Network design for optimal performance under consensus dynamics

Abstract

In this thesis we study the problem of optimizing consensus networks through network design. The performance of each network is determined by its topology. We consider how the performance of consensus networks can be optimized by selecting additional edges or leaders.

We close with a summary of our results and a discussion of remaining open questions and possible avenues for future research.

In our final set of results we study the problem of leader selection in noise-free social networks. We develop two new performance measures, both of which quantify the diversity of the follower's opinions. We then propose the problem of placing leaders with opinion one, given pre-existing leaders with opinion zero, so that the diversity of the followers' opinions is maximized. We find optimal solutions to this problem in a few simple graphs when the leader sets are both of size one.

We then extend our study of noisy leader-follower dynamics to include higher order systems. We prove under what conditions the higher order systems are stable, and then derive expressions for the coherence in second, third, and fourth order systems. We prove that the set functions based on the coherence are submodular and study the performance of those set functions through a few analytical examples.

We next consider first order systems with noisy leader-follower dynamics, where a subset of leader nodes dictate the desired state of the network and the remaining follower nodes execute noisy consensus dynamics. Both leader and follower agents in this setting are subject to stochastic external disturbances.The problem of interest is selecting the leader set that minimizes the system's coherence. We show that the performance measure for the leader selection problem can be expressed as a submodular set function over the nodes in the network. We then use this result to analyze the performance of two greedy, polynomial-time algorithms for leader selection, showing that the leader sets produced by the greedy algorithms are within provable bounds of optimal. We also consider a special case of a network of vehicle platoons where the noisy leaders are at the front of each platoon. Given these fixed leaders, we show how to connect these platoons into a network with optimal coherence.

We first study the problem of optimal network design in a Network of Networks, a graph composed of a set of disjoint subgraphs and a set of designed edges between them. Nodes obey noisy consensus dynamics, and our system model allows for both positive and negative edge weights. We quantify the system performance by its coherence, an $H_2$ norm that captures the steady-state variance of the deviation from consensus. We pose the problem of how to connect the subgraphs, by selecting a single connecting node in each subgraph, so that the resulting Network of Networks has optimal coherence. We then show that this problem can be solved by identifying the optimal connecting node in each subgraph independent of the other nodes and subgraphs. Thus, the problem can be solved in polynomial time in the order of largest subgraph. We prove that, when the connecting topology is a tree, this solution is optimal even under a more general model that allows for multiple connecting nodes per subgraph. We also derive bounds on the best and worst coherence for a general Network of Networks with all-positive edge weights and provide analytical and numerical examples that further explore coherence in a Network of Networks.