  We analyze generalized space-time fractional motions on undirected networks
and lattices. The continuous-time random walk (CTRW) approach of Montroll and
Weiss is employed to subordinate a space fractional walk to a generalization of
the time-fractional Poisson renewal process. This process introduces a
non-Markovian walk with long-time memory effects and fat-tailed characteristics
in the waiting time density. We analyze `generalized space-time fractional
diffusion' in the infinite $\it d$-dimensional integer lattice $\it
\mathbb{Z}^d$. We obtain in the diffusion limit a `macroscopic' space-time
fractional diffusion equation. Classical CTRW models such as with Laskin's
fractional Poisson process and standard Poisson process which occur as special
cases are also analyzed. The developed generalized space-time fractional CTRW
model contains a four-dimensional parameter space and offers therefore a great
flexibility to describe real-world situations in complex systems.
