On the Analysis of the DeGroot-Friedkin Model with Dynamic Relative Interaction Matrices

This paper analyses the DeGroot-Friedkin model for evolution of the individuals’ social powers in a social network when the network topology varies dynamically (described by dynamic relative interaction matrices). The DeGroot-Friedkin model describes how individual social power (self-appraisal, self-weight) evolves as a network of individuals discuss opinions on a sequence of issues. We seek to study dynamically changing relative interactions because interactions may change depending on the issue being discussed. Specifically, we study relative interaction matrices which vary periodically with respect to the issues. This may reflect a group of individuals, e.g. a government cabinet, that meet regularly to discuss a set of issues sequentially. It is shown that individuals’ social powers admit a periodic solution. Initially, we study a social network which varies periodically between two relative interaction matrices, and then generalise to an arbitrary number of relative interaction matrices.