Презентация Леонида Жукова: От микромотивов до...

Social Networks: from Micromotives to Macrobehavior

Leonid Zhukov Moscow, June 2014

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Micromotives and macrobehavior

Combined local actions -> global resultsSimple rules -> complex behaviorLocal structure -> global properties

Thomas C. Shelling, 1978!Nobel Prize in Economics 2005

Talk outlineNetwork structureNetwork modelsProcesses on networks

Complex networks:social networks, internet, WWW, citation networks, financial networks, trade networks, transportation networks, protein interaction networks, brain networks, food web, language networks, …

Network structure

Local structureNode degree = # of friends

Global: degree distributionHow many people with:

1 friend 2 friends 3 friends

!

Distribution: P(n)Power-law distribution

Power-law degree distribution

Local structureNode degree = # of friends!

Friend’s friends = triangles

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The strength of weak “The Strength of Weak Ties”. Mark Granovetter. 1973

Mark Granovetter. The strengtth of weak ties , American Journal of Sociology, 78(6):1360-1380, 1973

Triadic closerStrength of tiesTriadic closure: if A and B and A and C are strongly linked, then the tie between B and C is always presentClustering coefficient

Global: community structure

Local structureNode degree = # of friends!

Friend’s friends = triangles!

“Long connections” = small network diameter

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Small world

J. Travers and S. Milgram. An Experimental Study of the Small World Problem. Sociometry, vol 32, No 4, pp 425-433, 1969

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1969 Experiment296 volunteers, 217 sent

196 Nebraska (1300 miles)

100 Boston (25 miles)

target in Boston

vaguely ‘out there,’ on the Great Plains or somewhere.” There was littleconsensus about how many links it would take to connect people fromthese remote areas. Milgram himself pointed out in 1969, “Recently Iasked a person of intelligence how many steps he thought it would take,and he said that it would require 100 intermediate persons, or more, tomove from Nebraska to Sharon.”

Milgram’s experiment entailed sending letters to randomly chosenresidents of Wichita and Omaha asking them to participate in a studyof social contact in American society. The letter contained a shortsummary of the study’s purpose, a photograph, and the name and ad-dress of and other information about one of the target persons, alongwith the following four-step instructions:

HOW TO TAKE PART IN THIS STUDY

1. ADD YOUR NAME TO THE ROSTER AT THE BOT-TOM OF THIS SHEET, so that the next person who re-ceives this letter will know who it came from.

2. DETACH ONE POSTCARD. FILL IT OUT AND RE-TURN IT TO HARVARD UNIVERSITY. No stamp isneeded. The postcard is very important. It allows us to keeptrack of the progress of the folder as it moves toward the tar-get person.

3. IF YOU KNOW THE TARGET PERSON ON A PER-SONAL BASIS, MAIL THIS FOLDER DIRECTLY TOHIM (HER). Do this only if you have previously met thetarget person and know each other on a first name basis.

4. IF YOU DO NOT KNOW THE TARGET PERSON ON APERSONAL BASIS, DO NOT TRY TO CONTACT HIMDIRECTLY. INSTEAD, MAIL THIS FOLDER (POST-CARDS AND ALL) TO A PERSONAL ACQUAIN-TANCE WHO IS MORE LIKELY THAN YOU TOKNOW THE TARGET PERSON. You may send the folder

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to a friend, relative or acquaintance, but it must be someoneyou know on a first name basis.

Milgram had a pressing concern: Would any of the letters make itto the target? If the number of links was indeed around one hundred, ashis friend guessed, then the experiment would likely fail, since there isalways someone along such a long chain who does not cooperate. It wastherefore a pleasant surprise when within a few days the first letter ar-rived, passing through only two intermediate links! This would turn outto be the shortest path ever recorded, but eventually 42 of the 160 let-ters made it back, some requiring close to a dozen intermediates. Thesecompleted chains allowed Milgram to determine the number of peoplerequired to get the letter to the target. He found that the median num-ber of intermediate persons was 5.5, a very small number indeed—andcoincidentally, amazingly close to Karinthy’s suggestion. Round it up to6, however, and you get the famous “six degrees of separation.”

As Thomas Blass, a social psychologist who has devoted the last fif-teen years to in-depth research on the life and work of Stanley Milgram,pointed out to me, Milgram himself never used the phrase “six degrees ofseparation.” John Guare originated the term in his brilliant 1991 play ofthat title. After an extremely successful season on Broadway, the play wasmade into a movie with the same title. In the play, Ousa (played byStockard Channing in the movie), musing about our interconnectedness,tells her daughter, “Everybody on this planet is separated by only sixother people. Six degrees of separation. Between us and everybody elseon this planet. The president of the United States. A gondolier inVenice. . . . It’s not just the big names. It’s anyone. A native in a rain for-est. A Tierra del Fuegan. An Eskimo. I am bound to everyone on thisplanet by a trail of six people. It’s a profound thought. . . . How every per-son is a new door opening up into other worlds.”

