TOPOLOGICAL MODELING AND ANALYSIS OF COMPLEX STOCHASTIC NETWORKS
Abstract
A complex network is based on interactions of multiple actors. For a systematic understanding of complex networks, pairwise interactions in the network may not always be sufficient to characterize important properties, such as information propagation rate, redundancies in the network, network reliability, resistance to attacks, etc. In such situations, the model of a network must allow for interactions between more than two nodes. This means the existence of edges of dimension larger than 1, and networks with such higher dimensional edges are called hypergraphs. A simplicial complex is a special kind of a hypergraph, based on topological ideas, particularly those of simplicial homology. This proposal places special emphasis on three different types of complex network models. The first is a geometric simplicial complex, in which interactions are based on distances between elements in the network; it concentrates on the geometric and topological aspects of the network. The second is the simplicial complex, in which interactions within a part of the network depend combinatorically on the dimension of that part, i.e. a combinatorial simplicial complex; a general complex of this type is the dynamic multi-parameter simplicial complex. The third is a scale-free simplicial complex, in which the topology of the network is maintained within a large range of scales. For each of these classes of models we study the large deviations of the network from its average behavior with the goal of understanding the robustness of the network. Our approach combines probabilistic, topological and statistical ideas, many of which come from the fundamental tools in Topological Data Analysis, such as persistent homology and persistent Betti numbers. We aim to make these tools useful and efficient in analysis of the high dimensional complex system we propose to study.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Mar 07, 2023
- Source ID
- FA95502210238
Entities
People
- Takashi Owada
Organizations
- Air Force Office of Scientific Research
- Purdue University
- United States Air Force