Nonlinear asymptotes of dual information dimension of complex networks

Abstract

Extropy is a complementary measure to entropy, proposed as its informational dual. While entropy quantifies the uncertainty or disorder of a distribution, extropy quantifies the certainty or order of the distribution. Extropy has been used in information theory, statistical modeling, complex networks, adaptive systems, and decision-making under uncertainty. From this perspective, the dual information dimension is introduced. The dual information dimension provides an alternative way to measure certainty and network organization by exploiting extropy, offering a complementary view to entropy-based information dimensions.

In the present study, we propose a reformulation of the asymptotes for the dual information dimension, denoted by $D_{id}$, using the nonlinear Logistic4P (four-parameter logistic), Gompertz, Richards (4 and 5 parameters), and von Bertalanffy mathematical models. To validate the proposed reformulation, a representative set of 16 real-world networks, covering different areas of knowledge and practical applications, was used. The results indicate that the optimal nonlinear model depends on the dual $D_{id}$ used. Indeed, Richards4 and Richards5 excel in the Shannon and Tsallis $D_{id}$, respectively, while Logistic4P and Gompertz offer better fits in the Fractional $(q,q^{\prime })$ and Rényi $D_{id}$. However, no single model emerges as optimal across all cases, highlighting the need to select a model based on the specific characteristics of each network and the informational approach employed.