Hiroki Sayama [email protected] 2nd Annual French Complex Systems Summer School Continuous Space, Lattices and Swarms: Pattern Formation in Complex.

Hiroki [email protected]

2nd Annual French Complex Systems Summer SchoolContinuous Space, Lattices and Swarms:Pattern Formation in Complex Systems

Swarm Models

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Collective behavior of swarms• Particle/cell aggregation, fish

schools, bird flocks, insect swarms, ped. flows

• Creates macroscopic behavior out of autonomous individual motion and local kinetic interaction

•Acceleration•Alignment•Collision avoidance•Attraction to neighbors

• Modeled using “agents” © Iain Couzin© Iain Couzin

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Agent-based modeling

• One of the most generalized frameworks for modeling/simulation of complex systems

• You construct many virtual individuals, or “agents”, and simulate their behaviors explicitly in a computer

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Agents

• Have internal properties

• Spatially localized

• Perceive the environment

• Locally interact with other agents and behave based on predefined rules

• No central supervisor

• May learn autonomously

• May produce non-trivial “collective behavior” as a whole

{ {x, y},gender,

age,mood, …

}

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Implementing agent-based models in Mathematica

1. Decide what kind of data format you will use to represent the properties of each agent

2. Create a population of agents following that format

3. Write a code to visualize the current states of agents (and any other observation/analysis tools you may need)

4. Write a code to update the state of each agent based on the assumption of your model

Models with Fixed Number of Agents

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Example: Random walk and diffusion

• A small random vector is added to the location of each particle at every time step

• Each particle shows a Brownian motion

• The collection of random-walking particles show “diffusion” at a macroscopic scale

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Exercise

• Modify the “random walk” simulator code so that:– Each particle has its own velocity– Each particle is accelerated by the

attractive force toward the “center of mass” of the population of particles

This should create a simple swarming behavior

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Example: Diffusion limited aggregation (DLA)

• Two types of particles, “free” and “fixed”; only free ones can move

• When colliding into a fixed one, a free particle becomes fixed and loses mobility

• Starting with only one fixed particle, a complex self-similar pattern emerges (called “spatial fractal”)

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Exercise

• Carry out the DLA simulation runs with multiple “seeds” randomly positioned in the space

• Modify the simulator code so that (1) initially no “fixed” particles exist, but (2) the particles that touched the bottom edge of the space turn to “fixed”

Models with Agent-Environment Interaction

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Example: Garbage collection by ants

• Ants are wandering in a space where lots of garbage pieces are scattered

• When an ant finds garbage:– If the ant holds nothing, pick up a

piece of garbage from there– If the ant already holds garbage,

drop it off there

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Exercise

• Modify the ants’ garbage collection model so that:– Each ant produces pheromone

where it is– Pheromone diffuses and evaporates

at a constant rate– Ants are stochastically attracted

toward places where there are greater amount of pheromone

Models with Agent Replacement

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Models with birth/death of agents

• The updating function determines whether or not each of the current agents can survive to the next step– If yes, it will be added to the “next

agent population” list– If no, it will be simply disregarded

• The updating function also simulates the birth of new agents, which will be added to the list as well

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• Rabbits and foxes wander and reproduce in a space – Foxes move faster than

rabbits– A rabbit will survive if it is

not caught by foxes– A fox will survive if it ate rabbits

not long ago– Foxes’ reproduction is possible only

when they eat rabbits

Example: Predator-prey model

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Exercise

• Modify the simulator code so that it outputs the time series plots of rabbit and fox populations in addition to the visual image of simulation

• See if you actually have oscillatory behavior in these plots

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Exercise: Evolutionary adaptation

• Make the range of motion (motility) a heritable individual property of agents

• Introduce mutations of the agent motility in both foxes and rabbits

• See how the agent motilities spontaneously evolve in simulation

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Exercise: Evolutionary adaptation

• Make the reproduction of agent density-dependent– Too few neighbors / too many

neighbors will result in failure of reproduction

• See how the agent motilities spontaneously evolve in simulation

Pattern Formation in Heterogeneous Particle

Swarms: “Swarm Chemistry”

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Models of swarm behavior (1)• Continuous-time dynamical

models– Suzuki & Sakai (1973), Okubo (1977),

Shimoyama et al. (1996), Chuang et al. (2006) etc.

– Described by ODEs that include terms for•Self-propulsion•Friction •Pairwise kinetic interaction

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Models of swarm behavior (2)• Discrete-time kinematic models

– Reynolds (1987), Viscek et al. (1995), Couzin et al. (2002), Kunz & Hemelrijk (2003, 2004) etc.

– Each agent steers to•Approach local center of mass•Align with local average velocity •Avoid collisions•Then add updated velocity to its

position

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Diversity of swarm behavior• Distinct classes of behaviors

– Stationary clustering– Random swarming– Coherent linear motion– Amoeba-like structure– Milling– Dispersal

etc.

• Phase transitions between them

© Yao Li Chung© Yao Li Chung

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A question

• Earlier studies were largely focused on homogeneous particle swarms only– Some considered small intra-

population variations, but not kinetically distinct types

• What would happen if two or more types of self-propelled particles are mixed together?

