Phase Transitions in Spatial Networks as a Model of Cellular Symbiosis

Random Geometric or Spatial Graphs, are well studied models of networks where spatial embedding is an important consideration. However, the dynamic evolution of such spatial graphs is less well studied, at least analytically. Indeed when distance preference is included the principal studies have largely been simulations. An important class of spatial networks has application in the modeling of cell symbiosis in certain tumors, and, when modeled as a graph naturally introduces a distance preference characteristic of the range of cell to cell interaction. In this paper we present theoretical analysis, and, experimental simulations of such graphs, demonstrating that distance functions that model the mixing of the cells, can create phase transitions in connectivity, and thus cellular interactions. This is an important result that could provide analytical tools to model the transition of tumors from benign to malignant states, as well as a novel class of spatial network evolution.