title: Dynamics of influence on hierarchical structures.
abstract: Dichotomous spin dynamics on a pyramidal hierarchical structure (the Bethe lattice) are studied. The system embodies a number of classes, where a class comprises nodes that are equidistant from the root (head node). Weighted links exist between nodes from the same and different classes. The spin (hereafter state) of the head node is fixed. We solve for the dynamics of the system for different boundary conditions. We find necessary conditions so that the classes eventually repudiate or acquiesce in the state imposed by the head node. The results indicate that to reach unanimity across the hierarchy, it suffices that the bottommost class adopts the same state as the head node. Then the rest of the hierarchy will inevitably comply. This also sheds light on the importance of mass media as a means of synchronization between the topmost and bottommost classes. Surprisingly, in the case of discord between the head node and the bottommost classes, the average state over all nodes inclines towards that of the bottommost class regardless of the link weights and intraclass configurations. Hence the role of the bottommost class is signified.
authors.affiliation: Department of Electrical and Computer Engineering, McGill University, Montréal, Québec, Canada.
authors.forename: Babak
authors.initials: B
authors.lastname: Fotouhi
authors.forename: Michael G
authors.initials: MG
authors.lastname: Rabbat