The paper explores the topology of evolving networks, focusing on how local events such as the addition of new nodes and links, or rewiring of links, influence network structure. It shows that depending on the frequency of these processes, two distinct topologies can emerge: one with a power-law connectivity distribution and another with an exponential distribution. A continuum theory is proposed to predict these regimes, the scaling function, and the exponents, aligning well with numerical results. The model is applied to fit the connectivity distribution of the professional links between movie actors. Networks in social, biological, and communication systems are complex due to their interwoven structures. The study of random networks has been dominated by the Erdős–Rényi model, but recent findings suggest that real networks often exhibit scale-free behavior, where the probability distribution of node degrees follows a power-law. This is supported by measurements on the web, actor networks, and citation networks. The scale-free behavior is attributed to two mechanisms: the addition of new nodes connected to existing ones, and the preferential attachment of new links to highly connected nodes. The paper introduces an extended model of network evolution that incorporates additional local events, such as the addition of new links and rewiring. Using a continuum theory, it shows that the topology of a network depends on the relative frequency of these processes. In the scale-free regime, the connectivity distribution follows a power-law, with the exponent depending on the process frequencies. In the exponential regime, the distribution decays exponentially. A phase diagram is derived to predict the transition between these regimes. The model is applied to real networks, such as the collaboration graph of movie actors, where the connectivity distribution is fitted using the continuum theory. The results show that the majority of new links connect existing nodes, indicating high interconnectivity in the movie industry. The study highlights that the topology and connectivity distribution of networks are determined by the frequencies of local processes, suggesting that monitoring these rates can predict the large-scale structure of real networks. The findings also indicate that while scale-free networks exhibit non-universal scaling exponents, exponential networks show robustness, as seen in models like the Erdős–Rényi and Watts–Strogatz models.The paper explores the topology of evolving networks, focusing on how local events such as the addition of new nodes and links, or rewiring of links, influence network structure. It shows that depending on the frequency of these processes, two distinct topologies can emerge: one with a power-law connectivity distribution and another with an exponential distribution. A continuum theory is proposed to predict these regimes, the scaling function, and the exponents, aligning well with numerical results. The model is applied to fit the connectivity distribution of the professional links between movie actors. Networks in social, biological, and communication systems are complex due to their interwoven structures. The study of random networks has been dominated by the Erdős–Rényi model, but recent findings suggest that real networks often exhibit scale-free behavior, where the probability distribution of node degrees follows a power-law. This is supported by measurements on the web, actor networks, and citation networks. The scale-free behavior is attributed to two mechanisms: the addition of new nodes connected to existing ones, and the preferential attachment of new links to highly connected nodes. The paper introduces an extended model of network evolution that incorporates additional local events, such as the addition of new links and rewiring. Using a continuum theory, it shows that the topology of a network depends on the relative frequency of these processes. In the scale-free regime, the connectivity distribution follows a power-law, with the exponent depending on the process frequencies. In the exponential regime, the distribution decays exponentially. A phase diagram is derived to predict the transition between these regimes. The model is applied to real networks, such as the collaboration graph of movie actors, where the connectivity distribution is fitted using the continuum theory. The results show that the majority of new links connect existing nodes, indicating high interconnectivity in the movie industry. The study highlights that the topology and connectivity distribution of networks are determined by the frequencies of local processes, suggesting that monitoring these rates can predict the large-scale structure of real networks. The findings also indicate that while scale-free networks exhibit non-universal scaling exponents, exponential networks show robustness, as seen in models like the Erdős–Rényi and Watts–Strogatz models.