Emergent Crowd BehaviorChing-Shoei Chiang1

Christoph Hoffmann2

Sagar Mittal2

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 (rr , ro , ra)

– Field of perception (a)– Speed of motion (s)– Speed of turning (q)– Error (s)

• Focus on direction

Emergent Behavior

• Does the flock stay together?

• Higher-order group behavior?

Characterizing Flock Behavior

• Group polarization

• Group momentum

• where vk is the velocity vector, xk the position vector, and

the centroid’s position

1

1 Nk

k k

vp

N v

1

1k k

Nk k

k k

r x x

r vm

N r

x

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 Dro near zero with Dra large

Sample Run – Highly Parallel, N=100

take-off, t≈100

rr= 1ro= 8ra= 23t ≈ 200

Sample Run – Toroidal, N=100

organizational phase (at t≈50)

centroid track at t≈530

rr= 1ro= 5ra= 17t ≈500

Loss of Cohesion – N=100

rr= 1ro= 4ra= 9t = 37

individuals leavesubgroups 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, s=0, q=40o, a=270o

Region of breakup approximately Dra+Dro < 8

N=50, 100, 200, 400s=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?

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

Much more noise for low ra and high ro

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 97-2212-E-031-002• NSF grant DSC 03-25227• DOE award DE-FG52-06NA26290.