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DANNIE-WALKER

THE ID EA OF six degrees ofseparationthat is, that ev-

ery person in the world is no

more than six people awayfrom every other person on

earthhas fascinated social scientists

and laymen alike ever since Hungarian

writer Frigyes Karinthy introduced theconcept in 1929.

For the greater public, the cultural

touchstone of the theory was the 1990play entitled Six Degrees of Separationby John Guare. Although the drama

was not an exploration of the phe-nomenon by any means, it spawned

countless versions of parlor games.

For scientists, however, the wellspringof the six degrees phenomenon, also

called the small-world problem, was

a 1967 study undertaken by social psy-

chologist Stanley Milgram, in whicha selected group of volunteers in the

Midwestern U.S. were instructed to

forward messages to a target personin Boston. Milgrams results, pub-

lished in Psychology Today in 1967,

were that the messages were deliveredby chains that comprised anywhere

between two and 10 intermediaries,

with the mean being five.In the ensuing years, the problem

has become a perennial favorite among

researchers of many disciplines, fromcomputer scientists exploring proba-

bilistic algorithms for best use of net-

work resources to epidemiologistsexploring the interplay of infectious

diseases and network theory.

Most recently, the vast architectur-al resources of Facebook and Twitter

have supplied researchers with some-

thing they never possessed beforethe

capability to look at the small-world

problem from both the traditional al-gorithmic approach, which explores

the probabilities of how each person

(or network node) in a chain seeks out

the next messenger using only the lim-ited local knowledge they possess, and

the new topological approach, which

Science | DOI:10.1145/2209249.2209255 Gregory Goth

Degrees of SeparationResearchers now have the capability to look at thesmall-world problem from both the traditional algorithmicapproach and the new topological approach.

A study of 721 million Facebook users showed an average of 3.74 intermediaries between asource and target user, as opposed to social psychologist Stanley Milgrams mean of five.


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ing a message via terrestrial delivery

routesin order to fully appreciate the

similarity of results across the board.

While the Facebook experiment yield-ed approximately four degrees of sepa-

ration, work by distinguished scientist

Eric Horvitz of Microsoft Research andStanford University assistant profes-

sor Jure Leskovec, on connections be-tween users of the Microsoft InstantMessaging network, yielded an average

6.6 degrees of separation between any

two users. In their 2009 paper SocialSearch in Small-world Experiments

examining the algorithmic approach,

Watts, Sharad Goel, and Roby Muha-

mad discovered that roughly half of

all chains can be completed in six or

seven steps, thus supporting the sixdegrees of separation assertion, they

wrote, but on the other hand, esti-

mates of the mean are much longer,

suggesting that for at least some of thepopulation, the world is not small in

the algorithmic sense.

Discovering the reason why theworld is not small in the algorithmic

sense presents a wide swath of fertile

ground for those researchers, includ-ing Watts and Leskovec, who are still

plumbing the many vectors of net-

work navigation.One ironic, or counterintuitive,

factor in examining the small-world

problem as online communities grow

ever larger is that the experimentsattrition rates are also vastly greater

than in the past. For instance, Wattssays only 12% of those who signed upfor a joint small-world experiment at

Yahoo! and Facebook completed their

chains, compared with 75% of thosewho participated in Milgrams ex-

periment and the 35% who completed

chains in a 20012002 experimentrun by Watts.

However, Watts says the data they

have should allow them to still answer

the questions they care about most,which is exploring the efficiency of in-

termediary connections selected.We know how far you are from

the target, Facebook knows how far

your friends are from the target, and

we know who you picked, so we canestablish whether you made the right

can examine the entire structure of a

network as it also observes the progres-

sion of the algorithmic chains.Its amazing how far weve come,

says Duncan Watts, a founding part-

ner at Microsoft Research New York

City, who was until recently a seniorresearcher at Yahoo! Watts is one of

the worlds leading authorities on thesmall-world problem, dating to thepublication of Collective Dynam-

ics of Small-world Networks, co-

authored with Steven Strogatz, in Na-

ture in 1998. At that time, Watts says,

the largest available network, actors

listed in the Internet Movie Database,contained about 225,000 edge nodes

(individual actors). A recent study by

researchers from Facebook and the

University of Milan, however, looked

at 721 million Facebook users, whohad 69 billion unique friendships

among them, and revealed an aver-age of 3.74 intermediaries between a

source and target user, suggesting an

even smaller world than Milgramsoriginal study showed.

