  Motivated by the idea that some characteristics are specific to the relations
between individuals and not of the individuals themselves, we study a prototype
model for the dynamics of the states of the links in a fixed network of
interacting units. Each link in the network can be in one of two equivalent
states. A majority link-dynamics rule is implemented, so that in each dynamical
step the state of a randomly chosen link is updated to the state of the
majority of neighboring links. Nodes can be characterized by a link
heterogeneity index, giving a measure of the likelihood of a node to have a
link in one of the two states. We consider this link-dynamics model on fully
connected networks, square lattices and Erd \"os-Renyi random networks. In each
case we find and characterize a number of nontrivial asymptotic configurations,
as well as some of the mechanisms leading to them and the time evolution of the
link heterogeneity index distribution. For a fully connected network and random
networks there is a broad distribution of possible asymptotic configurations.
Most asymptotic configurations that result from link-dynamics have no
counterpart under traditional node dynamics in the same topologies.
