  We study the dynamical rheology of spring networks with a percolation model
constructed by bond dilution in a two-dimensional triangular lattice.
Hydrodynamic interactions are implemented by a Stokesian viscous coupling
between the network nodes and a uniformly deforming liquid. Our simulations
show that in a critical connectivity regime, these systems display weak power
law rheology in which the complex shear modulus scales with frequency as G^* ~
(i * omega)^Delta where Delta = 0.41, in discord with a mean field prediction
of Delta = 1/2. The weak power law rheology in the critical regime can be
understood from a simple scaling relation between the macroscopic rheology and
the nonaffine strain fluctuations, which diverge with vanishing frequency for
isostatic networks. We expand on a dynamic effective medium theory, showing
that it quantitatively describes the rheology of a diluted triangular lattice
far from isostaticity; although the EMT correctly predicts the scaling form for
the rheology of near-isostatic networks, there remains a quantitative disparity
due to the mean-field nature of the EMT. Surprisingly, by connecting this
critical scaling of the rheology with that of the strain fluctuations, we find
that the dynamical behavior of disordered spring networks is fully determined
by the critical exponents that govern the behavior of elastic network in the
absence of viscous interactions.
