Multiorder boundaries among discrete domains: Relative fractal dimension and others

In nature and society, most of competitions take place on the boundaries among a group of domains where different individuals or colonies share common resources; therefore, it is widely believed that domain boundaries play important roles in the evolution of many complex systems. Here, we first give a definition for multiorder boundaries among discrete domains and then propose a general method to calculate their relative fractal dimension, i.e., the ratio of the fractal dimension of the boundaries versus that of the domains themselves. Through analyzing three types of real-world discrete domains, several interesting results are revealed. For example, the limitation on the number of domains that an individual can join in may produce longer boundaries indicating more cruel competitions among the domains. Besides, the individuals with more social links are always considered more important in social networks, and it is found that these individuals as valuable resources of social domains are always centralized on the boundaries of higher order.