  Coordinated human behaviour takes place within a diverse range of social
organisational structures, which can be thought of as power structures with
"managers" who influence "subordinates". A change in policy in one part of the
organisation can cause cascades throughout the structure, which may or may not
be desirable. As organisations change in size, complexity and structure, the
system dynamics also change. Here, we consider majority rule dynamics on
organisations modelled as hierarchical directed graphs, where the directed
edges indicate influence. We utilise a topological measure called the trophic
incoherence parameter, q, which effectively gauges the stratification of power
structure in an organisation. We show that this measure bounds regimes of
behaviour. There is fast consensus at low q (e.g. tyranny), slow consensus at
mid q (e.g. democracy), and no consensus at high q (e.g. anarchy). These
regimes are investigated analytically, numerically and empirically with diverse
case studies in the Roman Army, US Government, and a healthcare organisation.
Our work demonstrates the usefulness of the trophic incoherence parameter when
considering models of social influence dynamics, with widespread consequences
in the design and analysis of organisations.
