The sandpile model and empire dynamics

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According to the self-organized criticality theory, the sandpile model is built to investigate the evolutionary dynamics of empires in the history of China. The methods of agent-based modeling and simulations are applied to capture empires’ mechanism of rising and falling cycles, and to obtain the observed life cycle pattern of empires in history. Under the self-organized criticality theory, natural systems and human empires systems have similar structures and mechanisms, which makes systems reaches the critical states automatically. Therefore, the rising and falling dynamics of empires can be reflected by the sandpile model as well. With the sandpile modeling and simulations, the optimal solution of parameters can be found, based on which the satisfactory fitness of results can be achieved. Under the optimal solution, we run the simulations for 1000 times to check the fitness and robustness. First, the number of empires can be matched. There were 22 empires in the history of China, and the same number of empires can be obtained via sandpile model simulations, and the amount of empires follows the normal distribution with the mean of 22 empires; Second, the distribution of empire durations follows the power-law distribution, for both simulated and historical empires; Third, for less than 22 simulated empires, we drop empires with tiny durations in history to compare the 19, 20, and 21 pairs of counterparts respectively, and the fitness can be guaranteed as well; finally, for more than 22 empires, we drop simulated empires with tiny durations to compare the 23, 24, and 25 pairs, the matching degree is satisfactory as well. It indicates that multiple simulations have more robust and stable outcomes than one, even the best, simulation.