Behavioral Graph Coloring “An Experimental Study of the Coloring Problem on Human Subject Networks” [Science 313, August 2006] Michael Kearns Computer and Information Science University of Pennsylvania Collaborators: Siddharth Suri Nick 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

Small Worlds Family Simple Cycle5-Chord Cycle20-Chord Cycle Leader Cycle Preferential 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

Science 11 August 2006: Vol no. 5788, pp DOI: /science PrevPrev | Table of Contents | NextTable of ContentsNext Graph statisticsAvg. experiment duration (s) and fraction solved Colors required Min. degree Max. degree Avg. degree S.D.Avg. distance Avg. duration & fraction solved Distributed heuristic Simple cycle / chord cycle / chord cycle /68265 Leader cycle /78797 Pref. att., newlinks= /61744 Pref. att., newlinks= /64703 smaller diameter  better performance preferential attachment much harder than cycle-based distributed heuristic gives reverse ordering

Small Worlds Family Simple Cycle5-Chord Cycle20-Chord Cycle Leader Cycle Preferential 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 color and 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 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

#opposite #same sumdeg, opp sumdeg, same maxdeg, opp maxdeg, same own degree fraction opp fraction same own degree > max opp own degree > max same constant (bias) Cycle-Based Model: Weights

(over 96 trials) mean soln time standard deviation Simple cycle chord cycle chord cycle11269 Leader cycle 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: – –web