Fig. 1: Dynamics of the model.

Agents are represented by the nodes of a social structure composed by interacting groups of different sizes. The vocabulary of the agents--for simplicity containing at most only two names (or conventions) {A, B}--is reflected in the colours of the nodes as shown in the legend. At each interaction, a group is chosen at random (highlighted in yellow in the figure) together with a speaker (node 1), while the remaining nodes act as hearers. Here we illustrate the unanimity rule (see model definition). a The speaker chooses a name at random from its vocabulary (here, A), and communicates it to the rest of the group. Since A is present in the vocabularies of all the hearers (nodes 2 and 3 support A, while node 4 knows both names), the group can reach an agreement. b With probability β the group agrees on the chosen name, and all nodes involved immediately update their vocabulary to A, erasing B. With probability 1 − β instead the agreement does not happen. c In this case, the speaker selects A, but node 3 does not possess A in its vocabulary. d Thus, there cannot be agreement in the group. Nevertheless, all hearers update their vocabularies by adding the heard name, i.e., node 3 switches from A to A,B.