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The propagation of signalling molecules within cellular networks is affectedby network topology, but also by the spatial arrangement of cells in thenetworks. Understanding the collective reaction--diffusion behaviour in spaceof signals propagating through cellular networks is an important considerationfor example for regenerative signals that convey positional information. Inthis work, we consider stochastic and deterministic random walk models ofsignalling molecules propagating and reacting within one-dimensional spatialnetworks with arbitrary node placement and connectivity. By taking a continuumlimit of the random walk models, we derive an inhomogeneousreaction--diffusion--advection equation, where diffusivity and advectivevelocity depend on local node density and connectivity within the network. Ourresults show that large spatial variations of molecule concentrations can beinduced by heterogeneous node distributions. Furthermore, we find that noisewithin the stochastic random walk model is directly influenced by node density.We apply our models to consider signal propagation within the osteocyte networkof bone, where signals propagating to the bone surface regulate bone formationand resorption processes. We investigate signal-to-noise ratios for differentdamage detection scenarios and show that the location of perturbations to thenetwork can be detected by signals received at the network boundaries.

Random walk models for the propagation of signalling molecules in one-dimensional spatial networks and their continuum limit