Systemic risk and topology

Schenck1312071 这个项目的重点是一个新的应用程序，最近开发的工具在计算拓扑的某些问题的风险，在金融网络。 复杂的问题往往受益于被视为在一个新的光;研究人员使用持续同源性来研究金融交易网络的行为。 持久同调关注一族单纯复形D（t）如何随真实的参数t变化。 通常，从对应于t=0的离散数据集（点云数据）开始，然后随着t的变化，点被以原始点为中心的半径为t的球取代。 因此，改变t会产生一个过滤的单纯复形族，即相关拓扑空间的Rips-Vietoris复形。 在交易网络的背景下，参数t代表保证金; t的低值对应于允许高杠杆交易。 该项目的目的是了解与交易网络相关的D（t）拓扑结构的变化如何以某种方式对应于系统性崩溃。 简而言之，调查人员探索导致局部事件导致全球蔓延的情况。 理解相互关联性的问题是当今的首要挑战之一。 连接性无处不在;互连通常使系统能够充分发挥其潜力并增强其鲁棒性。 连通性的另一面是，它为意外的紧急行为（“黑天鹅”事件）甚至系统性崩溃提供了途径。 世界充斥着数据：我们如何从中提取意义（并理解连通性）？ 一个答案是通过代数拓扑学的数学学科，这是一个工具，提取复杂的对象或空间的研究成一个简单的形式。 例如，我们如何区分橙子和甜甜圈？ 显而易见的答案是，甜甜圈有一个“洞”;代数拓扑使自然的直觉（这是不那么自然，在更高的维度！）严谨 特别是，代数拓扑是建立在探索局部和全局结构之间的关系，这使得它成为研究金融交易网络的理想工具。 扩散的问题，有一个汇合的地方/全球过渡和一个大的实验数据集已经引起了应用代数拓扑学领域，产生一个系统的方式来展开连接。 调查人员通过这种展开的视角解释了金融网络崩溃的各种概念。 这项工作的重点是清算网络，它对交易和负债进行编码，并将应用拓扑学中开发的工具用于揭示此类网络中的互连。 特别是，研究人员研究当地事件如何传播：在什么情况下，传染和系统性崩溃的结果，从一个地方的事件？

Schenck1312071 This project is focused on a novel application of recently developed tools in computational topology to certain problems of risk in financial networks. Complex problems often benefit from being viewed in a new light; the investigators use persistent homology to study behavior of financial trading networks. Persistent homology focuses on how a family of simplicial complexes D(t) vary with a real parameter t. Typically one starts with a discrete data set (point cloud data) corresponding to t=0, and then as t varies, points are replaced by balls of radius t, centered at the original point. Thus varying t yields a filtered family of simplicial complexes, the Rips-Vietoris complex of the associated topological space. In the context of trading networks, the parameter t represents margin; a low value for t corresponds to allowing highly leveraged trading. The aim of the project is to understand how changes in the topology of the D(t) associated to a trading network correspond in some way to systemic collapse. Put simply, the investigators explore the circumstances that cause a local event to cause global contagion. The problem of understanding interconnectedness is one of today's premier challenges. Connectivity is ubiquitous; interconnections often allow a system to be tapped to its full potential and enhance its robustness. The flip side of connectivity is that it gives rise to pathways for unexpected emergent behavior ("black swan" events) and even systemic collapse. The world is awash in data: how can we extract meaning (and understand connectivity) from it? One answer is via the mathematical discipline of algebraic topology, which is a tool to distill the study of complex objects or spaces into a simple form. For example, how can we distinguish between an orange and a donut? The obvious answer is that the donut has a "hole"; algebraic topology makes the natural intuition (which is not so natural in higher dimensions!) rigorous. In particular, algebraic topology is built to probe the relationship between local and global structures; this makes it an ideal tool for studying financial trading networks. The proliferation of problems where there is a confluence of local/global transitions and a large experimental data set has given rise to the field of applied algebraic topology, yielding a systematic way to unfold connections. The investigators interpret various notions of collapse of financial networks via this unfolding perspective. The work focuses on clearing networks, which encode trades and liabilities, and bring the tools developed in applied topology to bear on unfolding the interconnections in such networks. In particular, the investigators study how local events propagate: under what circumstances does contagion and systemic collapse result from a local event?