  People often face the challenge of choosing among different options with
similar attractiveness. To study the distribution of preferences that emerge in
such situations, a useful approach is to simulate opinion dynamics on top of
complex networks, composed by nodes (individuals) and their connections
(edges), where the state of each node can be one amongst several opinions
including the undecided state. We use two different dynamics rules: the one
proposed by Travieso-Fontoura (TF) and the plurality rule (PR), which are
paradigmatic of outflow and inflow dynamics, respectively. We are specially
interested in the impact of the network randomness on the final distribution of
opinions. For that purpose, we consider Watts-Strogatz networks, which possess
the small-world property, and where randomness is controlled by a probability
$p$ of adding random shortcuts to an initially regular network. Depending on
the value of $p$, the average connectivity $\langle k \rangle$, and the initial
conditions, the final distribution can be basically (i) consensus, (ii)
coexistence of different options, or (iii) predominance of indecision. We find
that, in both dynamics, the predominance of a winning opinion is favored by
increasing the number of reconnections (shortcuts), promoting consensus. In
contrast to the TF case, in the PR dynamics, a fraction of undecided nodes can
persist in the final state. In such cases, a maximum number of undecided nodes
occurs within the small-world region, due to ties in the decision group.
