Preferential attachment mechanism of complex network growth: "rich-gets-richer" or "fit-gets-richer"?

TL;DR

This paper demonstrates that preferential attachment and fitness models of network growth are fundamentally equivalent under general conditions, providing a microscopic basis for initial node attractiveness and explaining its empirical universality.

Contribution

It reveals the equivalence of preferential attachment and fitness models, offering a microscopic explanation for initial attractiveness in network growth.

Findings

Both models yield the same dynamic growth equation.

Initial attractiveness $K_0$ is mainly determined by fitness distribution width.

Empirical universality of $K_0$ around 1 is linked to lognormal fitness distribution.

Abstract

We analyze the growth models for complex networks including preferential attachment (A.-L. Barabasi and R. Albert, Science 286, 509 (1999)) and fitness model (Caldarelli et al., Phys. Rev. Lett. 89, 258702 (2002)) and demonstrate that, under very general conditions, these two models yield the same dynamic equation of network growth, $\frac{dK}{dt}=A(t)(K+K_{0})$, where $A(t)$ is the aging constant, $K$ is the node's degree, and $K_{0}$ is the initial attractivity. Basing on this result, we show that the fitness model provides an underlying microscopic basis for the preferential attachment mechanism. This approach yields long-sought explanation for the initial attractivity, an elusive parameter which was left unexplained within the framework of the preferential attachment model. We show that $K_{0}$ is mainly determined by the width of the fitness distribution. The measurements of $K_{0}$ in many complex networks usually yield the same $K_{0}\sim 1$. This empirical universality can be traced to frequently occurring lognormal fitness distribution with the width $\sigma\approx 1$.