behavior. Previous studies have emphasized the importance of network structure and dynamics in understanding the emergence of such statistical patterns and predicting extreme events. In this study, we investigate the emergence of power law behavior in delay distributions within a multi-level hierarchical network of agents governed by priority rules. Using railway systems as case study, we model the dynamics of high-speed and local trains agents assigned distinct priority levels. By introducing stochastic fluctuations into scheduled travel times, derived from empirical data, we observe that local trains exhibit a markedly higher incidence of larger delays than high-speed trains. We propose a queue-based dynamical model, calibrated using Italian railway data, and validate our findings through comparative analysis with Italian and German datasets. The model reproduces the empirically observed power law exponent associated with the Italian local train delays. Furthermore, we analyze the influence of operational policies, such as priority assignment and delay compensation thresholds, finding their effects both in data and in the model. These results underscore the capacity of simple hierarchical structures and rule-based dynamics to generate complex statistical behaviors without intricate interaction networks.