cemetery organization, sorting, graph partitioning and data...

Cemetery Organization, Sorting, Graph Partitioning and Data

Analysis

Introduction

• Ants cluster corpses to form cemeteries• Ants sort larvae• Not completely understood

– Simple model can predict behaviour• Model can be applied to:

– Data analysis– Graph partitioning – …

• Distributed clustering and sorting used as benchmark problems in robotics

Cemetary Organization• Several species form piles of corpses:

– Lasius Niger [Chretien]– Pheidole pallidula [Deneubourg]

• Basic mechanism is attraction between ants and corpses (Yikes!)– Small piles become bigger …– Positive feedback, once again

Swarm Intelligence: Bonabeau et al

Larval Sorting

• Leptothorax unifasciatus (and many other species) sort larvae

• Circular sorting– Small in middle– Large on outside– Individual space increases with distance from centre– Space and metabolic rate are correlated [Franks]

Model of Corpse Clustering

• General idea is that isolated items picked up and dropped at locations of higher density …

Lecture Notes

Aggregation and SegregationMechanisms

AM, EE141, Swarm Intelligence, W4-1

Outline

• Social insects (ants)• Algorithms (data clustering)• Robotics experiments

It defines a class of mechanisms exploited by social insects to coordinate andcontrol their activity via indirect interactions.

• Stigmergic mechanisms can be classified in two different categories:quantitative (or continuous) stigmergy and qualitative (or discrete)stigmergy

Stimulus

Answer

S1

R1

S2

R2

S3

R3

time

S 4

R4

S 5

R5

Stop

Definition

Stigmergy

Example of qualitative stigmergy Example of quantitative stigmergy

More detail in Week 6!• Duration of aggregation

process: 48 h!• Reduction of the spread of

infection? Chretien (1996)

Stigmergy

A Model of Corpse Clustering

Characteristics of the algorithm for individual behavior(Deneubourg et al., 1991)

• When an ant encounters a corpse, it will pick it up with aprobability which increases with the degree of isolation of thecorpse

• When an ant is carrying a corpse, it will drop it with a probabilitywhich increases with the number of corpses in the vicinity

• Modulation of pick up/drop probabilities as a function of thepheromone clouds around the cluster -> quantitative (continuous)stigmergy

The probability that an agent which is not carrying an item will pickup an item

Ppick up =K+

K+ + f

Probability that an agent carrying an item will drop the item

Pdrop =f

K- + f

f : fraction of neighborhood sites occupied by itemsK-, K+: threshold constants

Algorithm for individual behavior

A Model of Corpse Clustering

( )2

)2(

t = 15

t = 0

Short term memoryat t = 15

Probability of picking up an object:

Probability of dropping an objectwhich is being carried:

fi : fraction of neighboring sitesoccupied by objects of the same typeof the object i

K+, K- : constants

Pipick up = ( K+ / ( K+ + fi ) ) 2

Pidrop = ( fi / ( K- + fi ) ) 2

Individual behavioral algorithm

Model of HarvestSorting in Ants

Aggregation andSegregation Models

• Explanation of cemetery organization in Lasius niger,Pheidole pallidula, and Messor Sancta ant species ok

• Explanation of brood sorting in Leptothorax ants(concentric annular sorting) unexplained!

Algorithms

• We define a « distance » d (or a dissimilarity) between objects in theattribute space of the object.

• For instance, in the sorting problem previously mentioned, 2 objects oiand oj can be similar or different (binary dissimilarity):

If oi and oj are identical objects then: d(oi, oj) = 0If oi and oj are different objects then: d(oi, oj) = 1

• The problem (and the algorithm) can be extended to more compleobjects described by a finite number n of attributes, each attributerepresented by a real value for instance.

• These objects can be described as points in the Rn space and d(oi, oj) asthe eucledian distance between them.

