  Excitation waves are studied on trees and random networks of coupled active
elements. Undamped propagation of such waves is observed in those networks. It
represents an excursion from the resting state and a relaxation back to it for
each node. However, the degrees of the nodes influence drastically the
dynamics. Excitation propagates more slowly through nodes with larger degrees
and beyond some critical degree waves lose their stability and disappear. For
regular trees with a fixed branching ratio, the critical degree is determined
with an approximate analytical theory which also holds locally for the early
stage of excitation spreading in random networks.
