  Collective decision making processes lie at the heart of many social,
political and economic challenges. The classical voter model is a
well-established conceptual model to study such processes. In this work, we
define a new form of adaptive (or co-evolutionary) voter model posed on a
simplicial complex, i.e., on a certain class of hypernetworks/hypergraphs. We
use the persuasion rule along edges of the classical voter model and the
recently studied re-wiring rule of edges towards like-minded nodes, and
introduce a new peer pressure rule applied to three nodes connected via a
2-simplex. This simplicial adaptive voter model is studied via numerical
simulation. We show that adding the effect of peer pressure to an adaptive
voter model leaves its fragmentation transition, i.e., the transition upon
varying the re-wiring rate from a single majority state into to a fragmented
state of two different opinion subgraphs, intact. Yet, above and below the
fragmentation transition, we observe that the peer pressure has substantial
quantitative effects. It accelerates the transition to a single-opinion state
below the transition and also speeds up the system dynamics towards
fragmentation above the transition. Furthermore, we quantify that there is a
multiscale hierarchy in the model leading to the depletion of 2-simplices,
before the depletion of active edges. This leads to the conjecture that many
other dynamic network models on simplicial complexes may show a similar
behaviour with respect to the sequential evolution of simplicies of different
dimensions.
