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

## TL;DR

This paper develops a mean-field framework for evolving spatial networks with node properties, deriving an integro-PDE for the local state degree distribution, and applies it to osteocyte network formation, revealing how differentiation rates affect network density and dendrite count.

## Contribution

It introduces a novel mean-field approach for networks with evolving node properties and derives an integro-PDE for the local state degree distribution, validated through simulations.

## Key findings

- Higher differentiation rates increase osteocyte density.
- Increased differentiation leads to fewer dendrites per osteocyte.
- The model aligns well with stochastic simulation results.

## Abstract

We consider evolving networks in which each node can have various associated properties (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 more abstract internal property that describes something like social trait. Edges between nodes are created and destroyed, and new nodes enter the system. We introduce a "local state degree distribution" (LSDD) as the degree distribution at a particular point in state space. We then make a mean-field assumption and thereby derive an integro-partial differential equation that is satisfied by the LSDD. We perform numerical experiments and find good agreement between solutions of the integro-differential equation and the LSDD from stochastic simulations of the full model. To illustrate our theory, we apply it to a simple continuum model for osteocyte network formation within bones, with a view to understanding changes that may take place during cancer. Our results suggest that increased rates of differentiation lead to higher densities of osteocytes but with a lower number of dendrites. To help provide biological context, we also include an introduction to osteocytes, the formation of osteocyte networks, and the role of osteocytes in bona metastasis.