Modelling emergent networks: bridging discrete and continuum descriptions

The research project is about the modelling of emergent networks and particularly the description of the transition from a continuum to a network. Emergent networks are network structures that spontaneously appear in a previously spatially homogeneous medium. Most often emergent networks appear as the product of the interaction of agents through complex feedback reinforcement mechanisms and their structure is constantly evolving in time. Examples of such emergent networks are the vascular, lymphatic or neural networks in complex multicellular organisms, ant trail networks, social insect (ant, termite) nest structures, plant roots, leaf veins, fungus mycellium, or at a larger scale, geomorphological patterns such as estuaries or canyons, emergent cities such as slums in developing countries, etc. While network science has developed an impressive array of tools for networks having a definite topological structure such as graphs, weighted networks or random networks, or for phenomena occurring in networks such as flow in pipe networks, or TCP-IP dynamics on the Internet, there is little known about the mechanisms underlying the transition from a continuum to a network. The aim of this research project is to develop mathematical tools to investigate this transition and to apply it in a selection of case studies ranging from vascular network formation to erosion patterning through plant root dynamics. Understanding and controlling how emergent networks form has immense importance in biology (e.g. in cancer, cognition or tissue regeneration) and in social sciences. For instance, understanding better how blood capillary networks emerge will help fight angiogenesis (the recruitment of blood vessels by tumours) and treat cancer, a disease that hits more than 3 million people per year in Europe. In this project, a new representation of networks will be developed here coined 'field-based networks'. It considers a continuum director field (i.e. a vector field where at each point the vector has constant norm equal to one). Such vector fields carry singularities, such as point sources or sinks. Network nodes will be encoded in the singularities of this vector field while edges be specific vector field lines connecting singularities. This continuum director field can also be recovered from an Individual-Based Model consisting of discrete line segments, by averaging out the directions of the line segments in some neighbourhood of a given point. Therefore, this description of networks through a director field can be embedded into both discrete (Individual-Based) or continuum descriptions of a complex system. Often, the director field or the distribution of line segments are sufficient information and do not require network reconstruction. As connectivity is not hardwired (by contrast to earlier), it provides the required flexibility to allow for a transition from continuum to network. Some preliminary stages of this methodology have been previously applied to ant-trail formation, tissue self-organization and blood capillary formation. The goal of this project is to transform these preliminary demonstrators of the capabilities of smoothed networks into fully elaborated concepts and to apply them in a selection of case studies such as blood capillary formation, erosion patterning, and plant root development. These case studies will be conducted in collaboration with specialists such as biologists or geophysicists.

With the aim in mind of developing a new approach to better understand complex networks, we align our goals with those of the research area Complexity Sciences of the EPSRC. In particular, we will explore this methodology on vascularisation networks as well as on erosion patterns which are strongly linked with the following research areas: Fluid Dynamics and Aerodynamics, Mathematical Biology and Non-linear Systems.