This paper presents a mean-field theory for scale-free random networks, introduced by Albert-László Barabási and Réka Albert. The study focuses on the scaling properties of the scale-free model, which can account for the observed power-law distribution of connectivities in real-world networks. The authors develop a mean-field method to predict the growth dynamics of individual vertices and use this to calculate the connectivity distribution and scaling exponents. They also analyze two variants of the scale-free model that do not display power-law scaling. The paper begins by discussing the limitations of traditional random graph models, such as the Erdős–Rényi (ER) model, which assumes random connections between vertices. In contrast, real-world networks often exhibit scale-free properties, where the probability that a vertex has k connections follows a power-law distribution, $ P(k) \sim k^{-\gamma} $. This was observed in various real-world networks, including the collaboration graph of movie actors, the World Wide Web, and the electrical power grid. The authors then introduce the scale-free model, which incorporates two key features of real networks: (1) continuous growth by the addition of new vertices, and (2) preferential attachment, where new vertices connect preferentially to highly connected vertices. This model leads to a scale-invariant distribution of connectivities, with the probability that a vertex has k connections following a power-law with an exponent $ \gamma \approx 3 $. The paper also examines two variants of the scale-free model: Model A, which lacks preferential attachment and results in an exponential distribution of connectivities, and Model B, which lacks growth and results in a linear increase in connectivity over time. These models fail to reproduce the scale-free distribution observed in real networks, indicating that both growth and preferential attachment are necessary for the emergence of scale-free properties. The authors conclude that scale-free networks are a common feature of many complex systems, including social, business, and biological networks. The mean-field theory developed in this paper provides a framework for understanding the growth and topology of such networks, and highlights the importance of preferential attachment in the formation of scale-free structures. The study also suggests that the scale-free state is a generic property of many complex networks, with potential applications beyond the examples considered.This paper presents a mean-field theory for scale-free random networks, introduced by Albert-László Barabási and Réka Albert. The study focuses on the scaling properties of the scale-free model, which can account for the observed power-law distribution of connectivities in real-world networks. The authors develop a mean-field method to predict the growth dynamics of individual vertices and use this to calculate the connectivity distribution and scaling exponents. They also analyze two variants of the scale-free model that do not display power-law scaling. The paper begins by discussing the limitations of traditional random graph models, such as the Erdős–Rényi (ER) model, which assumes random connections between vertices. In contrast, real-world networks often exhibit scale-free properties, where the probability that a vertex has k connections follows a power-law distribution, $ P(k) \sim k^{-\gamma} $. This was observed in various real-world networks, including the collaboration graph of movie actors, the World Wide Web, and the electrical power grid. The authors then introduce the scale-free model, which incorporates two key features of real networks: (1) continuous growth by the addition of new vertices, and (2) preferential attachment, where new vertices connect preferentially to highly connected vertices. This model leads to a scale-invariant distribution of connectivities, with the probability that a vertex has k connections following a power-law with an exponent $ \gamma \approx 3 $. The paper also examines two variants of the scale-free model: Model A, which lacks preferential attachment and results in an exponential distribution of connectivities, and Model B, which lacks growth and results in a linear increase in connectivity over time. These models fail to reproduce the scale-free distribution observed in real networks, indicating that both growth and preferential attachment are necessary for the emergence of scale-free properties. The authors conclude that scale-free networks are a common feature of many complex systems, including social, business, and biological networks. The mean-field theory developed in this paper provides a framework for understanding the growth and topology of such networks, and highlights the importance of preferential attachment in the formation of scale-free structures. The study also suggests that the scale-free state is a generic property of many complex networks, with potential applications beyond the examples considered.