Edge Balance Ratio: Power Law from Vertices to Edges in Directed Complex Network

Abstract: Power law distribution is common in real-world networks including online social networks. Many studies on complex networks focus on the characteristics of vertices, which are always proved to follow the power law. However, few researches have been done on edges in directed networks. In this paper, edge balance ratio is firstly proposed to measure the balance property of edges in directed networks. Based on edge balance ratio, balance profile and positivity are put forward to describe the balance level of the whole network. Then the distribution of edge balance ratio is theoretically analyzed. In a directed network whose vertex in-degree follows the power law with scaling exponent $\gamma$, it is proved that the edge balance ratio follows a piecewise power law, with the scaling exponent of each section linearly dependents on $\gamma$. The theoretical analysis is verified by numerical simulations. Moreover, the theoretical analysis is confirmed by statistics of real-world online social networks, including Twitter network with 35 million users and Sina Weibo network with 110 million users.