  The effect of coordination on transport is investigated theoretically using
random networks of springs as model systems. An effective medium approximation
is made to compute the density of states of the vibrational modes, their energy
diffusivity (a spectral measure of transport) and their spatial correlations as
the network coordination $z$ is varied. Critical behaviors are obtained as
$z\to z_c$ where these networks lose rigidity. A sharp cross-over from a regime
where modes are plane-wave-like toward a regime of extended but
strongly-scattered modes occurs at some frequency $\omega^*\sim z-z_c$, which
does not correspond to the Ioffe-Regel criterion. Above $\omega^*$ both the
density of states and the diffusivity are nearly constant. These results agree
remarkably with recent numerical observations of repulsive particles near the
jamming threshold \cite{ning}. The analysis further predicts that the length
scale characterizing the correlation of displacements of the scattered modes
decays as $1/\sqrt{\omega}$ with frequency, whereas for $\omega<<\omega^*$
Rayleigh scattering is found with a scattering length $l_s\sim
(z-z_c)^3/\omega^4$. It is argued that this description applies to silica glass
where it compares well with thermal conductivity data, and to transverse
ultrasound propagation in granular matter.
