  We present a general framework for the study of coevolution in dynamical
systems. This phenomenon consists of the coexistence of two dynamical processes
on networks of interacting elements: node state change and rewiring of links
between nodes. The process of rewiring is described in terms of two basic
actions: disconnection and reconnection between nodes, both based on a
mechanism of comparison of their states. We assume that the process of rewiring
and node state change occur with probabilities Pr and Pc respectively,
independent of each other. The collective behavior of a coevolutionary system
can be characterized on the space of parameters (Pr, Pc). As an application,
for a voterlike node dynamics we find that reconnections between nodes with
similar states lead to network fragmentation. The critical boundaries for the
onset of fragmentation in networks with different properties are calculated on
this space. We show that coevolution models correspond to curves on this space
describing functional relations between Pr and Pc. The occurrence of a
one-large-domain phase and a fragmented phase in the network is predicted for
diverse models, and agreement is found with some earlier results. The
collective behavior of system is also characterized on the space of parameters
for the disconnection and reconnection actions. In a region of this space, we
find a behavior where different node states can coexist for very long times on
one large, connected network.
