A rate equation approach was presented for the exact computation of the three vertex degree correlations of the fixed act-size collaboration networks. Measurements of the three vertex degree correlations were based on a rate equation in the continuous degree and time approximation for the average degree of the nearest neighbors of vertices of degree k, with an appropriate boundary condition. The rate equation proposed can be generalized in more sophisticated growing network models, and also extended to deal with related correlation measurements. Finally, in order to check the theoretical prediction, a numerical example was solved to demonstrate the performance of the degree correlation function.