Statistical topology of radial networks: a case study of tree leaves

We describe a statistical method for characterizing the topological properties of radial networks based on the distribution of their 'energy' states, which are determined from the structural triangulation of the network. The partition function obtained from these energy states is used to calculate thermodynamic functions that embody the statistical properties of the network. The entropy, in particular, is a measure of the distribution of triangulated areas in the network, with a larger entropy corresponding to a higher symmetry in the branching structure. By varying the distribution parameter, which corresponds to an inverse 'temperature' in the statistical thermodynamic interpretation, we are able to vary the weight of the different generations of the network. This analysis identifies similar networks at their maturation state - the state when the system stops growing - as well as enabling the development of a network to be investigated. The latter feature is especially important for biological systems, where the details of the expansion of the network are not typically available. We illustrate our methodology with a model for the optimal transport of nutrients within tree leaves and show that statistical thermodynamic functions are capable of discriminating between various types of such radial networks. We conclude with a discussion about applications to the vasculature of the human placenta, which is our main motivation in developing this approach