The co-authorship network of scientists is a complex evolving network that reflects the structure and dynamics of scientific collaboration. By analyzing data from mathematics and neuroscience journals between 1991 and 1998, researchers identified key properties of this network, including its scale-free nature and the role of preferential attachment in its evolution. The network exhibits a power-law degree distribution, with the exponent γ being approximately 2.4 for mathematics and 2.1 for neuroscience. The average separation between nodes decreases over time, indicating a more interconnected network, while the clustering coefficient shows a slow convergence to an asymptotic value. The size of the largest cluster increases over time, suggesting the emergence of a giant connected component. The average degree also increases with time, deviating from the typical assumption of constant degree in evolving networks. Preferential attachment governs both the addition of new nodes and internal links, with new nodes more likely to connect to already well-connected nodes. Internal links, formed between existing authors, also follow preferential attachment, contributing significantly to the network's topology. A continuum theory was developed to model the network's evolution, showing that the degree distribution follows a power law with exponent γ = 2 in the asymptotic limit. Monte Carlo simulations confirmed these findings, demonstrating that the network's properties, such as average separation and clustering coefficient, depend on the completeness of the data. Nonlinear effects of preferential attachment were also explored, showing that internal attachment dominates the network's behavior, leading to a power-law degree distribution. The study highlights the importance of understanding the dynamics of complex networks and provides insights applicable to other evolving systems, such as the World Wide Web and social networks.The co-authorship network of scientists is a complex evolving network that reflects the structure and dynamics of scientific collaboration. By analyzing data from mathematics and neuroscience journals between 1991 and 1998, researchers identified key properties of this network, including its scale-free nature and the role of preferential attachment in its evolution. The network exhibits a power-law degree distribution, with the exponent γ being approximately 2.4 for mathematics and 2.1 for neuroscience. The average separation between nodes decreases over time, indicating a more interconnected network, while the clustering coefficient shows a slow convergence to an asymptotic value. The size of the largest cluster increases over time, suggesting the emergence of a giant connected component. The average degree also increases with time, deviating from the typical assumption of constant degree in evolving networks. Preferential attachment governs both the addition of new nodes and internal links, with new nodes more likely to connect to already well-connected nodes. Internal links, formed between existing authors, also follow preferential attachment, contributing significantly to the network's topology. A continuum theory was developed to model the network's evolution, showing that the degree distribution follows a power law with exponent γ = 2 in the asymptotic limit. Monte Carlo simulations confirmed these findings, demonstrating that the network's properties, such as average separation and clustering coefficient, depend on the completeness of the data. Nonlinear effects of preferential attachment were also explored, showing that internal attachment dominates the network's behavior, leading to a power-law degree distribution. The study highlights the importance of understanding the dynamics of complex networks and provides insights applicable to other evolving systems, such as the World Wide Web and social networks.