1. Flocks, Herds and Schools Modeling and Analytic Approaches

2. The Problem To Model: –Flocks of Birds in 3-D –Herds of animals in 2-D –Schools of Fish in 3-D To Calculate / Analyze: –How structured dynamic aggregates form and move, especially how a uniform heading is attained

3. History of Approaches 1986: Craig Reynold –“Flocks, Herds and Schools: A distributed behavioral model” 1995: T. Viscek –Discrete Equation-based simulation model 1998: John Toner and Yuhai Tu: –“Flocks, Herds, and Schools: A quantitative theory of flocking” 2002: Tanner and Jadbabaie –“Stable Flocking of Mobile Agents” 2002: Olfati-Saber

4. Reynold’s Boids The proof is in the watching: –http://www.red3d.com/cwr/boids/applet/ Point agents located in 3-D Euclidean space, with a variable heading vector: –A = (x1,x2,x3,h1,h2,h3) Two Basic components: –Individual Agent capabilities (geometric flight) –Inter-agent group behavior rules

5. Reynolds Boids We won’t really look at the first component The second component is key. The three “Reynolds rules” are: –Collision Avoidance –Velocity Matching –Flock centering

6. The Three Reynolds Rules Separation Coherence Alignment (Velocity matching)

7. Reynolds Boids The rules generate acceleration imperatives for the agent Accelerations are aggregated for each agent via weighting and priority (to reduce indecision) and then fed into a flight module Other goal-seeking and obstacle avoidance behaviors

8. Reynolds Boids Important in its time: in 1986 the power of local algorithms was not understood Made for graphics community, but taken up enthusiastically by “complexity” and A-Life community Flocking is impromptu: flocks “emerge” from randomized initial configurations

9. Reynolds Boids Between 1986 and 1998, there was a lot of interest in the model, and many variants of it: –Schooling (see video) –Ant behavior –Multiple species But very little theory.

10. A Theory of “Flocking” Toner + Tu decided to use statistical physics A continuum model in various dimensions Flocks are modeled as a compressible statistical fluid (!) Theory is related to –Landau-Ginsburg Ferromagnetism –Navier-Stokes theory

11. Flocks as Fluids “Birds” are particles without heading Large, large numbers available to meaningful statistics (i.e. the infinite limit) Flocks are isotropic Goal is to calculate correlations, symmetries, and phase changes

12. Flocks as Fluids The equations of the model are: where v is the bird velocity field and p is the bird density terms = Navier Stokes  and  terms = ensure movement P term = “Pressure” (collision avoidance) D terms = Diffusion (centering) f term = noise

13. Flocks as Fluids Main Results –There is an ordered phase in which given appropriate statistics of initial condition, group velocity spread tends to 0 –Flocks support Goldstone modes (symmetry breaking) –Flocks support sound-like wave modes, density waves, and shear waves –Much more long-range damping than a magnet in velocity space –Much more density fluctuation than in other condensed matter systems –Exact calculation of scaling exponents in 2-D, showing that the “flock” is truly long-range symmetry broken –Broken symmetry corresponds to flock direction

14. Flocks as Fluids A picture in the ordered state.

15. Flocks as Fluids Problems in this approach: –Assumptions are a bit weird (like horses as frictionless spheres) –Numbers are way too large and densities are too high –Statistics are too structured –Motion doesn’t look like flocks! It looks like a fluid or a weird weak magnet. –Equations of motion are somewhat unmotivated –Predictions are untestable/inapplicable

16. A Different Approach Consider N agents in the plane The important point is that there are control terms

17. Flocks as Graphs Neighboring agents are influence each other by non-zero controls. Nearest-neighbor relations generate a graph Let B(G) be the adjacency matrix of the graph (i.e. 0 if not neighbor, 1 if it is) L(G) = B(G) B(G)* is the graph Laplacian. Eigenvalues correspond to topological properties of the graph.

18. Now, dynamics are generated by potential function V which takes on a unique minimum at desired positions Control is determined via the equation:

19. Flocks as Graphs Assume that the control connection graph is connected at all times. Then, Tanner et. al. proved using graph Laplacians and non- smooth analysis that: –All pairwise velocity differences converge to 0 –And the system approaches a potential- minimizing configurations

21. Flocks as Graphs This model works at the right range of sizes It does not “overpredict” It gets geometry right It generates pictures that look pretty good.

22. Unresolved Questions Obstacle Avoidance (Olfati-Saber) Flock Emergence Flock Disruption General motion Collective leader choice Other non-potential based mechanisms