In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.

/pdf/consensus-problems-in-networks-of-agents-with-switching-25c3hqvtd8.pdf

Consensus problems in networks of agents with switching topology and time-delays

This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multivehicle formations

/pdf/consensus-and-cooperation-in-networked-multi-agent-systems-5ame7z72ny.pdf

Consensus and Cooperation in Networked Multi-Agent Systems

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

In a recent Physical Review Letters article, Vicsek et al. propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors." In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.

/pdf/coordination-of-groups-of-mobile-autonomous-agents-using-cqyatdq4iz.pdf

Coordination of groups of mobile autonomous agents using nearest neighbor rules

This note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies. Both discrete and continuous update schemes are proposed for information consensus. This note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves.

/pdf/consensus-seeking-in-multiagent-systems-under-dynamically-302xpjuq82.pdf

Consensus seeking in multiagent systems under dynamically changing interaction topologies

A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation $(\ensuremath{\eta})$ added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, $|{\mathbf{v}}_{a}|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous, since $|{\mathbf{v}}_{a}|$ is found to scale as $({\ensuremath{\eta}}_{c}\ensuremath{-}\ensuremath{\eta}{)}^{\ensuremath{\beta}}$ with $\ensuremath{\beta}\ensuremath{\simeq}0.45$.

https://arxiv.org/pdf/cond-mat/0611743.pdf

Novel Type of Phase Transition in a System of Self-Driven Particles

Bacterial colonies must often cope with unfavourable environmental conditions. To do so, they have developed sophisticated modes of cooperative behaviour. It has been found that such behaviour can cause bacterial colonies to exhibit complex growth patterns similar to those observed during non-equilibrium growth processes in non-living systems; some of the qualitative features of the latter may be invoked to account for the complex patterns of bacterial growth. Here we show that a simple model of bacterial growth can reproduce the salient features of the observed growth patterns. The model incorporates random walkers, representing aggregates of bacteria, which move in response to gradients in nutrient concentration and communicate with each other by means of chemotactic 'feedback'. These simple features allow the colony to respond efficiently to adverse growth conditions, and generate self-organization over a wide range of length scales.

Generic modelling of cooperative growth patterns in bacterial colonies

In nature, microorganisms must often cope with hostile environmental conditions. To do so they have developed sophisticated cooperative behaviour and intricate communication capabilities, such as: direct cell-cell physical interactions via extra-membrane polymers, collective production of extracellular 'wetting' fluid for movement on hard surfaces, long range chemical signalling such as quorum sensing and chemotactic (bias of movement according to gradient of chemical agent) signalling, collective activation and deactivation of genes and even exchange of genetic material. Utilizing these capabilities, the colonies develop complex spatio-temporal patterns in response to adverse growth conditions. We present a wealth of beautiful patterns formed during colony development of various microorganisms and for different environmental conditions. Invoking ideas from pattern formation in non-living systems and using 'generic' modelling we are able to reveal novel survival strategies which account for the salient fe...

Cooperative self-organization of microorganisms

We investigate a discrete model consisting of self-propelled particles that obey simple interaction rules. We show that this model can self-organize and exhibit coherent localized solutions in one- and in two-dimensions. In one-dimension, the self-organized solution is a localized flock of finite extent in which the density abruptly drops to zero at the edges. In two-dimensions, we focus on the vortex solution in which the particles rotate around a common center and show that this solution can be obtained from random initial conditions, even in the absence of a confining boundary. Furthermore, we develop a continuum version of our discrete model and demonstrate that the agreement between the discrete and the continuum model is excellent.

https://arxiv.org/pdf/cond-mat/0006477.pdf

Self-organization in systems of self-propelled particles.

Depending on the environmental conditions bacterial colonies growing on agar surfaces can exhibit complex colony formation and various types of collective motion Experimental results are presented concerning the hydrodynamics (vortices, migration of bacteria in clusters) and colony formation of a morphotype of Bacillus subtilis Some of these features are not specific to this morphotype but also have been observed in several other bacterial strains, suggesting the presence of universal effects A simple model of self-propelled particles is proposed, which is capable of describing the hydrodynamics on the intermediate level, including the experimentally observed rotating disks of bacteria The colony formation is captured by a complex generic model taking into account nutrient diffusion, reproduction, and sporulation of bacteria, extracellular slime deposition, chemoregulation, and inhomogeneous population Our model also sheds light on some possible biological benefits of this ``multicellular behavior'' \textcopyright{} 1996 The American Physical Society

Inon Cohen

Papers

Novel Type of Phase Transition in a System of Self-Driven Particles

Generic modelling of cooperative growth patterns in bacterial colonies

Cooperative self-organization of microorganisms

Self-organization in systems of self-propelled particles.

Formation of complex bacterial colonies via self-generated vortices