With the advent of multiple large-scale networks' such as biological networks, power networks, and many social networks' the control of these networks has become an increasingly important topic. The research objective of this proposal is to address fundamental questions that arise naturally in network control theory from a probabilistic perspective. In particular, the investigators will address questions such as: how likely is it that a broadcast signal can control a large network. Can media shape public opinion. Can a single leader move public opinion from one state to any arbitrary desired state? The investigators will use state-of-the-art tools in probability and random matrix theory to study such fundamental problems. One emphasis of the proposal is the synergistic relationship between probability and control theory. To encourage and assist other researchers, the investigators plan survey works that will emphasis particularly useful probabilistic tools and techniques that one needs to address these types of problems. Based on this duel relationship, many of the problems discussed by the investigators have the potential to lead to interesting new questions and directions in random matrix theory, which has many applications outside of mathematics including statistics, mathematical physics, combinatorics, and theoretical computer science. The proposal also has several educational components including support for graduate study in engineering and mathematics and promoting participation of minorities in higher education in science and engineering.
The proposed research aims to address fundamental problems and questions in network control theory using recently developed tools from probability and random matrix theory. Specifically, the investigators plan to analyze the general phenomenon that most systems "even those of a very discrete nature" are controllable. One specific motivating example is the question of controllability for Laplacian-based leader-follower dynamics on a large network. The investigators plan to analyze such systems by studying the case when the underlying graph or network is random. To address such problems, the investigators plan to use and extend a number of diverse and involved techniques from probability and random matrix theory including analytic techniques (e.g. resolvent techniques, concentration of measure), algebraic tools (e.g. linear algebra), and probabilistic methods (e.g. Littlewood-Offord theory). Additionally, the investigators also expect the proposed work to be useful to mathematicians, as it will highlight precisely what types of problems and applications exist in other scientific areas. Besides for the case of Laplacian-based leader-follower dynamics, the investigators also plan to address systems formed from directed and weighted graphs as well as a minimum-energy control problem for systems whose parameters are random.
