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 and allowing each layer to evolve according to its own topological temporal scale. When these time scales are the same the dynamics preserve the absorbing-fragmentation transition observed in a monolayer network at a critical value of the temporal scale that depends on interlayer connectivity. The time evolution equations obtained by pair approximation can be mapped to a coevolution voter model in a single layer with an effective average degree. 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 a critical level of interlayer coupling needed to prevent the fragmentation in a layer connected to a layer that does not fragment.