  Random walks are one of the best investigated dynamical processes on graphs.
A particularly fascinating phenomenon is the scaling relationship of
fluctuations $\sigma $ with the average flux $\langle f \rangle $. Here we
analyze how network topology and nodes with finite capacity lead to deviations
from a simple scaling law $\sigma \sim \langle f \rangle ^\alpha$. Sources of
randomness are the random walk itself (internal noise) and the fluctuation of
the number of walkers (external noise). We obtained exact results for the
extreme case of a star network which are indicative of the behavior of large
scale systems with a broad degree distribution.The latter are subsequently
studied using Monte Carlo simulations. We find that the network heterogeneity
amplifies the effects of external noise. By computing the `effective' scaling
of each node we show that multiple scaling relationships can coexist in a graph
with a heterogeneous degree distribution at an intermediate level of external
noise. Finally, we analyze the effect of a finite capacity of nodes for random
walkers and find that this also can lead to a heterogeneous scaling of
fluctuations.
