Analysis and stability of dynamical system models over networks

网络动力系统模型的分析和稳定性

• 批准号: 1211691
• 负责人: Matthew Macauley
• 金额: $ 8.5万
• 依托单位: Clemson University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211691&HistoricalAwards=false
• 关键词: Analysis; stability; dynamical; system; models

Discrete models are popular in a wide variety of scientific applications, such as gene-regulatory networks, epidemics over social contact graphs, algorithms for gene function inference, and many numerical methods. The framework of graph dynamical systems (GDSs) naturally captures many parallel and sequential, iterative models and algorithms in a mathematically precise way that is amenable to rigorous analysis. The underlying theme of this proposal is the further development of a theoretical framework and body of mathematical results of these time-discrete dynamical systems over networks. The mathematics should be of interest in its own right, yet help advance applications involving discrete models and iterative computational algorithms. It will allow for better insight into validation aspects and general properties of algorithms from computational systems biology, which should lead to the construction of improved models and algorithms in this field and beyond. With the current developments in scientific computing and applications, this non-traditional area of applied mathematics is in a keen need of further advances. Moreover, the analysis and the results constitute natural underpinnings for the theory of stochastic GDSs, a construct of great interest to many application areas that is gaining popularity. The time-discrete dynamical systems over networks that the investigator studies arise in systems biology and scientific computing from gene networks, to data mining, to epidemiology. Not only are these systems popular as models, but many algorithms are built on top of these models. Such algorithms are frequently poorly understood, and algorithm validation is usually approached by way of numerical experiments that shed little light on the fundamental analytic properties of the algorithms. One basic question is how to relate the constituents of the system (e.g., function type, network structure, update mechanism) to the resulting dynamics, especially with regard to stability analysis. In this project, the investigator will primarily focus on two specific system aspects, function structure and update sequence. This effectively comes down to two classes of graph dynamical systems in which the investigator has expertise: (1) Boolean networks (synchronous), and (2) sequential dynamical systems (asynchronous). For each of these areas, the investigator has a specific plan for the theory to be developed, as well as a specific application from computational systems biology in mind. For this first area, the newly introduced concept of nested canalyzing depth of functions will be studied, and with the application of reverse engineering gene networks in mind. For the second area, the dependence on the update sequence and the initial states on the phase space structure will be studied, with the goal of using this to develop quantitative stability measures that can be applied to iterative algorithms such as gene annotative methods.

离散模型在各种科学应用中很受欢迎，例如基因调控网络，社会联系图上的流行病，基因功能推断算法以及许多数值方法。图动态系统（GDS）的框架自然地以数学上精确的方式捕获了许多并行和顺序的迭代模型和算法，这些模型和算法可以进行严格的分析。这个建议的基本主题是进一步发展的理论框架和身体的数学结果，这些时间离散动力系统的网络。数学本身应该是有趣的，但有助于推进涉及离散模型和迭代计算算法的应用。它将允许更好地了解验证方面和计算系统生物学算法的一般属性，这将导致在这一领域和超越的改进模型和算法的建设。随着当前科学计算和应用的发展，这个非传统的应用数学领域迫切需要进一步的发展。此外，分析和结果构成了自然的基础理论的随机GDS，一个建设的极大兴趣，许多应用领域，越来越受欢迎。研究人员研究的网络上的时间离散动力系统出现在系统生物学和科学计算中，从基因网络到数据挖掘，再到流行病学。这些系统不仅作为模型而流行，而且许多算法都建立在这些模型之上。这样的算法往往是知之甚少，算法验证通常是接近的数值实验，揭示了算法的基本分析性质的光。一个基本问题是如何将系统的组成部分（例如，功能类型、网络结构、更新机制）对结果动态的影响，特别是关于稳定性分析。在这个项目中，研究者将主要关注两个具体的系统方面，功能结构和更新顺序。这有效地归结为两类图动力系统，其中研究者具有专业知识：（1）布尔网络（同步）和（2）序列动力系统（异步）。对于这些领域中的每一个，研究人员都有一个具体的理论开发计划，以及一个具体的应用程序，从计算系统生物学的头脑。对于这第一个领域，将研究新引入的嵌套canalysing功能深度的概念，并考虑到反向工程基因网络的应用。对于第二个领域，将研究更新序列和初始状态对相空间结构的依赖性，目标是使用此来开发可应用于迭代算法（如基因注释方法）的定量稳定性度量。