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Swarm Chemistry

• A novel artificial chemistry research project where artificial swarm populations are used as chemical reactantshttp://bingweb.binghamton.edu/~sayama/SwarmChemistry/

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Model assumptions• Kinematics partly based on Reynolds’

Boids• Simple semi-autonomous particles

moving in a continuous open 2-D space– Well-defined boundary of local perception– Kinetic interactions with local neighbors– No capability to distinguish different types

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Behavioral rules• If no particles are found within local

perception range, steer randomly (Straying)

• Otherwise:– Steer to move toward the average

position of local neighbors (Cohesion)– Steer towards the average velocity of

local neighbors (Alignment)– Steer to avoid collision with neighbors

(Separation)– Steer randomly with a given probability

(Randomness)

• Approximate its speed to its normal speed (Self-propulsion)

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Kinetic parameters(Assigned to each particle individually)

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Behavior of homogeneous swarms• Basic behavior of our model is

similar to that of earlier models when a population is homogeneous

Stationary Stationary clusteringclustering

Coherent Coherent linear linear

motionmotion

Amoeba-like Amoeba-like structurestructure

DispersalDispersal

[simulation][simulation]

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Dependence on kinetic parameters• A couple of distinct phase transitions

observed

• Not-so-significant dependence on c3

cc11: strength of cohesion: strength of cohesion

cc 22: st

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: str

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cc11: strength of cohesion: strength of cohesion cc11: strength of cohesion: strength of cohesion

cc 22: st

reng

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ignm

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: str

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ali

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ent

cc 22: st

reng

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: str

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<|v|> <|v|> : av. absolute vel.: av. absolute vel. <r><r>: av. distance from CoM: av. distance from CoM

(Results with (Results with cc3 3 ~ 50 ± 5)~ 50 ± 5)

<v><v>: average velocity: average velocity

Coherent linear motion

Random swarming

Oscill-ation

Dis

per

sal

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Exploring pairwise interactions• Monte Carlo simulations (> 50,000 runs)• Fixed parameters:

– N=300, R=200, Vn=10, Vm=40, c4=0, c5=0.5• Varied parameters:

– c1, c2, c3 (cohesion, alignment, separation)– Two types (A, B) of particles randomly

created; simulate each in isolation as well as in mixture

• Initial conditions:– Randomly positioned and oriented in a

300x300 space• Quantities measured after 200 time

steps:– Average linear/angular velocities, average

distance from center of mass, local homogeneity

[simulation][simulation]

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Local homogeneity

• Measured by averaging over all particles the probability of the same type of particles within 6 nearest neighbors

Local homogeneity for Local homogeneity for this particle = 4/6 = this particle = 4/6 = 0.670.67

HH ~ 1.0 ~ 1.0 Separated Separated

HH ~ 0.5 ~ 0.5 Well mixed Well mixed

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Spontaneous separation

• Commonly seen in heterogeneous swarms

• Often creates multilayer structures

(results with 4 types)(results with 4 types)

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Spontaneous separation

• Blending is possible if two types share the same cohesion-separation ratio (c1/c3)

((cc1 1 / / cc33))AA – ( – (cc1 1 / / cc33))BB

Loc

al h

omog

enei

ty

Loc

al h

omog

enei

ty HH

- Alignment (Alignment (cc22) ) doesn’t play much doesn’t play much rolerole

- Ratio Ratio cc11//cc33 determines pairwise determines pairwise equilibrium distance: equilibrium distance: rreqeq = ( = (cc11//cc33))–1/2 –1/2 (if (if cc22 is ignored) is ignored)

Spontaneous Spontaneous separation is caused separation is caused by the difference of by the difference of “personal space”“personal space”

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Emergent motion

• Mixing two types may generate new behavior not present in either of them

Linear motionLinear motion RotationRotation OscillationOscillation

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Interactions between >2 types?• Possibility space explodes

exponentially with # of types involved– Exhaustive parameter sweep not

feasible– Monte Carlo may not give enough

resolution

• It is no longer clear what kind of quantities should be measured to characterize resulting patterns– We don’t know what to expect

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Interactive evolutionary method• An alternative, more exploratory

approach• Heterogeneous swarms are

represented by lists of multiple parameter sets (recipes) which are evolved over many iterations

• A human participates in evolutionary processes of swarms by subjectively selecting and varying their recipes

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Recipe

• A list of kinetic parameter sets of different types in a swarm– Format: # of particles * (R, Vn, Vm, c1, c2,

c3, c4, c5)

– Each row represents one type

97 * (226.76, 3.11, 9.61, 0.15, 0.88, 43.35, 0.44, 1.0)38 * (57.47, 9.99, 35.18, 0.15, 0.37, 30.96, 0.05, 0.31)56 * (15.25, 13.58, 3.82, 0.3, 0.8, 39.51, 0.43, 0.65)31 * (113.21, 18.25, 38.21, 0.62, 0.46, 15.78, 0.49, 0.61)

[simulation][simulation]

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Evolutionary operators (1)

• Perturbation to relative frequencies of different parameter sets within a recipe

• Mixing of two swarms physically

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Evolutionary operators (2)

• Duplication of parameter sets

• Deletion of parameter sets

• Insertion of new parameter sets

• Point mutation of parameter values

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Complex structures/behaviors• More complex

structures/behaviors possible by mixing several types of swarms

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Summary

• Many physical/biological/ecological systems can be modeled as swarms using agent-based modeling– Can include local interaction,

heterogeneity, interaction with environment, birth/death of agents, etc.

• Heterogeneous swarms may self-organize and robustly produce complex patterns (structures and behaviors)