  In this paper, we propose a new model for continuous time opinion dynamics on
an evolving network. As opposed to existing models, in which the network
typically evolves by discretely adding or removing edges, we instead propose a
model for opinion formation which is coupled to a network evolving through a
system of ordinary differential equations for the edge weights. We interpret
each edge weight as the strength of the relationship between a pair of
individuals, with edges increasing in weight if pairs continually listen to
each others' opinions and decreasing if not. We investigate the impact of
various edge dynamics at different timescales on the opinion dynamic itself.
This is done partly through analytic results and partly through extensive
numerical simulations of two case studies: one using bounded confidence
interaction dynamics in the opinion formation process (as in the classical
Hegselmann-Krause model) and one using an exponentially decaying interaction
function. We find that the dynamic edge weights can have a significant impact
on the opinion formation process, since they may result in consensus formation
but can also reinforce polarisation. Overall, the proposed modelling approach
allows us to quantify and investigate how the network and opinion dynamics
influence each other.