Application of Clustering Algorithms to the

Classification of Objects

The attribute space is projected on a smaller dimension space (e.g. l=2)

• Assumption: the projection space has to be chosen so that the distances intra-clusters are smaller than distances inter-cluster

• We finally discretize the projected space (it can be seen as a sub-space of Z2), sothat many clusterizing agents can move around and operate on this space

Algorithm of Lumer et Faieta (1994)



The agents can locally perceive a certain number of cells around their position (areas2 around the current site r of the agent)

S

S

r




At time t, an agent at the site r finds an object oi, on this site f(oi) measures the meansimilarity of the object oi with the other objects oj which are in its neighborhood(within perception area sxs)

αααα : algorithm parameter which defines the dissimilarity scale


f(oi) = , if f > 01

s2[1– ]Σ

Oj ∈ Neigh(sxs) (r)

f(oi) = 0 , otherwise

α

d(oi, oj)



• If all the cells s2 around the agent at site r have similar objects to oi, weobtain :

∀ Oj ∈ Neigh(sxs) (r), d(oi, oj) = 0 and f(oi) = 1

• If all the cells s2 around the agent at site r have objects highly differentto oi, we obtain :

∀ Oj ∈ Neigh(sxs) (r), d(oi, oj) = α and f(oi) = 0




k1

k1 + f(oi)( )

2

{2f(oi), if f(oi) < k2

1, if f(oi) ≥ k2

If f(oi) = 1, oi has low probability to be picked upIf f(oi) = 0, oi has high probability to be picked up


Probability of an unloaded agent of picking up an object

Ppick up (oi) =

k1, k2 : threshold constants

Pdrop (oi) =

Probability of an loaded agent of dropping an object



Example of collective sorting

Attribute space: 4 gaussian distributions of real numbers

20.0

15.0

10.0

5.0

0.0

-5.0

-10.0-10.0 0.0 10.0 20.0

20.0

15.0

10.0

5.0

0.0

-5.0

-10.0-10.0 0.0 10.0 20.0



Points are randomly scattered on a grid of 52 X 52and clustering is performedwith 40 agents

t = 0 t = 500000

Example of collective sorting

Heuristic needed for better performances: heterogeneous agents (different speeds)and short term memory -> then 4 clusters …



Robots

The mission:From local actions to global tasks: stigmergy and collectiverobotics.

The plan:• Give the robot some means of moving some discrete items.• Give it a start by enabling it to make small clusters.• Think of some way of estimating local density so that it can

use the Deneubourg algorithm to make progressively largerclusters.

Puck Clustering (Beckers, Holland,and Deneubourg, 1994)

The behavior:



How does it works?

• probability of leaving a puck on a cluster increases with the size of the cluster• probability of taking a puck from a cluster decreases with the size of the cluster

so rate of growth increases with size

• adding a puck to a cluster increases its size• taking a puck from a cluster reduces its size

so the feedback is always positive

• The sum of the rates of growth over all clusters will be zero (conservation of pucks)• Therefore the rate of growth of at least the smallest cluster must be negative• So a group of n clusters will tend to become (n-1) clusters....and so on


Why a single cluster?

• clusters of 2 or less pucks are irreversibly eliminated• noise influence play a major role: algorithm is deterministic but

interactions robot-to-robot and robot-to-environment have a highstochastic component

• more quantitative analysis in the next lecture!


Other features of the robot system

• sensitive to friction and irregularities on floor• will form cluster around a suitable seed• robots are all different (small heterogeneities of the components)• robots change with time (battery life, general aging)• grippers entangling (mechanical interferences)• proximity sensors interferences (continuous emission)• puck lost by turning on the spot


Conclusion

• Robots can form clusters using a simpler algorithm than that proposed byDeneubourg: deterministic (but stochasticity in the interactions), strictly localsensing without memory (but memory = a little bit less local … ).

• Robots can show many of the advantageous features of ant behavior (robustnessto individual failure, robustness to environmental disturbance and interferences).

• The robot system is tightly coupled to the physics of the environment, andresponds coherently.

Frisbee Gathering and Sorting(Holland and Melhuish, 1998)

Bio-mimicking experiment: Leptothorax ants

• Live in cracks in rocks• Sort their brood• Perform nest migrations:

- find a new nest site- move the queen and brood there- build a surrounding wall- sort the brood

• Because of the 2D habitat and the interesting behaviors, ideal subjects forrepresenting in a land-robotic form ... but they build with particles of grit thesame width as their bodies (“blind bulldozing”)... so we need ‘building materials’the same width as the robots – Frisbees

• Arena’s area is 1760 times the area of a robot: same order of magnitude as theratio of the area of a Leptothorax nest to a single ant.

