  In this brief report, we present a disordered version of recursive networks.
Depending on the structural parameters $u$ and $v$, the networks are either
fractals with a finite fractal dimension $d_{f}$ or transfinite fractals
(transfractal) with a infinite fractal dimension. The scaling behavior of
degree and dimensionality are studied analytically and by simulations, which
are found to be different from those in ordered recursive networks. The
transfractal dimension $\tilde{d}_f$, which is recently introduced to
distinguish the differences between networks with infinite fractal dimension,
scales as $\tilde{d}_f\sim \frac{1}{u+v-1}$ for transfractal networks.
Interestingly, the fractal dimension for fractal networks with $u=v$ is found
to approach 3 in large limit of $u$, which is thought to be the effect of
disorder. We also investigate the diffusion process on this family of networks,
and the scaling behavior of diffusion time is observed numercally as $\tau\sim
N^{(d_{f}+1)/d_{f}}$ for fractal networks and $\tau\sim \frac{1}{\tilde{d}_f}N$
for transfractal ons. We think that the later relation will give a further
understanding of transfractal dimension.
