  The weighted planar stochastic (WPS) lattice introduces a topological
disorder that emerges from a multifractal structure. Its dual network has a
power-law degree distribution and is embedded in a two-dimensional space,
forming a planar network. We modify the original recipe to construct WPS
networks with degree distributions interpolating smoothly between the original
power-law tail, $P(q)\sim q^{-\alpha}$ with exponent $\alpha\approx 5.6$, and a
square lattice. We analyze the role of disorder in the modified WPS model,
considering the critical behavior of the contact process. We report a critical
scaling depending on the network degree distribution. The scaling exponents
differ from the standard mean-field behavior reported for CP on
infinite-dimensional (random) graphs with power-law degree distribution.
Furthermore, the disorder present in the WPS lattice model is in agreement with
the Luck-Harris criterion for the relevance of disorder in critical dynamics.
However, despite the same wandering exponent $\omega=1/2$, the disorder effects
observed for the WPS lattice are weaker than those found for uncorrelated
disorder.
