To develop a general mathematical model for social networks is one of the fundamental tasks currently on demand within social network research. Ignoring the strength of the relationships, existing social network models simply use a Boolean value to describe the existence of relationships between peers. This shortage can be overcome by importing repeated social interactions into the model and building the model on a path-based link analysis. In doing this, the authors developed a new semi-random graph model, which offers a general description of the evolution of social networks, with substantial power, to the well accepted hypothesis of preferential attachment in social networks. In addition to these theoretical results, the authors created a quantitative description of the bonding role of social relationship in networks, a parameter within the model denoted as V. Empirical results indicate that the presented model has a degree of distribution in line with those of real-world networks, which is superior to those of major existing models, and the parameter V, which essentially represents the cohesiveness of social networks, makes an ideal indicator for the cohesion in social networks.