Effects of long-range links on metastable states in a dynamic interaction network

We introduce a model for random-walking nodes on a periodic lattice, where the dynamic interaction network is defined from local interactions and E randomly-added long-range links. With periodic states for nodes and an interaction rule of repeated averaging, we numerically find two types of metastable states at low- and high-E limits, respectively, along with consensus states. If we apply this model to opinion dynamics, metastable states can be interpreted as sustainable diversities in our societies, and our result then implies that, while diversities decrease and eventually disappear with more long-range connections, another type of states of diversities can appear when networks are almost fully-connected.