Impact of the network structure on plurality rule opinion dynamics

People often face the challenge of choosing amongst
different options with similar attractiveness,
such as when choosing a parlamentary candidate,
a movie or buying a product in the supermarket.
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In order to study the distribution of preferences in such situations,
it is adequate to consider a plurality rule opinion dynamics on small-world networks,
where different options are available as well as the undecided state.
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Depending on the structural properties of the network and on
the initial conditions, the final distribution can range from
a wide distribution of preferences to consensus,
or also give rise to situations where indecision dominates.
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Decision making governed by the plurality rule
may yield ties, contributing to overchoice stagnation.
This is more pronounced in random networks than in scale-free ones.
Also, in the latter networks consensus is facilitated.
However, it is not still clear which are the contributions
of randomness and preferential attachment for those effects.
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In order to investigate that issue,
we develop the plurality rule dynamics in generalized Watts-Strogatz networks.
We start from regular networks and rewire their connections
to obtain different degrees of randomicity $p$.
Moreover, the rewired nodes can be selected by means of a parameter
$w$ which allows to tune from random selection to preferential attachment.
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Through a phase diagram in $(p,w)$ space,
we characterize the conditions that lead to the different final configurations.