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

One of my papers came out in final form today. Here are the details.

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

Authors: Jake P. Taylor-King, David Basanta, S. Jonathan Chapman, and Mason A. Porter

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 it can have some more abstract internal property that describes something like a 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 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 smaller 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 bone metastasis.

"A Unified Theory of Randomness"

This article is very cool. It talks about some exciting work that follows up that of people like Wendelin Werner, Oded Schramm, Martin Hairer, and others. The younger author of the pair who are doing many of these great things seems like somebody who is going to be considered strongly for a Fields Medal. (The article doesn't mention this, but this does appear to be at that level.)

In fact, several recent Fields Medals have been awarded for work with deep relationships to statistical mechanics (and related topics like kinetic theory and probability theory), and it took quite some time for these to be considered "acceptable" mathematics topics.