Collaborative Research: Random Dynamics on Networks

运输和分配网络有多种形式，从动物心血管和呼吸系统到通信和工业基础设施。实际问题比比皆是：局部尖峰的预测，灌注的估计，以及结构变化的影响，如血管闭塞。这种现象的复杂性可以用著名的Braess悖论来说明：增加交通网络的链接可能不会改善系统的运行！尽管最近取得了一些成功，但我们对网络流的理解通常仅限于小的确定性问题，而大多数应用程序对应于大的不确定性问题。该项目的目标是改进生物和技术运输和扩散网络的预测。例如，如果其中一条颈动脉变窄或堵塞，人们能否预测脑血流量将如何受到影响？血管系统会允许重新布线吗？如果是，概率有多大，速度有多快？本研究项目的主要挑战是不确定性的存在。对于许多应用程序，只有关于系统的部分信息可用。例如，特定船只的大小甚至存在可能是不确定的，或者路由器的状态未知。对这些问题的分析需要创造新的数学工具和数值方法来描述不确定性如何在庞大而复杂的网络中传播。要构建的计算工具将提供有关困难、昂贵或不可能测量的现象的信息，这些信息通常是概率性的。

Transport and distribution networks come in a number of forms, from animal cardiovascular and respiratory systems to communication and industrial infrastructures. Practical issues abound: prediction of local spikes, estimation of perfusion, and impact of structural changes such as vessel occlusion. The complexity of such phenomena can be illustrated by the well-known Braess' paradox: adding links to a transportation network might not improve the operation of the system! In spite of recent successes, our understanding of network flows is usually limited to small deterministic problems, while most applications correspond to large uncertain ones. The goal of this project is to enable improved predictions in biological and technological transport and diffusion networks. For instance, can one predict how cerebral blood flow will be affected if one of the carotids becomes narrow or blocked? Will the vasculature allow for re-routing? If so, with what probability and how fast? The main challenge in this research project is the presence of uncertainties. For many applications, only partial information about the systems is available. For instance the size or even the presence of a specific vessel might be uncertain or the status of a router unknown. The analysis of such problems requires the creation of novel mathematical tools and numerical methods to describe how uncertainties propagate through vast and complex networks. The computational tools to be constructed will provide information, usually probabilistic in nature, regarding phenomena that are difficult, expensive, or impossible to measure.