How fungi or plants invade a medium, how sexually transmitted diseases spread over a population, how cancer tumors grow in human bodies, how communication routes densify, are questions that may seem to refer to quite unrelated problems. However, the structure, dynamics and shape of the underlying network may rely on very similar models. The nature of such networks is not uniquely defined: some examples are informational networks (of relation between individuals, citation graphs,...), technological (power grids, public transportation, computer network,...), or biological (vascular, biochemical, neural network,...). In all the aforementioned examples, transformation arises from individuals, be it the development of a new connection between existing entities, as it often appears in neurons, or the introduction of a new individual in the system. All these contributions sum up to the evolution of the network as a unit on the macroscopic level. Modeling of such intricate processes ranges from simple explanatory toy-models to more realistic approaches, which need to be able to capture modifications at different scales. This can be achieved by linking microscopic objects, which describe individuals, with their collective mean behavior. Techniques borrowing from statistical physics for the analysis of nonlinear, non-equilibrium physical systems in the study of such collective behavior are of increasing use, in e.g. social, economical or biological systems.
The expansion of such networks may also be hindered by internal or external constraints which can significantly affect the observed results and patterns. When explicitly including the spatial dimension, the models considered may provide a pertinent description of the interaction processes at the small (micro) scale as well as the large (macro)scale featuring the emerging behavior, possibly under the form of a (thin) propagating front. The modeling and analysis of such dynamical processes within a multi-scale framework, where the different granularities of the system are to be considered, is a complex research field, that requires involving various disciplines.
In this project, we will specifically address the modeling and analysis of the expanding interconnected hyphal network (the vegetative filaments produced to form the \href{https://en.wikipedia.org/wiki/Mycelium}{\cyan{mycelium}}) of the fungus Podospora anserina.