Milgram’s study was confined to the United States, linking people“out there” in Wichita and Omaha to “over here” in Boston. ForGuare’s Ousa, however, six degrees applied to the whole world. Thus amyth was born. Because more people watch movies than read sociologypapers, Guare’s version has prevailed in popular thought.

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NAME, ADDRESS, OCCUPATION, JOB, HOMETOWN

1969 Experiment reached the target N = 64, 29%

ave chain length <l> = 5.2

channels:

hometown <l> = 6.1

business contacts <l> = 4.6

Location:

boston <l> = 4.4

nebraska <l> = 5.7

1994,Premiere magazine

Six degrees of Kevin Bacon

Universal properties of social networks

Power low degree distributionLarge clustering coefficientSmall world effect!

Gigantic connected componentTight core and peripheryHierarchical structure

Network models

Small world network

Duncan J. Watts and Steven H. Strogatz. Collective dynamics of ‘small-world’ networks. . Nature 393:440-42, 1998.

Preferential attachment

Preferential attachement

AL Barabasi and R. Albert. Emergence of Scaling in Random Networks. Science, 286, 1999.

Strategic network formation

Utility functionDistance based utilityCosts and benefits of connectionsMaximizing individual utility

Matthew O. Jackson, Asher Wolinsky, . A Strategic Model of Social and Economic Networks. Journal of Economic Theory, Vol 71, pp 44-74, 1996.

50 CHAPTER 2. REPRESENTING AND MEASURING NETWORKS

Figure 2.1.6 A Complete Network on Six Nodes and a Star Network onSix Nodes

A circle (also known as a cycle-graph) is a network that has a single cycle and such

that each node in the network has exactly two neighbors.

In the case of directed networks, there can be many di§erent stars involving the

same set of nodes and having the same center, depending on which directed links are

present between any two linked nodes. On occasion, it will be useful to distinguish

between these.

2.1.7 Neighborhood

The neighborhood of a node i is the set of nodes that i is linked to.8

Ni(g) = fj : gij = 1g:

8Note that whether i is in iís neighborhood depends on whether or not we have allowed gii = 1.

As I am following a default convention of gii = 0, i will generally not be considered to be in iís

neighborhood. This ensures that iís degree is the number of other nodes that i is linked to, which is

then the cardinality iís neighborhood.

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Strategic network formation

Pairwise stable small world network224 CHAPTER 6. STRATEGIC NETWORK FORMATION

Figure 6.5.1. A Pairwise Stable ìSmall Worldî in an Islands Version of theConnections Model

The intuition behind the proposition is relatively straightforward. Low costs of

connections to nearby players (those on the same island) lead to high clustering. The

high value to linking to other islands (accessing many other players) leads to low average

path length. The high cost to linking across islands means that there are only a few

links across islands.

These properties are illustrated in Figure 6.5.1. Figure 6.5.1 is for a case where

c < :04, 1 < C < 4:5, % = :95. players are grouped in sets of Öve who are completely

connected to and lie on the same island, and there are Öve separate islands.

This economic analysis of small worlds gives complementary insights to those of

rewiring analysis of Watts and Strogatz [623] discussed in Section ??. The randomrewiring analyzed by Watts and Strogatz shows that it is possible to have both high

clustering and short path length at the same time, whereas the above model gives more

insight into why we should expect this to be exhibited by social networks.

Another feature distinguishing an economic modeling and a random modeling of

these network characteristics concerns ìshortcutî links (i.e., those which link distant

parts of the network and if deleted would substantially alter the distance between the

connected nodes). In a random rewiring model shortcut links would at least occasion-

Processes on networks

Processes on networksInformation propagation

diffusivevirus like

Decision makingthreshold models

Diffusion modelmethod - diffusionvirus model“infected” on contactprobability depends on immunity can model

newsgossips

Diffusion modelStep 1

Diffusion modelStep 2

Diffusion model Step 3

Diffusion modelStep 4

Diffusion modelStep 5

!

Complete coverageProcess time depends on the sourceBased on connectivity pattern

Twitterretweetmention

“Truthy” Project. Center for Complex Networks and System Research. Indiana University.

#newsjp

#iraq

#ocra

Threshold modelneighbors “opinion”decision thresholdinformation cascademodel:

beliefs propagationpurchasing decisionsspread of innovations

A,B - types of behavior

q –decision threshold

If fraction of neighbors with A greater than q, accpet A

Threshold modelthreshold= 1/2

Threshold modelStep 1

2 sourcesthreshold= 2/5

Threshold modelStep 2

Threshold modelStep 3!

Threshold modelStep 4

incomplete cascadedepends on network topologystrongly depends on source selection

Cascade maximizationCascade maximization problem

Strategic source placementWell connected group of nodesInside various communities

Increase of competitive advantagereduce threshold level

Visual Complexitywww.visualcomplexity.com

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Textbooks

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