Does mathematical sociology contribute to the progress of sociology?

I propose that mathematical sociology should seek to formalize and explain the differentiation of roles in order for it to prevail in sociology. Mathematical sociology has established a niche within its parent discipline by demonstrating its explanatory power in such areas as the study of cooperation, trust, and social networks. However, it has not seriously examined the emergence, maintenance, and collapse of social order, a core sociological concept. Because social order is an abstract concept, I focus on social structure in this paper, role structure in particular, and propose an agent-based modeling framework to explain the differentiation of roles. If mathematical sociologists successfully develop models along these lines, they will succeed in explaining social order, and, as a result, mathematical sociology will proliferate in the field.