Geometric Network Theory with Applications to Heterogeneous Data Analysis
Abstract
In our proposed ARO work, we will develop a geometric approach for the study of a number of key properties of complex networks including their information content, which will allow us to fuse data from heterogeneous sources. This will be based on several extensions of Ricci curvature, a central concept of Riemannian Geometry, to discrete spaces in order to infer robustness and functionality in the networks of interest. We compare three possible discrete notions of Ricci curvature, the Olivier-Ricci (OR) curvature, the Bakry-Emery Ricci (BER) curvature, and the Forman-Ricci (FR) curvature, on some key real-world networks. While their exact relationship is still unknown, they do yield qualitatively similar results on our networks of interest. All these notions of ``discrete-space curvature are initially defined on undirected positively weighted graphs, although Forman-Ricci curvature admits a straightforward extension to the directed case. Accordingly, we will consider extensions of the other models as well. However, in real world applications, and specifically in those of possible interest to the U.S. Army, networks are considerably more complex. The corresponding graph structure may encode relations, transport and communication links, affinity and community membership among the entities/agents that are represented by the graph nodes. In addition, data attached to the nodes may be {em vectorial} and {em heterogeneous}, reflecting a multitude of features. The functionality of such networks depends in rather complicated ways on the geometry and quality of linkages (graph edges and corresponding weights). All these effect the transference of resources or information in different ways and at different time scales. Thus, in our research program we embark on developing a geometric framework that encompasses needed generalizations, i.e., a framework that can accommodate vector-valued features, graph edge directionality, and sign dependent interactions (reflecting activator/repressor qualities in transcription networks, positive/negative correlation between time series supported on the nodes, and so on). The framework aims at extending Riemannian-like constructs, and specifically, the {em Ricci curvature} as it uniquely captures features that relate to connectivity, functionality and robustness. We will highlight the fact that changes in Ricci curvature are directly correlated to changes in entropy, and thus gives a direct measure of how information flows in a dynamically evolving environment. In addition we explore dynamic notions of curvature that reveal salient features of a network that relate to time constants in propagation of resources or information across as well as the detection of communities in the given network. Finally, discrete Ricci curvature will be shown to be directly applicable to problems in machine learning. In particular, we will describe the curvature forest, where a vectorial version of curvature computed on vector-valued data may be employed as a set of features for random forest modeling.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Oct 12, 2022
- Source ID
- W911NF2210292
Entities
People
- Allen Tannenbaum
Organizations
- Army Contracting Command
- Research Foundation for the State University of New York
- United States Army