Absorbing and shattered fragmentation transitions in multilayer coevolution.

We introduce a coevolution voter model in a multilayer by coupling a fraction of nodes across two network layers (the degree of multiplexing) and allowing each layer to evolve according to its own topological temporal scale. When these time scales are the same, the time evolution equations can be mapped to a coevolution voter model in a single layer with an effective average degree. Thus the dynamics preserve the absorbing-fragmentation transition at a critical value that increases with the degree of multiplexing. When the two layers have different topological time scales, we find an anomalous transition, named shattered fragmentation, in which the network in one layer splits into two large components in opposite states and a multiplicity of isolated nodes. We identify the growth of the number of components as a signature of this anomalous transition. We also find the critical level of interlayer coupling needed to prevent the fragmentation in a layer connected to a layer that does not fragment.