In fact, the whole motivation of the

thing I did with Strogatz was preciselythat you couldnt do the exercise Face-

book just did, Watts says. Now the

empirical exercise is possible. Thats a

remarkable change.

A Similarity of Results

One must consider the large variety ofonline communities and compare the

small-world experiments performed

on them to Milgrams methodsend-

One ironic factor

in examiningthe small-worldproblem as onlinecommunities growever larger is thatthe experimentsattrition rates arealso vastly greater

than in the past.

A partial ly random string ofdigits can be amplified intototal randomness, according toa pair of theoretical physicists inSwitzerland.

Commercial randomnumber generators such asa beam splitter can send aphoton through one of two slits,generating either a 0 or a 1.But if someone has tamperedwith the generator so its outputis not perfectly random, theindividual could theoreticallydeduce patterns in the output

and thus break the code. Canwe turn this into a source of

perfect random numbers? asksRoger Colbeck of the Instituteof Theoretical Physics at ETHZurich. As Colbeck and fellowETH Zurich physicist RenatoRenner explain in an onlinepublication that appearedlast May in the journal NaturePhysics, Provided the adversarydoesnt know too much, theanswer is yes.

The solution relies onentanglement, the fact thattwo particles can be tiedtogether in such a way that

measuring a physical propertyof one immediately provides a

measurement of the other, evenif they are widely separated.Using the partially randomnumbers to decide whichproperty to measurethe angleof polarization of a photon,sayand then assigning a 0 or1 based on the outcome of thatmeasurement, produces a trulyrandom sequence.

If a high enough proportionof the initial sequence is notrandom, that affects the finalsequence in a detectable way.The ETH Zurich researchers

can run a statistical analysisof the outcomes of their

measurements and if thedistribution of outcomes straystoo far from the distributionpredicted by quantummechanics, they know thesystem is unreliable.

Colbeck would like tofind a way to generate perfectrandomness starting witheven tiny amounts of initialrandomness. That wouldprobably require performinganother, more complicatedprocedure, but just whatthat might be, the researchers

do not yet know.Neil Savage

Science

Quantum Mechanics Increases Randomness


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tion in everyday life and when youre

designing some online system.

Mozart Meets The Terminator

Recent research is beginning to use theshort-path principles of social search

in the online systems discussed by

Kleinberg. In Degrees of Separationin Social Networks, presented at the

Fourth International Symposium on

Combinatorial Search 2011, research-

ers from Shiraz University, CarnegieMellon University, and the University

of Alberta designed a search algorithm,

tested on Twitter, intended for uses be-yond social search.

For example, they reported in Voice

over Internet Protocol (VoIP) networks,when a user calls another user in the

network, he or she is first connected

to a VoIP carrier, a main node in thenetwork. The VoIP carrier connects the

call to the destination either directly

or, more commonly, through another

VoIP carrier.The length of the path from the

caller to the receiver is important since

it affects both the quality and price ofthe call, the researchers noted. The

algorithms that are developed in this

paper can be used to find a short path(fewest carriers) between the initial

(sender) and the goal (receiver) nodes

in the network.These algorithms, such as greedy

algorithms enhanced by geographic

heuristics, or probabalistic bidirection-

al methods, have the potential to cutsome of the overhead, and cost, of net-

work search sessions such as the sam-

ple VoIP session, the authors believe.

Leskovecs most recent work based

on small-world algorithms explores

the paths that humans take in connect-

ing concepts that, on the surface, seemrather disparate, such as Wolfgang

Amadeus Mozart and the Termina-

tor character from the science-fictionfilms starring Arnold Schwarzenegger.

