Consensus of Complex Multi-agent Systems

Abstract

This entry provides a broad overview of the basic elements of consensus dynamics. It describes the classical Perron-Frobenius theorem that provides the main theoretical tool to study the convergence properties of such systems. Classes of consensus models that are treated include simple random walks on grid-like graphs and in graphs with a bottleneck, consensus on graphs with intermittently randomly appearing edges between nodes (gossip models), and models with nodes that do not modify their state over time (stubborn agent models). Application to cooperative control, sensor networks, and socioeconomic models are mentioned. Multi-agent Systems and Consensus Multi-agent systems constitute one of the fundamental paradigms of science and technology of the present century (Castellano et al. 2009; Strogatz 2003). The main idea is that of creating complex dynamical evolutions from the interactions of many simple units. Indeed such collective behaviors are quite evident in biological and social systems and were indeed considered in earlier times. More recently, the digital revolution and the miniaturization in electronics have made possible the creation of man-made complex architectures of interconnected simple devices (computers, sensors, cameras). Moreover, the creation of the Internet has opened a totally new form of social and economic aggregation. This has strongly pushed towards a systematic and deep study of multi-agent dynamical systems. Mathematically they typically consist of a graph where each node possesses a state variable; states are coupled at the dynamical level through dependences determined by the edges in the graph. One of the challenging problems in the field of multi-agent systems is to analyze the emergence of complex collective phenomena from the interactions of the units which are typically quite simple. Complexity is typically the outcome of the topology and the nature of interconnections. Consensus dynamics (also known as average dynamics) (Jadbabaie et al. 2003; Carli et al. 2008) is one of the most popular and simplest multi-agent dynamics. One convenient way to introduce it is with the language of social sciences. Imagine that a number of independent units possess an information represented by a real number, for instance, such number can represent an opinion on a given fact. Units interact and change their opinion by averaging with the opinions of other units. Under certain assumptions, this will lead the all community to converge to a consensus opinion which takes into consideration all the initial opinion of the agents. In social sciences, empiric evidences (Galton 1907) have shown how such aggregate opinion may give a very good estimation of unknown quantities: such phenomenon has been proposed in the literature as wisdom of crowds (Surowiecki 2004). E-mail: fabio.fagnani@polito.it