A mathematical model of dynamic social networks

A mathematical model for dynamic networks is developed that is based on closed, rather than open, sets. For a social network, it seems appropriate to use a neighborhood concept to establish these sets. We then define a rigorous concept of continuous change, and show that it shares some of the properties associated with the continuity of the calculus. We demonstrate that continuity is local in nature, in that if the network change is discontinuous, it will be so at a single point and the discontinuity will be apparent in that point’s immediate neighborhood. Necessary and sufficient criteria for continuity are provided when the change involves only the addition, or deletion, of individual nodes or connections (edges). To illustrate large scale continuous change, we choose a practical process which reduces a complex network to its fundamental cycles, in the course of which most triadically closed subportions are removed. Finally, we explore several variants of the neighborhood concept, and prove that a rigorous notion of fuzzy closure can be defined.