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The analysis of single particle trajectories plays an important role in elucidating dynamics within complex environments such as those found in living cells. However, the characterization of intracellular particle motion is often confounded by confinement of the particles within non-trivial subcellular geometries. Here, we focus specifically on the case of particles undergoing Brownian motion within a tubular network, as found in some cellular organelles. An unraveling algorithm is developed to uncouple particle motion from the confining network structure, allowing for an accurate extraction of the diffusion coefficient, as well as differentiating between Brownian and fractional Brownian dynamics. We validate the algorithm with simulated trajectories and then highlight its application to an example system: analyzing the motion of membrane proteins confined in the tubules of the peripheral endoplasmic reticulum in mammalian cells. We show that these proteins undergo diffusive motion with a well-characterized diffusivity. Our algorithm provides a generally applicable approach for disentangling geometric morphology and particle dynamics in networked architectures.