Behavioral Graph Coloring

“An Experimental Study of the Coloring Problem on Human Subject Networks”

[Science 313, August 2006]

Michael KearnsComputer and Information Science

University of Pennsylvania

Collaborators:Siddharth SuriNick Montfort

Special Thanks: Colin Camerer, Duncan Watts, Huanlei Ni

Background and Motivation

• Network Science: Structure, Dynamics and Behavior– sociology, economics, computer science, biology…– network universals and generative models– empirical studies: network is given, hard to explore alternatives

• Navigation and the Six Degrees– Travers & Milgram Watts, Kleinberg– distributed all-pairs shortest paths– what about other problems?

• Behavioral Economics and Game Theory– human rationality in the lab– typically subjects in pairs

• This Work:– human subject experiments in distributed graph coloring– controlled variation of network structure (and other variables)

(Behavioral) Graph Coloring

• Undirected graph; imagine a person “playing” each vertex• Finite vocabulary of colors; each person picks a color• Goal: no pair connected by an edge have the same color• Computationally well-understood and challenging…

– no efficient centralized algorithm known (exponential scaling)

– strong evidence for computational intractability (NP-hard)

– even extremely weak approximations are just as hard

• …Yet simple and locally verifiable

solved not solved

The Experiments: Overview

• Designed and built a system for distributed graph coloring• Designed specific sequence of experiments• Obtained human subjects review (IRB) approval• Recruited human subjects (n = 38, two sessions)• Ran experiments!• Analyzed findings

Experimental Design Variables

• Network Structure– six different topologies – inspired by recent generative models

• Information View– three different views

• Incentive Scheme– two different mechanisms

• Design space: 6 x 3 x 2 = 36 combinations• Ran all 36 of them (+2)

Research Questions

• Can large groups of people solve these problems at all?• What role does network structure play?

– information view, incentives?• What behavioral heuristics do individuals adopt?• Can we do collective modeling and prediction?

– some interesting machine learning challenges

Choices of Network Structure

SmallWorldsFamily

Simple Cycle 5-Chord Cycle 20-Chord Cycle

Leader CyclePreferential Attachment,

= 2

Preferential Attachment,

= 3

Choices of Information Views

Choices of Incentive Schemes

• Collective incentives:– all 38 participants paid if and only if entire graph is properly colored– payment: $5 per person for each properly colored graph– a “team” mechanism

• Individual incentives– each participant paid if they have no conflicts at the end of an experiment– payment: $5 per person per graph– a “selfish” mechanism

• Minimum payout per subject per session: $0• Maximum: 19*5 = $95

The Experiments: Some Details

• 5 minute (300 second) time limit for each experiment• Population demographics: Penn CSE 112 students• Handout and intro lecture to establish understanding• Intro and exit surveys• No communication allowed except through system• Experiments performed Jan 24 & 25, 2006

– Spring 2005: CSE 112 paper & pencil face-to-face experiments– Sep 2005: system launch, first controlled experiments

• Jan 24 session: collective incentives; Jan 25 session: individual incentives• Randomized order of 18 experiments within each session• First experiment repeated as last to give 19 total per session

The Results:Overview

• 31 of 38 experiments solved

• mean completion time of solved = 82s

•median = 44s

• exceeded subject expectations (52 of 76)

Effects of Network Structure

Colors required

Min. degree

Max. degree

Avg. degree

S.D. Avg. distance

Avg. duration & fraction solved

Distributed heuristic

Simple cycle

2 2 2 2 0 9.76 144.17 5/6 378

5-chord cycle

2 2 4 2.26 0.60 5.63 121.14 7/7 687

20-chord cycle

2 2 7 3.05 1.01 3.34 65.67 6/6 8265

Leader cycle

2 3 19 3.84 3.62 2.31 40.86 7/7 8797

Pref. att., newlinks=2

3 2 13 3.84 2.44 2.63 219.67 2/6 1744

Pref. att., newlinks=3

4 3 22 5.68 4.22 2.08 154.83 4/6 4703

• smaller diameter better performance• preferential attachment much harder than cycle-based• distributed heuristic gives reverse ordering

SmallWorldsFamily

Simple Cycle 5-Chord Cycle 20-Chord Cycle

Leader CyclePreferential Attachment,

= 2

Preferential Attachment,

= 3

Effects of Information View

Effects of Incentive Scheme

Towards Behavioral Modeling

Prioritize color matches to high degree nodes. That is, I tried to arrange it so that the high degree nodes had to change colors the least often. So if I was connected to a very high degree node I would always change to avoid a conflict, and vice versa, if I was higher degree than the others I was connected to I would usually stay put and avoid changing colors. [many similar comments]

Strategies in the local view: I would wait a little before changing my color to be sure that the nodes in my neighborhood were certain to stay with their color. I would sometimes toggle my colors impatiently (to get the attention of other nodes) if we were stuck in an unresolved graph and no one was changing their color.

Strategies in the global view: I would look outside my local area to find spots of conflict that were affecting the choices around me. I would be more patient in choices because I could see what was going on beyond the neighborhood. I tried to solve my color before my neighbors did.

I tried to turn myself the color that would have the least conflict with my neighbors (if the choices were green, blue, red and my neighbors were 2 red, 3 green, 1 blue I would turn blue). I also tried to get people to change colors by "signaling" that I was in conflict by changing back and forth.

If we seemed to have reached a period of stasis in our progress, I would change colorand create conflicts in my area in an attempt to find new solutions to the problem.

When I had two or three neighbors all of whom had the same color, I would go back and forth between the two unused colors in order to inform my neighbors that they could use either one if they had to.

Algorithmic Introspection(Sep 2005 comments)

(Sep 2005 data)

signaling behaviors

Machine Learning for the Collective(work in progress)

• Natural approach to develop a model of individual behavior:– treat each subject & time step as an example– develop a set of state features believed to be salient (neighbor conflicts, degrees, history,…)– transform data to <features,action> where action is new color or no change– learn a conditional model: Pr[action|features]

• Some model details:– weight vector for each action– take inner product with feature values– run through sigmoid squashing function– normalize output values to obtain conditional distribution

• Some learning details:– aggregate all subject data to learn a single model– gradient descent on log-loss

• Standard ML evaluation: log-loss on the test data– still care about this, but…

• New and interesting additional evaluation: collective behavior– run 38 copies of the model in simulation on graphs– can the learned model explain/reconstruct the ordering of the human subjects?– makes collective predictions as well

Cycle-Based Model: Training

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Cycle-Based Model: Weights

(over 96 trials) mean soln time

standard deviation

Simple cycle 14559 13661

5-chord cycle 1730 1683

20-chord cycle 112 69

Leader cycle 220 254

Cycle-Based Model: Collective Behavior

Summary

• Human groups can solve rather complex coloring problems– including from very limited, local information

• Network structure has clear effects– within cycle-based family, solution time decreases with diameter– preferential attachment appears considerably harder

• More info helpful for cycle-based, harmful for preferential attachment• Individuals adopt sensible and natural heuristics

– inverse dependence of activity on degree– signaling behaviors– injection of “randomization” to escape local minima

Future Work• More human subject experiments!

– wider variety of graph topologies– larger subject pools

• controlled vs. web-based– approximations and the “behavioral price of anarchy”– imposed vs. “natural” network structure– richer communication channels– other collective problems (independent set, consensus vs. differentiation,…)– etc. etc. etc.

• Currently designing and developing portable Java-based system• Contact:

– email mkearns@cis.upenn.edu– web www.cis.upenn.edu/~mkearns