CIF: Small: Data Analysis in Higher-Order Complex Networks

通信、生物和社交网络是我们日常生活的重要组成部分。可以被建模为网络的系统的例子包括为我们提供蜂窝通信的无线基础设施和允许我们彼此保持联系的在线社交平台。在这些网络中，代理（如无线站或人）被建模为节点，这些代理之间的交互（如数据传输或友谊）发生在网络的边缘。这种方法的一个固有的局限性是假设全球网络动态出现完全从代理之间的成对交互。然而，存在大量与成对相互作用不足以建模元素的系统相关的数据。这个项目将从成对网络的分析出发，发展从多关系连接中学习所需的理论。具体来说，该项目将产生从数据中推断这些复杂连接的方法和算法，并利用推断的关系结构来更好地理解所表示的系统。此外，该项目将通过为本科生和研究生提供包容性的指导、新颖的教学和令人兴奋的研究机会，对莱斯大学的学生产生真实的和持久的影响。将通过组织辅导和特别会议促进研究的传播。最后，为了扩大西班牙裔在计算方面的参与，将在当地高中举办一系列与网络相关的双语讲习班，该项目的主要研究目标是开发一种原则性的理论来处理和学习高阶网络上定义的数据。更准确地说，数据被建模为定义在单纯复形的结构拓扑元素上的信号，单纯复形是超图的一个特定子类。这使得来自代数拓扑的概念的实现能够在这些结构中定义来自信号处理的概念，例如频率和滤波，以及来自深度学习的概念，例如卷积神经网络和深度生成模型。为了实现主要研究目标，提出了三个重点，其中共同点是将信号处理和深度学习技术与拓扑数据域相结合。首先，（非高阶）图将被视为数据域，但重点将集中在感兴趣的数据位于边缘（如质量流，能量流或信息流）而不是图的节点上的情况。第二个重点将推导出Hodge Laplacian的谱特征与单纯复形的拓扑特征之间的关系，以奠定高阶网络的信号处理和深度学习的基础。上述方向利用图和高阶网络的结构来更好地处理定义在其上的数据。然而，这的先决条件是首先能够访问网络。通常情况下，人们只能观察高阶结构的某些部分中的数据，并希望推断结构的其余部分，以最终将其用于处理。因此，高阶网络拓扑推理的新工具将在第三个研究重点中开发。从长远来看，该研究者旨在建立一个基于信号处理和机器学习与拓扑概念相结合的丰富的数据科学框架。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。

Communication, biological, and social networks are a key part of our everyday lives. Examples of systems that can be modeled as networks include the wireless infrastructure providing us with cellular communications and the online social platforms allowing us to stay in contact with each other. In these networks, agents, such as wireless stations or people, are modeled as nodes, and interactions between these agents (such as data transmissions or friendships) take place over the edges of the network. An inherent limitation of this method is the assumption that global network dynamics emerge exclusively from pairwise interactions between agents. Nevertheless, there exists a wealth of data associated with systems where pairwise interactions are insufficient modeling elements. This project will depart from the analysis of pairwise networks and develop the theory needed to learn from multi-relational connections. Specifically, this project will result in methods and algorithms to infer these complex connections from data and to leverage the inferred relational structures to better understand the systems that are being represented. Moreover, the project will achieve a real and lasting impact on students at Rice University via inclusive mentoring, novel teaching, and exciting research opportunities for the undergraduate and graduate populations. Research dissemination will be promoted through the organization of tutorials and special sessions. Finally, with the objective of broadening the participation of Hispanics in computing, a series of bilingual network-related workshops will be delivered at local high-schools.The primary research goal of this project is to develop a principled theory to process and learn from data defined on higher-order networks. More precisely, the data is modeled as signals defined on structural topological elements of simplicial complexes, which are a specific subclass of hypergraphs. This enables the implementation of concepts from algebraic topology to define in these structures notions from signal processing, such as frequency and filtering, and from deep learning, such as convolutional neural networks and deep generative models. In order to achieve the primary research goal, three thrusts are proposed, where the common denominator is the integration of signal processing and deep learning techniques with topological data domains. First, a (non higher-order) graph will be considered as the data domain but the thrust will focus on the case where the data of interest is located on the edges (such as flows of mass, energy, or information) as opposed to the nodes of the graph. The second thrust will derive relations between the spectral features of the Hodge Laplacian and topological characteristics of simplicial complexes to lay out the fundamentals of signal processing and deep learning for higher-order networks. The aforementioned directions leverage the structure of graphs and higher-order networks to better process data defined on them. However, a prerequisite for this is to have access to the network in the first place. Oftentimes, one only gets to observe data in some portions of the higher-order structure and wants to infer the rest of the structure to ultimately use it for processing. Accordingly, novel tools for higher-order network topology inference will be developed in the third research thrust. In the longer run, the investigator seeks to establish a fertile data science framework grounded on the combination of signal processing and machine learning with topological notions.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.