Analysis and stability of dynamical system models over networks

离散模型在广泛的科学应用中很受欢迎，如基因调控网络、社会接触图上的流行病、基因功能推断的算法以及许多数值方法。图动力系统(GDSS)的框架以一种数学上精确的方式自然地捕获了许多并行和顺序的迭代模型和算法，并且易于进行严格的分析。这一提议的基本主题是进一步发展网络上这些时间离散动力系统的理论框架和数学结果。数学本身应该很有趣，但也有助于推进涉及离散模型和迭代计算算法的应用。它将允许从计算系统生物学更好地洞察算法的验证方面和一般特性，这应该会导致在这一领域和更远的领域建立改进的模型和算法。随着当前科学计算和应用的发展，这一非传统的应用数学领域迫切需要进一步的发展。此外，这些分析和结果构成了随机GDSS理论的自然基础，随机GDSS是许多应用领域非常感兴趣的一种结构，正在得到普及。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很受欢迎，而且许多算法都建立在这些模型的基础上。这样的算法通常很难被理解，算法验证通常是通过数值实验的方式进行的，这些实验几乎没有阐明算法的基本分析性质。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)与由此产生的动态联系起来，特别是在稳定性分析方面。在这个项目中，调查者将主要关注两个具体的系统方面，功能结构和更新顺序。这实际上归结为两类研究人员有专长的图动态系统：(1)布尔网络(同步)和(2)顺序动态系统(异步)。对于这些领域中的每一个领域，研究人员都有一个具体的理论发展计划，以及来自计算系统生物学的具体应用。对于第一个领域，将研究新引入的嵌套分析函数深度的概念，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，目的是利用这一点来开发可应用于迭代算法(如基因注释法)的定量稳定性度量。

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.