Abstract: We have investigated the influence of the average degree $\langle k \rangle$ of network on the location of an order--disorder transition in opinion dynamics. For this purpose, a variant of majority rule (VMR) model is applied to Watts--Strogatz (WS) small-world networks and Barabási--Albert (BA) scale-free networks which may describe some non-trivial properties of social systems. Using Monte Carlo simulations, we find that the order--disorder transition point of the VMR model is greatly affected by the average degree $\langle k \rangle$ of the networks; a larger value of $\langle k \rangle$ results in a more ordered state of the system. Comparing WS networks with BA networks, we find WS networks have better orderliness than BA networks when the average degree $\langle k \rangle$ is small. With the increase of $\langle k \rangle$, BA networks have a more ordered state. By implementing finite-size scaling analysis, we also obtain critical exponents $\beta/\nu$, $\gamma/\nu$ and $1/\nu$ for several values of average degree $\langle k \rangle$. Our results may be helpful to understand structural effects on order--disorder phase transition in the context of the majority rule model.

Key words: complex networks, majority rule