  Randomness is known to affect the dynamical behaviour of many systems to a
large extent. In this paper we investigate how the nature of randomness affects
the dynamics in a zero temperature quench of Ising model on two types of random
networks. In both the networks, which are embedded in a one dimensional space,
the first neighbour connections exist and the average degree is four per node.
In the random model A, the second neighbour connections are rewired with a
probability $p$ while in the random model B, additional connections between
neighbours at Euclidean distance $l ~ (l >1)$ are introduced with a probability
$P(l) \propto l^{-\alpha}$. We find that for both models, the dynamics leads to
freezing such that the system gets locked in a disordered state. The point at
which the disorder of the nonequilibrium steady state is maximum is located.
Behaviour of dynamical quantities like residual energy, order parameter and
persistence are discussed and compared. Overall, the behaviour of physical
quantities are similar although subtle differences are observed due to the
difference in the nature of randomness.
