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。该研究项目研究了这些网络的集合，即所谓的多层网络，其中每个单独的网络（层）具有最广泛的组织，但具有一些允许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，以及各种商品的国家之间的贸易网络。目前的项目提出了合并应用程序和理论的整体努力，并将开发技术，将适用于解决各种现实生活中的问题，涉及图形结构的数据。这项研究的结果将有利于许多领域的知识，依赖于分析的随机网络，其中层属于不同的群体：a）脑科学研究，提供工具，分析大脑网络和它们的变化在各种条件下; B）医学研究和实践，提供基于模型的解释是什么使大脑网络与正常的特定疾病不同; c）分子生物学，通过开发技术来分析与各种功能有关的蛋白质之间的酶影响; d）金融和国际关系，通过分析与各种模式相对应的世界贸易和金融网络; e）社会科学，通过分析与不同类型的社会联系有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充足的培训机会，包括指导博士，M.S.教授专题研究生课程，组织跨学科研讨会，促进跨学科研究和多样性。 更详细地说，该项目将研究复用网络模型，其中所有层都具有相同的节点集，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。该研究将建立在网络层中节点之间连接的概率矩阵遵循最通用的广义点积图（GDPG）模型的概念上。GDPG包括所有流行的块网络模型作为其特定情况。虽然已经有一些努力，以扩大GDPGs的多层场景，多层GDPG配方已被限制在非常严格的情况下，所有的网络是由相同的不变子空间。后者是配备SBM的多路复用网络的直接延伸，其中社区在所有层中都存在。上述公式的不足之处在于，它阻止了根据某些自然条件将网络的层划分为组。因此，必须将复用GDPG推进到层组嵌入到不同子空间中的情况。找到这些层的集群将允许提供基于模型的评估网络之间的差异对应于不同的条件。此外，GDPG将进一步推广到多重签名GDPG（SGDPG）网络设置，这允许更灵活地建模各种真实的生活网络。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大脑网络的分析。此外，该奖项旨在通过Oracle不等式和极大极小研究，为统计程序提供精确度保证。该奖项反映了NSF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。

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.