Modelling emergent networks: bridging discrete and continuum descriptions

新兴网络建模：桥接离散和连续描述

• 批准号: 1943921
• 金额: --
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1943921
• 关键词: Modelling; emergent; networks; bridging; discrete

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.

该研究项目是关于新兴网络的建模，特别是从连续体到网络的过渡描述。涌现网络是自发地出现在先前空间均匀介质中的网络结构。大多数情况下，新兴网络是代理人通过复杂的反馈强化机制相互作用的产物，它们的结构随着时间的推移而不断演变。这种新兴网络的例子是复杂多细胞生物体中的血管、淋巴或神经网络，蚂蚁踪迹网络，社会昆虫网络，（蚂蚁、白蚁）巢穴结构、植物根系、叶脉、真菌菌丝体，或在更大的尺度上，河口或峡谷等地貌形态，发展中国家贫民窟等新兴城市，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，例如图、加权网络或随机网络，或者用于网络中出现的现象，例如管道网络中的流动，或TCP-IP动态在互联网上，很少有人知道的机制，从一个连续的过渡到一个网络。本研究项目的目的是开发数学工具来研究这种转变，并将其应用于选择的案例研究，从维管网络的形成，侵蚀图案，通过植物根系动力学。理解和控制涌现网络的形成方式在生物学（例如癌症、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如何出现将有助于对抗血管生成（肿瘤招募血管）和治疗癌症，这种疾病每年在欧洲袭击300多万人。在这个项目中，一个新的网络表示将在这里创造“基于场的网络”。它考虑了一个连续指向矢场（即一个向量场，其中在每一点上向量的范数都等于1）。这样的向量场带有奇点，例如点源或点汇。网络节点将被编码在这个向量场的奇点中，而边是连接奇点的特定向量场线。这个连续指向矢场也可以从由离散线段组成的基于个体的模型中恢复，通过在给定点的某个邻域中平均线段的方向。因此，这种通过导向场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，指向矢场或线段的分布是足够的信息，并且不需要网络重构。由于连接不是硬连线的（与以前相比），它提供了从连续体到网络过渡所需的灵活性。这种方法的一些初步阶段以前已被应用于蚂蚁踪迹形成，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功能的初步演示转化为充分阐述的概念，并将其应用于选择的案例研究，如毛细血管形成，侵蚀模式和植物根系发育。这些案例研究将与生物学家或生物学家等专家合作进行。为了开发更好地理解复杂网络的新方法，我们将我们的目标与EPSRC复杂性科学研究领域的目标相结合。特别是，我们将探讨这种方法的血管化网络以及侵蚀模式，这是密切相关的以下研究领域：流体动力学和空气动力学，数学生物学和非线性系统。