Matrix Functions and Network Analysis

网络是生物学、社会科学和安全等多个领域的重要研究课题。网络的两个重要特性经常被忽视，即它们是动态的（随时间变化），并且具有在不同尺度上相关的特征。因此，从动态和多尺度的角度研究网络是非常重要的。一个自然的模型是假设我们不仅有一个网络，而是一个网络族，由一个参数（或可能更多）索引。这个参数可以随着时间的推移，或者随着一个人将焦点改变到不同的尺度而跟踪变化。在这个项目中，PI将开发分析工具和技术，提供有用的见解，但也可以有效地计算大型网络，即使在处理参数的许多值时。他们专注于网络分析文献中的几个感兴趣的指标，如中心性，枢纽和权威，良好的广播和良好的接收器的措施。对于更大的网络，这些度量的研究变得更加困难。发现网络重要特征的方法通常基于矩阵函数。其中两个是矩阵指数和预解式;这些可以通过连接节点的路径的加权和来测量节点之间的连通性，并且可以很容易地设计这些函数的修改以强调网络的某些特征。PI已经开发了近似计算和误差估计的方法，可以应用于广泛的矩阵函数。他们将调整和开发近似计算方法，并确定与项目设置相关的这些方法的误差估计。网络特征的一些量化涉及矩阵分解，如特征值或奇异值分解，而其他量化仅涉及简单的矩阵求和，如对角项或行和列和。PI将设计用于执行分解的方法，这些方法在参数接近时利用矩阵之间的接近性，以减少计算负担，并更好地跟踪随着参数变化而重要性上升或下降的分解部分。

Networks are important subjects of study in several fields, including biology, social sciences, and security. Two important properties of networks that are often overlooked are that they are dynamic (change over time), and have features that are relevant at different scales. Thus, it is important to study networks from a dynamic and multiscale point of view. A natural model is to assume that we have not just one network, but a family of networks, indexed by one parameter (or may be more). That parameter can track changes as time passes, or as one change one's focus to different scales. In this project the PIs will develop analysis tools and techniques that provide useful insights, but are also efficiently computable for large networks, even when dealing with many values for the parameter. They concentrate on several measures of interest in the network analysis literature, like measures of centrality, hubs and authorities, good broadcasters and good receivers. The study of these measures becomes more difficult for larger networks.Methods for discovering important features of a network are often based on matrix functions. Two of them are the matrix exponential and the resolvent; these can be heuristically justified as measuring connectivity between nodes by weighted sums of paths connecting them, and modifications of these functions can be easily devised to stress some features of the network over others. The PIs have developed methods for approximate computation and error estimation that can be applied to a wide class matrix functions. They will adapt and develop methods of approximate computation, and determine error estimates for these methods, as relevant to the project setting. Some quantifications of network features involve matrix decompositions, like the eigenvalue or singular value decompositions, while others involve only simple matrix summaries, like the diagonal entries, or row and column sums. The PIs will devise approaches for carrying out decompositions that take advantage of the closeness between the matrices when the parameters are close, in order to reduce the computational burden and to better keep track of the parts of the decomposition that rise or fall in importance as the parameter changes.