  The propagation of signalling molecules within cellular networks is affected
by network topology, but also by the spatial arrangement of cells in the
networks. Understanding the collective reaction--diffusion behaviour in space
of signals propagating through cellular networks is an important consideration
for example for regenerative signals that convey positional information. In
this work, we consider stochastic and deterministic versions of random walk
models of signalling molecules propagating and reacting within one-dimensional
spatial networks with arbitrary node placement and connectivity. By taking a
continuum limit of the random walk models, we derive an inhomogeneous
reaction--diffusion--advection equation, where diffusivity and advective
velocity depend on local node density and connectivity within the network. Our
results show that large spatial variations of molecule concentrations can be
induced by heterogeneous node distributions. Furthermore, we find that noise
within the stochastic random walk model is directly influenced by node density.
We apply our models to consider signal propagation within the osteocyte network
of bone, where signals propagating to the bone surface regulate bone formation
and resorption processes. We investigate signal-to-noise ratios for different
damage detection scenarios and show that the location of perturbations to the
network can be detected by signals received at the network boundaries.