As a human, I sort of know howthe knowledge fits together, Lesk-ovec says. If I want to go from Mozart

to Terminator and I know Mozart was

from Austria and Schwarzenegger wasfrom Austria, maybe I can go through

the Austrian connection. A computer

that is truly decentralized has no clue,

it has no conception that getting toSchwarzenegger is good enough.

Interestingly enough, Leskovec

says, computers fared better than

humans on average on solving suchsearch chains, but humans also were

less likely to get totally lost and were ca-pable of forming backup plans, which

the Web-crawling agents could not do.

Effectively, he says, the payoff of suchresearch is understanding how hu-

mans do this, what kind of cues are we

using, and how to make the cues more

efficient or help us recognize them, tohelp us understand where we are, right

now, in this global network.

Further Reading

Backstrom, L., Boldi, P., Rosa, M.,Ugander, J., and Vigna, S.

Four degrees of separation, http://arxiv.org/

abs/1111.4570, Jan. 6, 2012.

Bakhshandeh, R., Samadi, M.,Azimifar, Z., and Schaeffer, J.

Degrees of separation in social networks,

Proceedings of the Fourth International

Symposium on Combinatorial Search,

Barcelona, Spain, July 1516, 2011.

Goel, S., Muhamad, R., and Watts, D.

Social search in small-world

experiments, 18th

International WorldWide Web Conference, Madrid, Spain,

April 2024, 2009.

Kleinberg, J.

The small-world phenomenon: an

algorithmic perspective, 32ndACM

Symposium on Theory of Computing,

Portland, OR, May 2123, 2000.

West, R., and Leskovec, J.

Human wayfinding in information networks,

22ndInternational World Wide Web

Conference, Lyon, France, April 1620, 2012.

Gregory Goth is an Oakville, CT-based writer whospecializes in science and technology.

2012 ACM 0001-0782/12/07 $15.00

choice, Watts says. So we can get the

most science out of it, its just a little

bummer that the attrition was so bad.

The logic behind finding the mostefficient paths may produce payoffs

unforeseen for both theoretical mod-

eling and production networks suchas search engine optimization. Find-

ing the best ways to determine thosepaths, though, will necessitate a leapfrom the known models of small-world

networks to a better understanding of

the intermediary steps between anytwo endpoints of a chain.

Leskovec says, given constants from

graph theory, the diameter of any given

network will grow logarithmically withits size; that is, the difference between

five and six degrees of separation man-

dates a graph an order of magnitude

larger or denser. Jon Kleinberg, TischUniversity professor in the department

of computer science at Cornell Univer-sity, whose The Small-World Phenom-

enon: An Algorithmic Perspective is re-

garded as one of the problems seminalmodeling documents, says this basic

property is precisely what makes the

small-world theory so appealing while

also presenting the research communi-ty the greatest challenge inherent in it.

Its something that still feels coun-

terintuitive when you first encounter

it, Kleinberg says. It makes sense inthe end: I know 1,000 people and my

friend knows 1,000 peopleand youdont have to multiply 1,000 by itself

too many times for it to make sense.

However, this logarithmic progres-

sion also precludes the ability to ex-amine or design intermediate levels

of scale, Kleinberg says. We thought

the right definition of distance wasgoing to be Here I am, and how many

steps do I have to go to get to you?

but that turns out not to be. We need

some other measure and I think thatremains an interesting open question,

that people are actively looking at: Isthere some kind of smoother scale

here? Who are the 10,000 people clos-

est to me? The 100,000?

We need a much more subtle wayto do that and it is going to require

some sophisticated mathematical

ideas and sophisticated combination-al ideaswhat is the right definition of

distance when youre looking at social

networks? Its not just how many stepsI have to go. Thats an important ques-

What is the rightdefinition of distancewhen youre lookingat social networks?

asks Jon Kleinberg.Its not justhow many stepsI have to go.


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