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

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

In this paper, we present a theoretical framework for design and analysis of distributed flocking algorithms. Two cases of flocking in free-space and presence of multiple obstacles are considered. We present three flocking algorithms: two for free-flocking and one for constrained flocking. A comprehensive analysis of the first two algorithms is provided. We demonstrate the first algorithm embodies all three rules of Reynolds. This is a formal approach to extraction of interaction rules that lead to the emergence of collective behavior. We show that the first algorithm generically leads to regular fragmentation, whereas the second and third algorithms both lead to flocking. A systematic method is provided for construction of cost functions (or collective potentials) for flocking. These collective potentials penalize deviation from a class of lattice-shape objects called /spl alpha/-lattices. We use a multi-species framework for construction of collective potentials that consist of flock-members, or /spl alpha/-agents, and virtual agents associated with /spl alpha/-agents called /spl beta/- and /spl gamma/-agents. We show that migration of flocks can be performed using a peer-to-peer network of agents, i.e., "flocks need no leaders." A "universal" definition of flocking for particle systems with similarities to Lyapunov stability is given. Several simulation results are provided that demonstrate performing 2-D and 3-D flocking, split/rejoin maneuver, and squeezing maneuver for hundreds of agents using the proposed algorithms.

/pdf/flocking-for-multi-agent-dynamic-systems-algorithms-and-5efqu9aidp.pdf

Flocking for multi-agent dynamic systems: algorithms and theory

This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Spin glasses: Experimental facts, theoretical concepts, and open questions

Statistical physics has proven to be a fruitful framework to describe phenomena outside the realm of traditional physics. Recent years have witnessed an attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. A wide list of topics are reviewed ranging from opinion and cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, and social spreading. The connections between these problems and other, more traditional, topics of statistical physics are highlighted. Comparison of model results with empirical data from social systems are also emphasized.

Statistical physics of social dynamics

We present a quantitative continuum theory of ``flocking'': the collective coherent motion of large numbers of self-propelled organisms. In agreement with everyday experience, our model predicts the existence of an ``ordered phase'' of flocks, in which all members of even an arbitrarily large flock move together with the same mean velocity $〈\stackrel{\ensuremath{\rightarrow}}{v}〉\ensuremath{\ne}0.$ This coherent motion of the flock is an example of spontaneously broken symmetry: no preferred direction for the motion is picked out a priori in the model; rather, each flock is allowed to, and does, spontaneously pick out some completely arbitrary direction to move in. By analyzing our model we can make detailed, quantitative predictions for the long-distance, long-time behavior of this ``broken symmetry state.'' The ``Goldstone modes'' associated with this ``spontaneously broken rotational symmetry'' are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole. These ``Goldstone modes'' mix with modes associated with conservation of bird number to produce propagating sound modes. These sound modes lead to enormous fluctuations of the density of the flock, far larger, at long wavelengths, than those in, e.g., an equilibrium gas. Our model is similar in many ways to the Navier-Stokes equations for a simple compressible fluid; in other ways, it resembles a relaxational time-dependent Ginsburg-Landau theory for an $n=d$ component isotropic ferromagnet. In spatial dimensions $dg4,$ the long-distance behavior is correctly described by a linearized theory, and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems. For $dl4,$ nonlinear fluctuation effects radically alter the long distance behavior, making it different from that of any known equilibrium model. In particular, we find that in $d=2,$ where we can calculate the scaling exponents exactly, flocks exhibit a true, long-range ordered, spontaneously broken symmetry state, in contrast to equilibrium systems, which cannot spontaneously break a continuous symmetry in $d=2$ (the ``Mermin-Wagner'' theorem). We make detailed predictions for various correlation functions that could be measured either in simulations, or by quantitative imaging of real flocks. We also consider an anisotropic model, in which the birds move preferentially in an ``easy'' (e.g., horizontal) plane, and make analogous, but quantitatively different, predictions for that model as well. For this anisotropic model, we obtain exact scaling exponents for all spatial dimensions, including the physically relevant case $d=3.$

Flocks, herds, and schools: A quantitative theory of flocking

We propose a nonequilibrium continuum dynamical model for the collective motion of large groups of biological organisms (e.g., flocks of birds, slime molds, etc.) Our model becomes highly nontrivial, and different from the equilibrium model, for $dl{d}_{c}=4$; nonetheless, we are able to determine its scaling exponents exactly in $d=2$ and show that, unlike equilibrium systems, our model exhibits a broken continuous symmetry even in $d=2$. Our model describes a large universality class of microscopic rules, including those recently simulated by Vicsek et al.

Long-Range Order in a Two-Dimensional Dynamical XY Model: How Birds Fly Together.

Hydrodynamics and phases of flocks

Theories for elasticity and dislocation defects in two-dimensional pentagonal and three-dimensional icosahedral quasicrystals are presented.

Elasticity and dislocations in pentagonal and icosahedral quasicrystals.

We construct the equations of motion for the coupled dynamics of order parameter and concentration for the nematic phase of driven particles on a solid surface, and show that they imply i) giant number fluctuations, with a standard deviation proportional to the mean in dimension d = 2 of primary relevance to experiment, and ii) long-time tails $\sim t^{-d/2}$in the autocorrelation of the particle velocities despite the absence of a hydrodynamic velocity field. Our predictions can be tested in experiments on aggregates of amoeboid cells as well as on layers of agitated granular matter.

https://iopscience.iop.org/article/10.1209/epl/i2003-00346-7/pdf

John Toner

Papers

Flocks, herds, and schools: A quantitative theory of flocking

Long-Range Order in a Two-Dimensional Dynamical XY Model: How Birds Fly Together.

Hydrodynamics and phases of flocks

Elasticity and dislocations in pentagonal and icosahedral quasicrystals.

Active nematics on a substrate: Giant number fluctuations and long-time tails