• Qualitative/feasibility rather than quantitative/efficiency study

The Robots

• 23 cm in diameter

• 3h battery autonomy

• Motorola 68332, 16 Mb RAM

• 4 continuous emitting IRproximity sensors

• Microswitch on the gripper (threshold between 1 and 2 fresbees),locking-unlocking frisbee operations controlled by an actuator

• Double optical sensor (color detection) for center and periphery colordetection of the carried frisbee


Experimental set-up


Exp. 1: behavior for basic clustering

Rule 1:if (gripper pressed & Object ahead) then make random turn away from object Rule 2:if (gripper pressed & no Object ahead) then reverse small distance make random turn left or right Rule 3: go forward

Modified Beckers et al. basic behavior for rigid walls/robots!


Frisbee Gathering (Holland andMelhuish, 1998)

Exp. 1: Results for basic clustering (10 robots, 44 frisbees)

8h 25 min, endcriterium:90% offrisbees gathered(40)

Exp. 1: comparison with performance Beckers et al. 94

• Beckers et al. 1994: 81 pucks and 3 robots in 1h 45 min

• Holland et al. 1998: 40 frisbees and 10 robots in 8h 25 min

Why? No quantitative comparison, modelling up to date …

• Different arena’s area, different robot speed

• Cluster of 1 object (Holland ‘98) vs. cluster of 2 objects(Beckers ‘94) irreversibly removed

• Noise in cluster shape (compacity of the clusters)

• …


Exp. 2: Could we form clusters at the edges of the arena?

"It must be emphasized that a very large arena was necessary inDeneubourg et al's experiments to obtain "bulk" clusters: in effect,ants are attracted towards the edges of the experimental arena if theseare too close to the nest, resulting in clusters almost exclusively alongthe edges."Eric Bonabeau, 1998

Test:• Vary the algorithm ok• Vary the sensors ok• Vary the arena size?



Exp. 2: Boundary clustering, algorithm

Rule 1:if (gripper pressed & Object ahead) then with probability p make random turn away from object else with probability (1-p) reverse small distance (dropping the frisbee) make random turn left or right Rule 2:if (gripper pressed & no Object ahead) then reverse small distance (dropping the frisbee) make random turn left or right Rule 3: go forward

Algorithm modified:

Rule 1 probabilistic!


Exp. 2: Boundary clustering, parameter bifurcationprobabilityof retention p

RESULTS

1.0 leads to a central cluster after 6 hours 35 minutes

0.95 leads to a central cluster, stopped when 2 main central clusters formed. Stopped~2.5hours

0.9 leads to a central cluster, stopped when 2 main central clusters formed. Stopped~5hours

0.88 1 cluster formed at edge. 40/44 at 9hrs 5m continued to be stable up to 11hrs20min.

0.85 1 major cluster formed at edge and approx. 15 singletons around the periphery.stopped after 1110hours

0.8 1 major cluster formed at edge and approx. 15 singletons around the periphery.Stopped after 11hrs

0.5 All pucks taken to periphery (frame 8, 0hr40mins)but no single cluster formedStopped at 11 hrs

0.0 All pucks taken to periphery (frame 3 0hr15m) but no single cluster formed.Stopped at 3hrs.


Exp. 2: Boundary clustering, parameter bifurcation

p = 0.0p = 0.5p = 0.8

p = 0.88p = 0.9p = 1.0


Exp. 2: Boundary clustering, sensory modification

Drop puck/Leave puck

Drop puck/Leave puck

wall

Mean 100.3 0

Pick up/ Retain puck

Obstacle detection only with central sensor (instead of the 3 frontal sensors)!

p =100.3/180 = 0.56

But robot movementsnot really uniformlydistributed(trajectories, no wallfollowing ) …


Exp. 2: Boundary clustering, sensory modification

Similar to p = 0.88 in the algorithmic version!

Frisbee Sorting (Holland andMelhuish, 1998)

Exp. 3: The pull back algorithm

Rule 1:if (gripper pressed & Object ahead) then make random turn away from object Rule 2:if (gripper pressed & no Object ahead) then if plain then lower pin and reverse for pull-back distance raise pin endif reverse small distance make random turn left or right

• Initial idea: change the compacity (pull back distance is a sensitive parameter) forspeeding up aggregation process

• Robot behavior different with ring or plain frisbees


Exp. 3: The pull back algorithm, results (pullback distance 2.6frisbees, 6 robots)

t = 0h00 t = 1h45 t = 8h05

Annular sorting like in Leptothorax ants!


