  We study the role of connectivity on the linear and nonlinear elastic
behavior of amorphous systems using a two-dimensional random network of
harmonic springs as a model system. A natural characterization of these systems
arises in terms of the network coordination relative to that of an isostatic
network $\delta z$; a floppy network has $\delta z<0$, while a stiff network
has $\delta z>0$. Under the influence of an externally applied load we observe
that the response of both floppy and rigid network are controlled by the same
critical point, corresponding to the onset of rigidity. We use numerical
simulations to compute the exponents which characterize the shear modulus, the
amplitude of non-affine displacements, and the network stiffening as a function
of $\delta z$, derive these theoretically and make predictions for the
mechanical response of glasses and fibrous networks.
