  A certain degree of inhibition is a common trait of dynamical networks in
nature, ranging from neuronal and biochemical networks, to social and
technological networks. We study here the role of inhibition in a
representative dynamical network model, characterizing the dynamics of random
threshold networks with both excitatory and inhibitory links. Varying the
fraction of excitatory links has a strong effect on the network's population
activity and its sensitivity to perturbation. The average degree $K$, known to
have a strong effect on the dynamics when small, loses its influence on the
dynamics as its value increases. Instead, the strength of inhibition is a
determinant of dynamics and sensitivity here, allowing for criticality only in
a specific corridor of inhibition. This criticality corridor requires that
excitation dominates, while the balance region corresponds to maximum
sensitivity to perturbation. We develop mean-field approximations of the
population activity and sensitivity and find that the network dynamics is
independent of degree distribution for high $K$. In a minimal model of an
adaptive threshold network we demonstrate how the dynamics remains robust
against changes in the topology. This adaptive model can be extended in order
to generate networks with a controllable activity distribution and specific
topologies.