Exp. 3: The pull back algorithm, results (pullback distance 2.6frisbees, 6 robots, end criterium: first cluster of 20 frisbees)

Trial 1 2 3 4 5

Time in hours 7.58 2.75 25.3 11.7 4.50

Number ofplains

11 12 11 10 12

High std dev

Low std dev


Conclusion

• Robots can form clusters using a simpler algorithm than that proposed byDeneubourg: deterministic (but stochasticity in the interactions), strictly localsensing without memory (but memory = a little bit less local … ).

• Robots can show many of the advantageous features of ant behavior (robustnessto individual failure, robustness to environmental disturbance and interferences).

• The robot system is tightly coupled to the physics of the environment, andresponds coherently.

But how about qualitative considerations? How can we improve the systemefficiency, how can we reduce the variability of the team performance, what arethe key parameters of the experiment?

Next lecture an answer attempt …

Generalization• Function f, replaced by fA and fB

– Memory enhanced to store quantities of A and B seen in time window• Gutowitz:

– Use entropy to track clustering dynamics:

– Pl fraction of all objects found in patch nxn• Oprisan

– All more recent objects greater weighting than old• Bonabeau

– Looked at various weighting factors, including short term activation, long term inhibition

∑−∈

=)(log

patchesslll PPE

Back to pdf

Random Placement

Unladen rule

Laden rule

Move rule


Improvements to LF algorithm• Agents move with different speeds, v

• v in [1, vmax]• Fast ants not as selected as slow ants• Clusters form over different time scales• Variable speed like temperature

>

−+

−= ∑∈

otherwise 0

0 f if 11

),(11

)( )(

max

2rNeigho

ji

i sxsj

vvood

sofα

Improvements to LF Algorithm

• Add short term memory– Store last m objects and their locations– Pick up item, compares with m objects and

moves to most similar location– Forces cluster coalescence

• Behavioral switches– Add ability to destroy clusters– Other operators are possible too (forced merge)

Comments

• Cannot expect perfect clustering:– Cluster positions in plane arbitrary– Distances cannot be correlated due to projection– Nonparametric

• Algorithm is “halfway” between:– Cluster analysis– Multidimensional scaling

• Essentially, iterative merging of clusters

Application of LF Algorithm

• Applied technique to 1650 bank records• Found interesting underlying correlations

Single, ~20, Interest checking and mostlylive at home.

Married, possibly widowed, female

owners of a house …

Married male tenants, average

age 44.


But …

• Algorithm is VERY computationally expensive

• Isn’t really worth the effort • It’s really a simple way to implement

multidimensional scaling

Graph Partitioning

• Kuntz et al– Goal is to identify clusters within graph– Can be used as a graph-drawing algorithm– Useful for VLSI layout problems

• So: start with vertices with random coordinates• Goal is to:

– Clusters should be located in same region of space– Intercluster edges are minimized– Clusters are clearly separated

• Problem is NP-complete

Distance measure

• Algorithm is similar to that of LF – Vertices move in plane– Attracted or repelled

by other vertices in cluster

– Similarity is replaced by a distance measure

∑ ∑

∑

= =

=

+

−= n

k

n

kjkik

n

kjkik

ji

aa

aavvd

1 1

1),(

Density Measure

• f(vi) a measure of average distance within graph of ith vertex to other vertices

>

−

= ∑∈

otherwise 0

0 f if ),(

11)( )(

2rNeigho

ji

i sxsj

vvdsvf α

2

2

2

1

1

)()()(

)()(

+

=

+

=

i

iid

iip

vfkvfvp

vfkkvp


Works well for smallGraphs, decays with N = 200 (see Fig. 4.13 and 4.14)


Random Placement

Unladen rule

Laden rule

Move rule

Correlations

Initial

After 2,000,000 steps

Summary

• Corpse aggregation commonly observed behaviors in ants– Simple agent model (pickup/drop) works

• Model can be extended:– Multi-attribute, real domains– Maps high dimensional spaces onto 2D

• Application to graph drawing/layout described• Has been used in collective robotics

cemetery organization, sorting, graph partitioning and data...

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