Dynamics of Influence on Hierarchical Structures

TL;DR

This paper analyzes how influence propagates in hierarchical structures with fixed top influence, revealing conditions for consensus and highlighting the bottom class's pivotal role in the overall state.

Contribution

It provides a mathematical framework for understanding influence dynamics in hierarchical networks, emphasizing the bottom class’s impact on the system's state.

Findings

Unanimity is achieved if the bottom class adopts the head node’s state.

In case of discord, the average state tends toward the bottom class's state.

Mass media can synchronize the hierarchy effectively.

Abstract

Dichotomous spin dynamics on a pyramidal hierarchical structure (the Bethe lattice) are studied. The system embodies a number of \emph{classes}, where a class comprises of nodes that are equidistant from the root (head node). Weighted links exist between nodes from the same and different classes. The spin (hereafter, \emph{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 bottom-most 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 top-most and bottom-most classes. Surprisingly, in the case of discord between the head node and the bottom-most classes, the average state over all nodes inclines towards that of the bottom-most class regardless of the link weights and intra-class configurations. Hence the role of the bottom-most class is signified.