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Introductory talk about history, structure, dynamics and processes in social networks

Introductory talk about history, structure, dynamics and processes in social networks

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Social Networks Social Networks Presentation Transcript

  • SOCIAL NETWORKS FIVE SHORT STORIES LEONID ZHUKOV NATIONAL RESEARCH UNIVERSITY HIGHER SCHOOL OF ECONOMICS LZHUKOV@HSE.RU 1
  • FIVE SHORT STORIESSCIENTISTS AND POETS THIS IS A SMALL WORLD RICH GET RICHER STRENGTH OF WEAK TIES ECONOMICS OF FRIENDSHIP FOLLOWING THE CROWD 2
  • SCIENTISTS AND POETS INTRODUCTORY STORY 3
  • THE VERY BEGINNING1736: LEONARD EULER. KOENIGSBERG BRIDGES1929: FRIGYES KARINTHY “CHAINS - LANCSZEMEK” 4
  • 60TH AND 70TH 1959: PAUL ERDOS, RANDOM NETWORKS1967: STANLEY MILGRAM, SMALL WORLD 1973: MARK GRANOVETER, STRENGTH OF WEAK TIES 5
  • LAST 10 YEARSALBERT-LÁSZLÓ BARABÁSI, NORHEASTERN, PHYSICS.DUNKAN WATTS, COLUMBIA, SOCIOLOGYPAUL NEWMAN, UNIV OF MICHIGAN, PHYSICSJOHN KLEINBERG, CORNELL, COMPUTER SCIENCEMATTHEW JACKSON, STANFORD, ECONOMICS 6
  • SUBJECTSCOMPUTER SCIENCE: ALGORITHMS, GRAPH THEORY,SEARCH ON GRAPH, PATHS LENGTH, CONNECTEDCOMPONENTS, CLIQUES, GRAPH COLORING ETCSOCIOLOGY: SOCIAL ROLES, STATUS, IDENTITY,COMMUNITIES, INFLUENCE, COHESIVENESSPHYSICS: STATISTICS, PHASE TRANSITIONS, EVOLUTIONMODELS, DYNAMICAL SYSTEMECONOMICS: NETWORK GAMES, OPTIMALITY,EQUILIBRIUM 7
  • COMPLEX NETWORKSNETWORK(GRAPH) : NODESAND CONNECTIONS (EDGES)COMPLEXNOT REGULAR NOR RANDOMVARIOUSUNIVERSAL CLASS C NETWORKS IMAGE BY BARRETT LYON 8
  • COMPLEX NETWORKSPROTEIN - PROTEIN INTERACTION MAP OF SCIENTIFIC JPORNALS IMAGE BY HAWOONG JEONG IMAGE BY JOHAN BOLLEN 9
  • COMPLEX NETWORKS TWITTER FOLLOWERS IMAGE BY BURAK ARIKAN 10
  • THIS IS A SMALL WORLD FIRST STORY 11
  • SMALL WORLD“THE SMALL-WORLD PROBLEM”. STANLEY MILGRAM. 1967.“AN EXPERIMENTAL STUDY OF THE SMALL WORLD PROBLEM”, J. TRAVERS, S. MILGRAM,1969 12
  • these remote areas. Milgram himself pointed out in 1969, “Recently I asked a person of intelligence how many steps he thought it would take, and he said that it would require 100 intermediate persons, or more, to move from Nebraska to Sharon.” Milgram’s experiment entailed sending letters to randomly chosen residents of Wichita and Omaha asking them to participate in a study 1969 EXPERIMENT of social contact in American society. The letter contained a short summary of the study’s purpose, a photograph, and the name and ad- dress of and other information about one of the target persons, along with the following four-step instructions: HOW TO TAKE PART IN THIS STUDY296 VOLUNTEERS, 217 SENT 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. 196 NEBRASKA (1300 MILES) 2. DETACH ONE POSTCARD. FILL IT OUT AND RE- TURN IT TO HARVARD UNIVERSITY. No stamp is needed. The postcard is very important. It allows us to keep 100 BOSTON (25 MILES) track of the progress of the folder as it moves toward the tar- get person. 3. IF YOU KNOW THE TARGET PERSON ON A PER-TARGET IN BOSTON SONAL BASIS, MAIL THIS FOLDER DIRECTLY TO HIM (HER). Do this only if you have previously met the target person and know each other on a first name basis. 0738206679-01.qxd 3/13/02 2:08 PM Page 29 4. IF YOU DO NOT KNOW THE TARGET PERSON ON A PERSONAL BASIS, DO NOT TRY TO CONTACT HIM DIRECTLY. INSTEAD, MAIL THIS FOLDER (POST- CARDS AND ALL) TO A PERSONAL ACQUAIN- TANCE WHO IS MORE LIKELY THAN YOU TO Six Degrees of Separation 29 KNOW THE TARGET PERSON. You may send the folder to a friend, relative or acquaintance, but it must be someone you know on a first name basis. Milgram had a pressing concern: Would any of the letters make it to the target? If the number of links was indeed around one hundred, as NAME, ADDRESS, the experiment would likely fail,HOMETOWN his friend guessed, then OCCUPATION, JOB, since there is always someone along such a long chain who does not cooperate. It was therefore a pleasant surprise when within a few days the first letter ar- rived, passing through only two intermediate links! This would turn out to be the shortest path ever recorded, but eventually 42 of the 160 let- ters made it back, some requiring close to a dozen intermediates. These 13 completed chains allowed Milgram to determine the number of people
