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

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 $\sigma\approx 1$.