Modelling emergent networks: bridging discrete and continuum descriptions

该研究项目是关于紧急网络的建模，特别是描述从连续体到网络的过渡。涌现网络是指在先前的空间同质介质中自发出现的网络结构。突现网络通常是agent通过复杂的反馈强化机制相互作用的产物，其结构随时间不断演变。这种新兴网络的例子有复杂多细胞生物中的维管、淋巴或神经网络、蚂蚁踪迹网络、群居昆虫（蚂蚁、白蚁）巢结构、植物根、叶脉、真菌菌丝体，或者在更大的范围内，地貌模式，如河口或峡谷，新兴城市，如发展中国家的贫民窟等。虽然网络科学已经开发了一系列令人印象深刻的工具，用于具有明确拓扑结构的网络，如图、加权网络或随机网络，或用于网络中发生的现象，如管道网络中的流量，或互联网上的TCP-IP动态，但人们对从连续体到网络转换的机制知之甚少。本研究项目的目的是开发数学工具来研究这种转变，并将其应用于从维管网络形成到植物根系动力学侵蚀模式的一系列案例研究中。理解和控制突现网络如何形成在生物学（如癌症、认知或组织再生）和社会科学中具有巨大的重要性。例如，更好地了解毛细血管网络是如何形成的，将有助于对抗血管生成（肿瘤招募血管）和治疗癌症。在欧洲，每年有300多万人罹患癌症。在这个项目中，将开发一种新的网络表示形式，称为“基于现场的网络”。它考虑一个连续方向场（即一个向量场，在每个点上向量的常范数等于1）。这样的向量场带有奇点，例如点源或点汇。网络节点将编码在该向量场的奇异点中，而边是连接奇异点的特定向量场线。这种连续方向场也可以从由离散线段组成的基于个体的模型中恢复，方法是在给定点的某些邻域中平均线段的方向。因此，通过指挥域对网络的描述可以嵌入到复杂系统的离散（基于个体）或连续描述中。通常，厂长场或线段的分布就有足够的信息，不需要重建网络。由于连接性不是硬连接的（与以前相比），因此它提供了从连续体到网络转换所需的灵活性。这种方法的一些初步阶段以前已应用于抗trail形成，组织自组织和毛细血管形成。该项目的目标是将这些平滑网络能力的初步演示转化为充分阐述的概念，并将其应用于一系列案例研究，如毛细血管形成、侵蚀模式和植物根系发育。这些个案研究将与生物学家或地球物理学家等专家合作进行。我们的目标是开发一种新的方法来更好地理解复杂网络，我们的目标与EPSRC的复杂性科学研究领域保持一致。特别是，我们将在血管化网络以及侵蚀模式上探索这种方法，这些方法与以下研究领域密切相关：流体动力学和空气动力学，数学生物学和非线性系统。

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.