Statistical Inference for Multilayer Network Data with Applications

随机网络的分析是非常重要的，并被用于各种应用，如社会学，生物学，遗传学，生态学，信息技术和国家安全。网络对于描述节点之间的关系非常方便，这些节点可能代表社交网络中的人或一个人的大脑区域。大多数网络的特性之一是，它们可以被划分为具有不同特性和连接模式的社区。这些分区可以用来回答各种各样的问题，比如寻找紧密联系的社会群体，或者识别与疾病相关的大脑区域的连接模式。虽然最初的努力集中在对单个网络模型的分析上，但在过去的几年里，网络科学中最重要的方向之一已经转移到对单个网络集的研究，即所谓的多层网络，因为多层网络的通用性和使用这一概念可以解决的各种应用。本研究的目的是开发对此类网络进行理论和算法分析的工具。这些理论可以应用于分析癫痫手术可能影响的与语言相关的大脑网络。本研究将与Advent Health Hospital for Children的脑功能测绘和脑机接口实验室合作进行。该项目产生的技术可以应用于依赖于多层随机网络数据分析的各种领域：a)医疗实践，因为更好地了解与语言相关的大脑区域之间的连接将导致更安全有效的癫痫治疗方案；B)医学研究，为考虑与特定疾病相关的大脑区域之间联系的个体差异提供工具；C)脑科学研究，提供分析大脑网络及其变化的工具；D)分子生物学，通过提出分析与各种功能相关的蛋白质之间的酶影响的技术；E)统计遗传学，通过制定与几种疾病有关的基因网络同时研究的程序；F)国际关系和金融，通过分析与各种模式相对应的世界贸易和金融网络；G)社会科学，通过分析与各种类型的社会联系相关的社区的异同。资金还将用于培训劳动力，开展各种教育活动，促进跨学科研究和多样性。本提案的研究议程将大大推动非参数统计领域的发展，特别是新兴的网络数据分析领域。对多层网络兴趣的火花导致了关于该主题的一系列出版物。然而，这些出版物分为两种截然不同的类别：一种是应用驱动的论文，对结果没有理论保证；另一种是统计论文，提供了这些保证，但在非常严格的假设下。虽然在许多应用程序中，主要目标是确定不同层或层集中社区之间的差异，但统计论文完全关注所有层中社区相同的情况。由于缺乏相关的理论结果，在应用中，作者要么使用临时技术，要么被迫做出一个可疑的假设，即所有层的社区结构都是相同的。因此，迫切需要建立坚实的理论基础和开发高效的计算算法来分析具有不同社区结构的多层网络。特别是，本研究的目的是构建非参数技术，用于对多层网络进行估计和聚类，其中每层都遵循流行的随机块模型，并且社区结构在某些层中重合而在其他层中不同。此外，统计程序将通过oracle不等式和极小极大研究来补充精度保证。这将通过PI最近使用的现代代数技术的应用来完成。总之，该研究将通过开发非参数估计和聚类技术，显著拓宽适用于多层网络数据分析的方法库，这些技术需要很少的假设，在计算上可行，并且还伴随着理论上的精度保证。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Analysis of stochastic networks is extremely important and is used in a variety of applications such as sociology, biology, genetics, ecology, information technology and national security. Networks are very convenient for describing relationship between nodes that may represent people in a social network or brain regions of a person. One of the properties of the majority of networks is that they can be divided into communities with distinct properties and connection patterns. These partitions can be used for answering a variety of questions such as finding tightly connected social groups, or identifying brain regions' connection patterns associated with a disease. While initial efforts were focused on analysis of a single network model, in the last few years one of the most important directions in network science has shifted to the study of sets of individual networks, the so-called multilayer networks, due to both the versatility of the multilayer networks and the variety of applications that can be addressed using this concept. The objective of this research is to develop tools for theoretical and algorithmic analysis of such networks. The theories developed can be applied to analysis of speech-related brain networks that can be affected by epilepsy surgery. This research will be carried out in collaboration with the Functional Brain Mapping and Brain Computer Interface Lab of Advent Health Hospital for Children. The techniques resulting from this project could be applied in a variety of fields that rely on analysis of multilayer stochastic network data: a) medical practice, since a better understanding of connections between brain regions associated with speech will result in more safe and efficient epileptic treatment options; b) medical research, by providing tools for taking into account individual variations of connections between brain regions associated with particular diseases; c) brain science research, by providing tools for analysis of brain networks and their variations; d) molecular biology, by proposing techniques for analyzing the enzymatic influences between proteins related to various functions; e) statistical genetics, by developing procedures for simultaneous studies of gene networks related to several diseases; f) international relations and finance, by analyzing world trade and financial networks corresponding to various modalities; g) social sciences, by analyzing the similarities and the differences in communities related to various types of social connections. Funding will also be used for training work force by carrying out various educational activities, and promoting interdisciplinary research and diversity.The research agenda of this proposal will substantially advance the fields of non-parametric statistics in general, and the emerging field of network data analysis in particular. The spark of the interest in multilayer networks has led to a stream of publications on the subject. However, these publications fall into two very distinct categories: applications driven papers with no theoretical guarantees of the results and statistical papers where those guarantees are provided, but under very restrictive assumptions. While in many applications the main goal is to determine the differences between communities in different layers or sets of layers, the statistical papers focus entirely on the case where the communities are the same in all layers. Due to the absence of relevant theoretical results, in applications, the authors either utilize ad hoc techniques or are forced to make a questionable assumption that the community structure is the same for all the layers. For this reason, there is an overwhelming need for laying solid theoretical foundations and developing efficient computational algorithms for analysis of multilayer networks with diverse community structures. In particular, the objective of this research is the construction of non-parametric techniques that carry out estimation and clustering of multilayer networks where each layer follows the popular Stochastic Block Model and the community structures coincide for some layers and differ for the others. Furthermore, statistical procedures will be supplemented with the precision guarantees via oracle inequalities and minimax studies. This will be accomplished by application of modern algebraic techniques recently employed by the PI. In summary, the research will significantly broaden the arsenal of methods applicable to analysis of multilayer network data by developing techniques for non-parametric estimation and clustering that require few assumptions, are computationally viable, and are also accompanied by theoretical precision guarantees.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.