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.
