  To better understand the temporal characteristics and the lifetime of
fluctuations in stochastic processes in networks, we investigated diffusive
persistence in various graphs. Global diffusive persistence is defined as the
fraction of nodes for which the diffusive field at a site (or node) has not
changed sign up to time $t$ (or in general, that the node remained
active/inactive in discrete models). Here we investigate disordered and random
networks and show that the behavior of the persistence depends on the topology
of the network. In two-dimensional (2D) disordered networks, we find that above
the percolation threshold diffusive persistence scales similarly as in the
original 2D regular lattice, according to a power law $P(t,L)\sim t^{-\theta}$
with an exponent $\theta \simeq 0.186$, in the limit of large linear system
size $L$. At the percolation threshold, however, the scaling exponent changes
to $\theta \simeq 0.141$, as the result of the interplay of diffusive
persistence and the underlying structural transition in the disordered lattice
at the percolation threshold. Moreover, studying finite-size effects for 2D
lattices at and above the percolation threshold, we find that at the
percolation threshold, the long-time asymptotic value obeys a power-law
$P(t,L)\sim L^{-z\theta}$ with $z\simeq 2.86$ instead of the value of $z=2$
normally associated with finite-size effects on 2D regular lattices. In
contrast, we observe that in random networks without a local regular structure,
such as Erd\H{o}s-R\'enyi networks, no simple power-law scaling behavior exists
above the percolation threshold.
