The paper by Albert-László Barabási, Réka Albert, and Hawoong Jeong explores the scaling properties of scale-free networks, which are characterized by a power-law distribution of vertex connectivities. They introduce a mean-field theory to predict the growth dynamics of individual vertices and calculate the connectivity distribution and scaling exponents. The mean-field method is applied to two variants of the scale-free model, one without growth and the other without preferential attachment, to understand their properties. The authors find that both ingredients—growth and preferential attachment—are essential for the observed scale-free behavior. They also discuss extensions of the model, including the possibility of nonlinear attachment rules, continuous edge addition, and link rewiring, and highlight the potential for universality classes in random network models. The study aims to provide a better understanding of the generic properties of complex networks and their applications in various fields.The paper by Albert-László Barabási, Réka Albert, and Hawoong Jeong explores the scaling properties of scale-free networks, which are characterized by a power-law distribution of vertex connectivities. They introduce a mean-field theory to predict the growth dynamics of individual vertices and calculate the connectivity distribution and scaling exponents. The mean-field method is applied to two variants of the scale-free model, one without growth and the other without preferential attachment, to understand their properties. The authors find that both ingredients—growth and preferential attachment—are essential for the observed scale-free behavior. They also discuss extensions of the model, including the possibility of nonlinear attachment rules, continuous edge addition, and link rewiring, and highlight the potential for universality classes in random network models. The study aims to provide a better understanding of the generic properties of complex networks and their applications in various fields.