Multiplex Generalized Dot Product Graph networks: theory and applications

随机网络模型出现在各种应用中，包括遗传学、蛋白质组学、医学成像、国际关系、脑科学等等。这项研究项目考察了这些网络的集合，即所谓的多层网络，其中每个单独的网络(层)具有尽可能广泛的组织，但又具有一些允许进行有意义的随机推理的共同特征。例子包括不同个体的大脑网络、蛋白质相互作用网络以及不同商品的国家之间的贸易网络。目前的项目是一项融合应用和理论的整体努力，并将开发适用于解决涉及图形结构数据的各种现实问题的技术。这项研究的结果将有益于许多依赖随机网络分析的知识领域，其中层属于不同的组：a)脑科学研究，提供分析脑网络及其在不同条件下的变化的工具；b)医学研究和实践，提供基于模型的解释，解释是什么使与特定疾病相关的脑网络不同于正常；c)分子生物学，通过开发技术，分析与不同功能相关的蛋白质之间的酶影响；d)金融与国际关系，通过分析世界贸易和金融网络，对应于各种形式；e)社会科学，分析与不同类型的社会联系相关的社区的相似性和差异。此外，该项目将通过各种教育活动提供大量的培训机会，包括指导博士、硕士和本科生，教授专题研究生课程，组织跨学科研讨会，以及促进跨学科研究和多样性。更详细地说，该项目将研究多路复用网络模型，其中所有层都具有相同的节点集，并且节点之间的所有边都在层内绘制，这在上面讨论的应用中是真实的。这项研究将建立在网络各层节点之间的连接概率矩阵遵循最通用的广义点乘积图(GDPG)模型的概念上。GDPG将所有流行的区块网络模型作为其特殊情况包括在内。虽然已经有一些努力将GDPG扩展到多层场景，但多层GDPG公式仅限于所有网络由相同的不变子空间生成的非常有限的情况。后者是配备SBM的多路传输网络的直接延伸，其中社区持续存在于所有层。上述公式的不足之处在于，它阻止根据某些自然条件找到网络的层到组的划分。因此，将复合GDPG推进到层组嵌入不同的子空间的情况下是势在必行的。找到这些层的集群将允许对对应于不同条件的网络之间的差异提供基于模型的评估。此外，GDPG还将进一步推广到多路符号GDPG(SGDPG)网络环境，从而可以更灵活地对各种现实网络进行建模。该项目的目标是为多层GDPG模型提供各种扩展，开发可扩展的算法和理论工具进行分析，并将这些发现应用于脑网络的分析。此外，它的目标是通过甲骨文不平等和极小极大研究来补充统计程序的精确度保证。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Stochastic network models appear in a variety of applications, including genetics, proteomics, medical imaging, international relationships, brain science and many more. This research project examines collections of such networks, the so called multilayer network, where each of the individual networks (layers) have the broadest possible organization and yet possess some common features that allow meaningful stochastic inference. Examples include brain networks of different individuals, protein interaction networks, and trade networks between countries in various commodities. The current project presents an integral effort of merging applications and theory and will develop techniques that will be applicable for solution of a variety of real-life problems involving graph-structured data. Results of this research will be beneficial for many domains of knowledge that rely on analysis of stochastic networks where layers belong to different groups: a) brain science research by providing tools for analysis of brain networks and their variations under various conditions; b) medical research and practice by providing model-based explanations on what makes brain networks associated with particular diseases different from normal; c) molecular biology by developing techniques for analyzing the enzymatic influences between proteins related to various functions; d) finance and international relations by analyzing world’s trade and financial networks corresponding to various modalities; e) social sciences by analyzing the similarities and the differences in communities related to different types of social connections. In addition, the project will provide ample opportunities for training through various educational activities, including mentoring Ph.D., M.S. and undergraduate students, teaching a Special Topics graduate courses, organizing interdisciplinary seminars, and promoting interdisciplinary research and diversity. In more detail, the project will study the multiplex network model where all layers have the same set of nodes, and all the edges between nodes are drawn within layers, which is true in the applications discussed above. The research will be built on the notion that the matrices of probabilities of connections between nodes in layers of the network follow the most versatile Generalized Dot Product Graph (GDPG) model. GDPG includes all popular block network models as its particular cases. Although there have been some efforts to extend GDPGs to multilayer scenarios, the multilayer GDPG formulations have been limited to the very restrictive case where all networks are generated by the same invariant subspace. The latter is a direct extension of the SBM-equipped multiplex network where communities persist in all layers. The deficiency of the above formulation is that it prevents finding partitions of layers of the network into groups according to some natural condition. Hence, it is imperative to advance the multiplex GDPG to the case where groups of layers are embedded into different subspaces. Finding those clusters of layers will allow to provide model-based assessments of the differences between networks corresponding to different conditions. In addition, GDPG will be further generalized to the multiplex Signed GDPG (SGDPG) network setting, which allows more flexible modeling of a variety of real life networks. The objective of the project is to provide various extensions to the multilayer GDPG models, to develop scalable algorithms and theoretical tools for their analysis, and to apply those findings to analysis of brain networks. Furthermore, it aims to supplement statistical procedures with precision guarantees via oracle inequalities and minimax studies.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.