Based on the verified empirical models of urbanization level and traffic network, a set of mathematical models of traffic network development are proposed. With the help of mathematical deduction, a Boltzmann equation is derived for the *β* index of traffic networks from the common logistic function of urbanization level growth, and a quadratic Boltzmann equation is derived for the *β* index from the quadratic logistic function of urbanization curve. The former is applicable to European and American countries and China’s southeast coastal areas, while the latter is applicable to mainland China, especially northern China. The main functions of the models are as follows: the first is to explain and predict the development of traffic networks, and the second is to research the stage division of a growth process or even the spatial dynamics of a traffic network.

%U https://xbna.pku.edu.cn/EN/10.13209/j.0479-8023.2022.118