We study the evolution of binary opinions on a simple adaptive network of $N$ nodes. At each time step, a randomly selected node updates its state (“opinion”) according to the majority opinion of the nodes that it is linked to; subsequently, all links are reassigned with probability $\tilde{p}$ $\left(\tilde{q}\right)$ if they connect nodes with equal (opposite) opinions. In contrast to earlier work, we ensure that the average connectivity (“degree”) of each node is independent of the system size (“intensive”), by choosing $\tilde{p}$ and $\tilde{q}$ to be of $O\left(1/N\right)$. Using simulations and analytic arguments, we determine the final steady states and the relaxation into these states for different system sizes. We find two absorbing states, characterized by perfect consensus, and one metastable state, characterized by a population split evenly between the two opinions. The relaxation time of this state grows exponentially with the number of nodes, $N$. A second metastable state, found in the earlier studies, is no longer observed.

• Received 15 August 2010

DOI:https://doi.org/10.1103/PhysRevE.82.066104