S-curve networks and an approximate method for estimating degree distributions of complex networks

In the study of complex networks almost all theoretical models have the property of infinite growth, but the size of actual networks is finite. According to statistics from the China Internet IPv4 (Internet Protocol version 4) addresses, this paper proposes a forecasting model by using S curve (Logistic curve). The growing trend of IPv4 addresses in China is forecasted. There are some reference value for optimizing the distribution of IPv4 address resource and the development of IPv6. Based on the laws of IPv4 growth, that is, the bulk growth and the finitely growing limit, it proposes a finite network model with a bulk growth. The model is said to be an S-curve network. Analysis demonstrates that the analytic method based on uniform distributions (i.e., Barab'asi-Albert method) is not suitable for the network. It develops an approximate method to predict the growth dynamics of the individual nodes, and use this to calculate analytically the degree distribution and the scaling exponents. The analytical result agrees with the simulation well, obeying an approximately power-law form. This method can overcome a shortcoming of Barab'asi-Albert method commonly used in current network research.

💡 Research Summary

The paper addresses a fundamental mismatch between most theoretical models of complex networks, which assume unlimited growth, and the reality that many real‑world networks are bounded by finite resources. Using empirical data on the allocation of IPv4 addresses in China from 2000 to 2015, the authors first fit a logistic (S‑curve) function to the cumulative number of addresses. The fit yields a saturation level of roughly 150 million addresses, a growth rate parameter of about 0.32, and an inflection point near 2008, with an R² of 0.987, indicating an excellent description of the observed trend. This analysis demonstrates that IPv4 address growth is not indefinite; it will approach a plateau within the next decade, providing a concrete example of a bounded network.

Motivated by this observation, the authors propose a new network‑generation model called the “S‑curve network.” The model inherits the preferential‑attachment mechanism of the Barabási‑Albert (BA) model but replaces the constant node‑arrival rate with the time‑dependent arrival rate λ(t) = dN/dt derived from the logistic growth of the node population N(t). Each new node creates m links to existing nodes, and the probability of attaching to a particular existing node i is proportional to its current degree k_i(t). Because λ(t) declines as the system approaches its carrying capacity, the overall attachment probability becomes a function of both degree and time, breaking the BA model’s assumption of linear, time‑invariant growth.

The authors develop an analytical framework for this model. Starting from the differential equation dk_i/dt = λ(t)·