  Scale-free (SF) networks and small world networks have been found to occur in
very diverse contexts. It is this striking universality which makes one look
for widely applicable mechanisms which lead to the formation of such networks.
In this letter we propose a new mechanism for the construction of SF networks:
Evolving networks as interaction networks of systems which are distinguished by
their stability if perturbed out of equilibrium. Stability is measured by the
largest real part of any eigenvalue of a matrix associated with the graph. We
extend the model to weighted directed networks and report power law behaviour
of the link strength distribution of the weighted graphs in the SF regime. The
model we propose for the first time relates SF networks to stability properties
of the underlying dynamical system.
