  It has been shown recently that a specific class of path-dependent stochastic
processes, which reduce their sample space as they unfold, lead to exact
scaling laws in frequency and rank distributions. Such Sample Space Reducing
processes (SSRP) offer an alternative new mechanism to understand the emergence
of scaling in countless processes. The corresponding power law exponents were
shown to be related to noise levels in the process. Here we show that the
emergence of scaling is not limited to the simplest SSRPs, but holds for a huge
domain of stochastic processes that are characterized by non-uniform prior
distributions. We demonstrate mathematically that in the absence of noise the
scaling exponents converge to $-1$ (Zipf's law) for almost all prior
distributions. As a consequence it becomes possible to fully understand
targeted diffusion on weighted directed networks and its associated scaling
laws law in node visit distributions. The presence of cycles can be properly
interpreted as playing the same role as noise in SSRPs and, accordingly,
determine the scaling exponents. The result that Zipf's law emerges as a
generic feature of diffusion on networks, regardless of its details, and that
the exponent of visiting times is related to the amount of cycles in a network
could be relevant for a series of applications in traffic-, transport- and
supply chain management.
