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

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.

💡 Research Summary

The paper addresses a gap in the study of directed complex networks: while vertex‑centric properties such as degree distributions are well known to follow power‑law scaling, the statistical behavior of edges has received little attention. To fill this void, the authors introduce the edge balance ratio (R), defined for a directed edge as the ratio of the in‑degree of its source node to the in‑degree of its target node. An R value greater than one indicates a flow from a high‑in‑degree node to a lower‑in‑degree node, whereas R < 1 signals the opposite direction. This simple definition captures the asymmetry of information or influence flow at the edge level, which is invisible when only vertex statistics are examined.

From R the authors derive two aggregate descriptors. The balance profile is the histogram of R values plotted on log‑log axes; it reveals that the distribution of R is not a single power law but a piecewise power‑law with several linear segments. The second descriptor, positivity, is the fraction of edges with R > 1, i.e., the proportion of edges that point from higher‑in‑degree to lower‑in‑degree nodes. Positivity therefore quantifies the overall directional bias of the network.

The theoretical core of the work assumes that the in‑degree distribution of vertices follows a power law (P(k)\propto k^{-\gamma}). Under this assumption, the probability that a randomly chosen edge connects a source of degree (k_s) to a target of degree (k_t) is proportional to (k_s P(k_s) \times k_t P(k_t)). By changing variables to the ratio (R = k_s/k_t) and integrating over all admissible degree pairs, the authors obtain an analytical expression for the probability density (f(R)). The result is a piecewise power‑law: for each interval (