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We consider evolving networks in which each node can have various associatedproperties (a state) in addition to those that arise from network structure.For example, each node can have a spatial location and a velocity, or some moreabstract internal property that describes something like social trait. Edgesbetween nodes are created and destroyed, and new nodes enter the system. Weintroduce a "local state degree distribution" (LSDD) as the degree distributionat a particular point in state space. We then make a mean-field assumption andthereby derive an integro-partial differential equation that is satisfied bythe LSDD. We perform numerical experiments and find good agreement betweensolutions of the integro-differential equation and the LSDD from stochasticsimulations of the full model. To illustrate our theory, we apply it to asimple continuum model for osteocyte network formation within bones, with aview to understanding changes that may take place during cancer. Our resultssuggest that increased rates of differentiation lead to higher densities ofosteocytes but with a lower number of dendrites. To help provide biologicalcontext, we also include an introduction to osteocytes, the formation ofosteocyte networks, and the role of osteocytes in bona metastasis.

A Mean-Field Approach to Evolving Spatial Networks, with an Application to Osteocyte Network Formation