We search for conditions under which a characteristic time scale for ordering dynamics towards either of two absorbing states in a finite complex network of interactions does not exist. With this aim, we study random networks and networks with mesosc
ale community structure built up from randomly connected cliques. We find that large heterogeneity at the mesoscale level of the network appears to be a sufficient mechanism for the absence of a characteristic time for the dynamics. Such heterogeneity results in dynamical metastable states that survive at any time scale.
We address the role of community structure of an interaction network in ordering dynamics, as well as associated forms of metastability. We consider the voter and AB model dynamics in a network model which mimics social interactions. The AB model inc
ludes an intermediate state between the two excluding options of the voter model. For the voter model we find dynamical metastable disordered states with a characteristic mean lifetime. However, for the AB dynamics we find a power law distribution of the lifetime of metastable states, so that the mean lifetime is not representative of the dynamics. These trapped metastable states, which can order at all time scales, originate in the mesoscopic network structure.
