On the Herdability of Linear Time-Invariant Systems with Special Topological Structures

In this paper, we investigate the herdability property, namely the capability of a system to be driven towards the (interior of the) positive orthant, for linear time-invariant state-space models. Herdability of certain matrix pairs (A,B), where A is the adjacency matrix of a multi-agent network, and B is a selection matrix that singles out a subset of the agents (the "network leaders"), is explored. The cases when the graph associated with A, G(A), is directed and clustering balanced (in particular, structurally balanced), or it has a tree topology and there is a single leader, are investigated.Comment: Provisionally accepted in Automatica, currently under review. arXiv admin note: substantial text overlap with arXiv:2108.0157

A control theoretic perspective on social networks

This thesis discusses the application of control theory to the study of complex networks, drawing inspiration from the behavior of social networks. There are three topic areas covered by the thesis. The first area considers the ability to control a dynamical system which evolves over a network. Specifically, this thesis introduces a network controllability notion known as herdability. Herdability quantifies the ability to encourage general behavioral change in a system via a set-based reachability condition, which describes a class of desirable behaviors for the application of control in a social network setting. The notion is closely related to the classical notion of controllability, however ensuring complete controllability of large complex networks is often unnecessary for certain beneficial behaviors to be achieved. The basic theory of herdability is developed in this thesis. The second area of study, which builds directly on the first, is the application of herdability to the study of complex networks. Specifically, this thesis explores how to make a network herdable, an extension of the input selection problem which is often discussed in the context of controllability. The input selection problem in this case considers which nodes to select to ensure the maximal number of nodes in the system are herdable. When there are multiple single node sets which can be used to make a system completely herdable, a herdability centrality measure is introduced to differentiate between them. The herdability centrality measure, a measure of importance with respect to the ability to herd the network with minimum energy, is compared to existing centrality measures. The third area explores modeling the spread of the adoption of a beneficial behavior or an idea, in which the spread is encouraged by the action of a social network. A novel model of awareness-coupled epidemic spread is introduced, where agents in a network are aware of a virus (here representing something which should be spread) moving through the network. If the agents have a high opinion of the virus, they are more likely to adopt it. The behavior of this viral model is considered both analytically and in simulation.Ph.D