Emergent Crowd Behavior Ching-Shoei Chiang 1 Christoph Hoffmann 2 Sagar Mittal 2 1 ) Computer Science, Soochow University, Taipei, R.O.C. 2 ) Computer Science, Purdue University, West Lafayette, IN

Problem Many crowds have no central control Individual decisions, based on limited cognition, create an emergent crowd behavior How can we script the collective behavior by prescribing the limited individual behavior?

Applications?

Robotics

Fish Vortex

Starlings flocking

Modeling Crowds

Some Prior Art Reynolds, 1988 and 1999 – Three core rules (separation, alignment, cohesion) – Behavior hierarchy Couzin, 2002 and 2005 – Investigate core rules – Determine leadership fraction Bajec et al., 2005 – Fuzzy logic Cucker and Smale, 2007 – Convergence results Itoh and Chua, 2007 – Chaotic trajectories

Core Rules (Reynolds ‘88) First to articulate these rules Centroid used for attraction Limited perception

Couzin’s Model Seven parameters – Zonal radii (r r, r o, r a ) – Field of perception (  ) – Speed of motion (s) – Speed of turning (  ) – Error (  ) Focus on direction

Emergent Behavior Does the flock stay together? Higher-order group behavior?

Characterizing Flock Behavior Group polarization Group momentum where v k is the velocity vector, x k the position vector, and the centroid’s position

Couzin’s Formation Types Swarm (A): m ≈ 0, p ≈ 0 Torus (B): m > 0.7, p ≈ 0 Dynamic parallel (C): m ≈ 0, p ≈ 0.8 Highly parallel (D): m ≈ 0, p ≈ 1

Swarm Behavior Random milling around Start behavior for random initial position/orientation Stable for  r o near zero with  r a large

Sample Run – Highly Parallel, N=100 take-off, t≈100 r r = 1 r o = 8 r a = 23 t ≈ 200

Sample Run – Toroidal, N=100 organizational phase (at t≈50) centroid track at t≈530 r r = 1 r o = 5 r a = 17 t ≈500

Loss of Cohesion – N=100 r r = 1 r o = 4 r a = 9 t = 37 individuals leave subgroups form

Our Questions How does the choice of the zonal parameters and the initial configuration affect: – Cohesion of the flock ? – Formation type ? Is this behavior scale-independent ? Do the answers in 3D differ from 2D ?

N=100,  =0,  =40 o,  =270 o Region of breakup approximately  r a +  r o < 8

N=50, 100, 200, 400  =0, 0.05 rad, 0.10 rad

2D Vs. 3D

The 2D graph could almost be the 3D graph, but doubled in size… but why?

Much more noise for low r a and high r o

Configuration Dependence

3D: 5x5x4 grid 3D: plane hexagon, 30 trials 2D: plane hexagon, 48 trials 2D: R=5, random Initial Configuration in 2D and 3D Cohesion

Some Observations 2D and 3D scenarios differ in how they evolve Cohesion and swarm type is not scale- invariant – In triangle: subgroup development – In saw-tooth notch: individuals take off Cohesion and swarm type has dependence on initial configurations― the collective memory. No dynamic parallel behavior

Acknowledgements NSC Taiwan grant NSC E NSF grant DSC DOE award DE-FG52-06NA26290.