  We study the effects of adding loops to a critical percolation cluster on the
diffusional, and equivalently, (scalar) elastic properties of the fractal
network. From the numerical calculations of the eigenspectrum of the transition
probability matrix, we find that the spectral dimension $d_s$ and the walk
dimension $d_w$ change suddenly as soon as the floppy ends of a critical
percolation cluster are connected together to form relatively large loops, and
that the additional inclusion of successively smaller loops only change these
exponents little if at all. This suggests that there is a new universality
class associated with the loop-enhanced percolation problem.
