Possible origin of scaling laws in preferential attachment growth networks

Abstract

The degree distribution of network nodes exhibits scale-free features, which is a universal phenomenon in natural and social networks. It is commonly believed that growth and linear preferential attachment are the evolutionary mechanisms of scaling laws. However, the underlying cause behind these mechanisms remains unknown. In this paper, we propose the concept of the topology state of network nodes, and then define the network homomorphism ratio. Combining the total energy of the network defined in previous studies, we systematically investigate the connection between these two quantities in preferential attachment growth networks. Our results show that under the thermodynamic limit condition, the emergence of the scaling laws of the preferential attachment growth network is dominated by two principles. One principle is that the homomorphism ratio gradually tends to zero as the network grows, and the other principle is that the total energy of the system is the lowest under the condition of satisfying principle one. Interestingly, these two principles of the network are similar to the arrangement rules of electrons outside the nucleus which dominated by the Pauli exclusion principle and the principle of minimum energy.