Differential Inequalities in Multi-Agent Coordination and Opinion Dynamics Modeling

Distributed algorithms of multi-agent coordination have attracted substantial attention from the research community; the simplest and most thoroughly studied of them are consensus protocols in the form of differential or difference equations over general time-varying weighted graphs. These graphs are usually characterized algebraically by their associated Laplacian matrices. Network algorithms with similar algebraic graph theoretic structures, called being of Laplacian-type in this paper, also arise in other related multi-agent control problems, such as aggregation and containment control, target surrounding, distributed optimization and modeling of opinion evolution in social groups. In spite of their similarities, each of such algorithms has often been studied using separate mathematical techniques. In this paper, a novel approach is offered, allowing a unified and elegant way to examine many Laplacian-type algorithms for multi-agent coordination. This approach is based on the analysis of some differential or difference inequalities that have to be satisfied by the some "outputs" of the agents (e.g. the distances to the desired set in aggregation problems). Although such inequalities may have many unbounded solutions, under natural graphic connectivity conditions all their bounded solutions converge (and even reach consensus), entailing the convergence of the corresponding distributed algorithms. In the theory of differential equations the absence of bounded non-convergent solutions is referred to as the equation's dichotomy. In this paper, we establish the dichotomy criteria of Laplacian-type differential and difference inequalities and show that these criteria enable one to extend a number of recent results, concerned with Laplacian-type algorithms for multi-agent coordination and modeling opinion formation in social groups.Comment: accepted to Automatic

Distributed Optimization: Convergence Conditions from a Dynamical System Perspective

This paper explores the fundamental properties of distributed minimization of a sum of functions with each function only known to one node, and a pre-specified level of node knowledge and computational capacity. We define the optimization information each node receives from its objective function, the neighboring information each node receives from its neighbors, and the computational capacity each node can take advantage of in controlling its state. It is proven that there exist a neighboring information way and a control law that guarantee global optimal consensus if and only if the solution sets of the local objective functions admit a nonempty intersection set for fixed strongly connected graphs. Then we show that for any tolerated error, we can find a control law that guarantees global optimal consensus within this error for fixed, bidirectional, and connected graphs under mild conditions. For time-varying graphs, we show that optimal consensus can always be achieved as long as the graph is uniformly jointly strongly connected and the nonempty intersection condition holds. The results illustrate that nonempty intersection for the local optimal solution sets is a critical condition for successful distributed optimization for a large class of algorithms

Opinion Dynamics in Social Networks with Hostile Camps: Consensus vs. Polarization

Most of the distributed protocols for multi-agent consensus assume that the agents are mutually cooperative and "trustful," and so the couplings among the agents bring the values of their states closer. Opinion dynamics in social groups, however, require beyond these conventional models due to ubiquitous competition and distrust between some pairs of agents, which are usually characterized by repulsive couplings and may lead to clustering of the opinions. A simple yet insightful model of opinion dynamics with both attractive and repulsive couplings was proposed recently by C. Altafini, who examined first-order consensus algorithms over static signed graphs. This protocol establishes modulus consensus, where the opinions become the same in modulus but may differ in signs. In this paper, we extend the modulus consensus model to the case where the network topology is an arbitrary time-varying signed graph and prove reaching modulus consensus under mild sufficient conditions of uniform connectivity of the graph. For cut-balanced graphs, not only sufficient, but also necessary conditions for modulus consensus are given.Comment: scheduled for publication in IEEE Transactions on Automatic Control, 2016, vol. 61, no. 7 (accepted in August 2015

The Total s-Energy of a Multiagent System

We introduce the "total s-energy" of a multiagent system with time-dependent links. This provides a new analytical lens on bidirectional agreement dynamics, which we use to bound the convergence rates of dynamical systems for synchronization, flocking, opinion dynamics, and social epistemology