Network science has emerged as a powerful conceptual paradigm in science and engineering. Constructs and phenomena such as interconnected networks, random and small–world networks, and phase transition nowadays appear in a wide variety of research literature, ranging across social networks, statistical physics, sensor networks, economics, and of course multiagent coordination and control. The reason for this unprecedented attention to network science is twofold. On the one hand, in a number of disciplines—particularly in biological and material sciences–it has become vital to gain a deeper understanding of the role that inter–elemental interactions play in the collective functionality...

Graph-based abstractions of networked systems contain virtually no information about what exactly is shared by the agents, through what protocol the exchange takes place, or what is subsequently done with the received information. Instead, the graph-based abstraction contains high-level descriptions of the network topology in terms of objects referred to as vertices and edges. In this chapter, we provide a brief overview of graph theory. Of particular focus will be the area of algebraic graph theory, which will provide the tools needed in later chapters for tying together inherently dynamic objects (such as multi-agent robotic systems)...

Consider a situation where a group of sensors are to measure the temperature of a given area. Although the temperature measured by each sensor will vary according to its location, it is required that the sensor group– using an information sharing network–agree on a single value which represents the temperature of the area. For this, the sensor group needs aprotocolover the network, allowing it to reach consensus on what the common sensor measurement value should be.

In this first chapter devoted to theagreement–or theconsensus–protocol over static networks, we explore the interdependency between the...

Lyapunov theory is an intuitive framework for the analysis of asymptotic properties of dynamical systems–one with far-reaching consequences. The power and convenience of using this framework is the relative ease by which one can analyze the stability of dynamical systems with nonlinearities, noise, and delays, and to incorporate control inputs to improve the nominal performance of the system.¹ In the first part of this chapter, we will explore the utility of the basic Lyapunov machinery in the realm of the agreement protocol.

Using Lyapunov theory for analyzing the agreement protocol (3.2), at first, seems like bringing in a...

Allowing the underlying network in the agreement protocol to switch among a finite number of topologies can be by design or necessity. In the latter case, the analysis that allows us to ascertain that the protocol retains its convergence properties can be categorized asrobustness analysis. In this chapter, we first consider yet another facet associated with the robustness of the agreement protocol, this time, by allowing random failures in the edges of the network.

In the Erdős-Rényi model of random graphs onnvertices, the existence of an edge between a pair of vertices in the setV=...

Formation control is one of the first problems one typically addresses when controlling multiple mobile agents. In this chapter, we present this topic by first discussing how formations can be specified, and then proceed by presenting a suite of graph–based formation control strategies.

Regardless of the particulars of a given target formation problem, these problems all share the general property of involving moving the agents in such a way that they satisfy a particularshapeorrelative stateand a certain aspect of assigning roles (targets in the shape or the relative state) to individual agents. In fact, formation...

Arguably, a large portion of this book can be thought of as being about teams of networked mobile robots. However, in the previous chapters we have thought of the underlying graph structure as being either static or dynamic without explicit geometric conditions on the existence of edges between vertices. In this chapter, we focus on the situation in which the edges have a direct, geometric interpretation in terms of limiting sensing capabilities, as is the case when the network consists of mobile robots. In particular, we will focus on the case when the graph is a$\Delta $-disk proximity graph, that...

Estimation theory is a truth–seeking endeavor; it is the scientific means of designing processes by which a static or dynamic variable of interest can be uncovered by processing a noisy signal that functionally depends on it. Estimation is a rich discipline with a wide range of applications in signal processing and control. Our emphasis in this chapter is naturally on the distributed and networked aspects of certain discrete–time estimation algorithms, namely, distributed linear least squares and distributed Kalman filtering.

We start our discussion by examining how linear least squares can be viewed and analyzed in the distributed setting....

Our first example, which has a strong resemblance to the agreement problem, is inspired by considering a social network of friends, viewed as nodes in an undirected graphG, that adopt a certain level of “fashionability,” measured in terms of a real number on the unit interval [0, 1]. Thus${x_i}(k) = 1$and${x_j}(k) = 1$refer to the scenario where nodesiandjare, respectively, the most and least fashionable a member of the social group can be at time indexk. Let us initialize the group fashionability state atk= 0 by choosing$0 \leqslant {x_i}(0) \leqslant 1,i = 1,2,...,n,$which can be done, for...

The agreement protocol, as introduced in Chapter 3, provides the ambient setting for the evolution of a set of dynamic agents. Just as a stabilizing controller is typically a first step in the control design phase, the agreement protocol will provide the underlying cohesion of the network. In this chapter, we consider situations where the agreement protocol over a fixed network is also influenced by external inputs, injected at particular nodes. We also consider the case where the correspondinglinear systemcan be observed. Although, in principle, one can designate network inputs and outputs at distinct nodes, we will be...

In the first part of this chapter, we examine a candidate notion, where agents in the network can assess what local structures are desirable to them. Not surprisingly, the local nature of decisions and information structure for this part of our discussion assumes a game theoretic flavor. We then turn our attention to cases where the network structure is driven by a centralized algorithm that relies on a global information about the entire network structure.

In this chapter, we consider processes by which agents in a network—aided by local or global knowledge of the network structure—can make decisions...

In certain classes of distributed systems, the existence and the quality of the information-exchange mechanism between a pair of dynamic elements is determined—either fully or partially—by their respective states. We refer to the resulting structure, reflecting the dynamic nature of the agents’ states on one hand, and the combinatorial character of their interactions on the other, asstate-dependent dynamic graphs, orstate-dependent graphs. As we will see in this chapter, state-dependent graphs not only put the study of dynamic networks in the realm of system theory, but also invite us to consider a host of new problems in...

In this chapter, we point out how the concepts developed in this book can be generalized beyond graphs to higher–order structures. However, a disclaimer is already in order at this point; we do not present a particularly mature body of work, as it pertains to networked systems. Rather, we simply point out some possible (and certainly fascinating) extensions.

To take the step from graph models, where the key objects are nodes and edges, to richer structures, one first needs to turn to algebraic topology. In fact, a graph can be generalized to a more expressive combinatorial object known as...