Link-based social dynamics in complex networks: time evolution and heterogeneous asymptotic states

Motivated by the idea that some characteristics are specific of the
relations between individuals and not of the individuals themselves, we
model them as states on the links in a network of interacting agents.
Each link in the network can be in one of two equivalent states and, in
each dynamical step a link is chosen, and this one takes the state of
the majority of links surrounding it. We can characterize the nodes by
the link heterogeneity index, which measures the tendency of a node to
have links in one state or the other. On a fully connected network and a
square lattice we fully characterize the non-trivial asymptotic
configurations accessible from random initial conditions. For
Erdös-Renyi random networks we also characterize the asymptotic
configurations and the mechanisms leading to them. All these non-trivial
asymptotic configurations are quite heterogeneous for a fully connected
network and random networks. Furthermore those configurations are not
stable for the node majority rule, which highlights the importance of
modeling link-based dynamics. Once the asymptotic states are understood
we can explain the evolution of the distributions of link heterogeneity
indices on the different networks.