Matrix Functions and Network Analysis

网络是几个领域的重要研究课题，包括生物学、社会科学和安全。网络的两个经常被忽视的重要属性是它们是动态的(随时间变化)，并且具有在不同规模上相关的功能。因此，从动态和多尺度的角度来研究网络是很重要的。一个自然的模型是假设我们不只有一个网络，而是一个由一个参数(或可能更多)索引的网络家族。该参数可以跟踪随着时间推移的变化，或者当一个人将自己的注意力转移到不同的尺度时。在这个项目中，PI将开发分析工具和技术，提供有用的见解，但对于大型网络也是有效的计算，即使在处理参数的许多值时也是如此。他们集中在网络分析文献中的几个有趣的衡量标准上，比如中心性的衡量标准、枢纽和权威机构、好的广播者和好的接收者。对于较大的网络，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。它们中的两个是矩阵指数和预解式；它们可以被启发式地证明为通过连接它们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，以强调网络的一些特征。PI已开发出可应用于大类矩阵函数的近似计算和误差估计方法。他们将适应和发展近似计算的方法，并根据项目设置确定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，而其他仅涉及简单的矩阵汇总，如对角线条目，或行和列和。投资促进机构将设计方法进行分解，以便在参数接近时利用矩阵之间的接近，以减少计算负担，并更好地跟踪分解中随着参数变化而重要性上升或下降的部分。

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.