  • 1969 EXPERIMENTREACHED THE TARGET N = 64, 29%AVE CHAIN LENGTH <L> = 5.2CHANNELS: HOMETOWN <L> = 6.1 BUSINESS CONTACTS <L> = 4.6LOCATION: BOSTON <L> = 4.4 NEBRASKA <L> = 5.7 14
  • SIX DEGREES OF SEPARATION DUNCAN WATTS, 2001, EMAIL, 48,000 SENDERS, <L> ~ 6 JURE LESKOVEC AND ERIC HORVITZ, 2007, MSN MESSENGER 240 MLN USERS, < L> = 6.6 USERS YAHOO, 2011, “YAHOO RESEARCH SMALL WORLD EXPERIMENT” ON FACEBOOK :) GRAPH DIAMETER DAVE PATH LENGTH <L> CO-AUTHORSHIP NETWORK IMAGE BY LOTHAR KREMPEL 15
  • CAYLEY TREE (MOORE GRAPH) 6 26 106 A ROUGH ESTIMATE: EXACT: EACH HAS D FRIENDS D^K = N K = LOG N/LOG D 6 BLN 50 FRIENDS K~ 5.8 16
  • SMALL WORLD MODEL WATTS-STROGATZ MODEL SOLVABLE MODEL SMALL WORLD: <L>~ LOG(N)“COLLECTIVE DYNAMICS OF SMALL-WORLD NETWORK”, D.J STROGATZ, S.H. WATTS. 1998 17
  • RICH GET RICHER SECOND STORY 18
  • RANDOMNETWORKS, which resemble the U.S.highway system nodes with a very high number of links. In such networks, the (simplified in left map), consist of nodes with randomly placed distribution of node linkages follows a power law [center graph) connections. In such systems, a plot of the distribution of node in that most nodes have just a few connections and some have linkages will follow a bell-shaped curve (left graph), with most a tremendous number of links. In that sense, the system has no nodes having approximately the same number of links. "scale." The defining characteristic of such networks is that the SIMPLE HYPOTHESIS In contrast, scale-free networks, which resemble,the U.S. airline system (simplified in right map). contain hubs [red)- distribution [right graph), of links, if plotted on a double-logarithmic results in a straight line. scale RandomNetwork Scale-Free NetworkWEB SEARCH 1999: LYCOS, 1994; ALTAVISTA 1995, YAHOO, 1995; INKTOMI, 1996; GOOGLE 1998.... RAMBLER 1996; YANDEX 1997 EACH PAGE LINKS INDEPENDENTLY AT RANDOM, CLT -> NORMAL DISTRIBUTION Bell Curve ~istribution of Node Linkages PowerLaw Distribution of Node Linkages L ~ ~ If) QJ -c 0 . ~~ Z 0 0 QJ 0 Z zCij ""0 0 0 If) c;; ..c ~ ~ 011 E ~ ~~ :::J E E~ Z :::J. :::J Z Z Number of Links Number of Links Number of Links (log scale)Specifically, a power OF SCALING IN “EMERGENCE law does not have a RANDOM NETWORKS”.Abound some social ALBERT. 1999 Scale-Free Networks A-L BARABASI, R networks are scale-free. A col-peak, as a bell curve does, but is instead de- OVER THE PAST several years, re- laboration between scientists from Bostonscribed by a continuously decreasing func- searchers have uncovered scale-free struc" University and Stockholm University, fortion. When plotted on a double-logarith- tures in a stunning range of systems. instance, has shown that a netWork ofmic scale, a power law is a straight line 19 When we studied the World Wide Web, we looked at the virtual network of Web sexual relationships among people in
  • vertices decays as a power law, following atively modest size of the network, contain- common featur P(k) k . This result indicates that large ing only 4941 vertices, the scaling region is is that the prob networks self-organize into a scale-free state, less prominent but is nevertheless approxi- connected verte a feature unpredicted by all existing random mated by a power law with an exponent es exponentiall POWER LAW DISTRIBUTION network models. To explain the origin of this scale invariance, we show that existing net- power 4 (Fig. 1C). Finally, a rather large complex network is formed by the citation large connectiv contrast, the po work models fail to incorporate growth and patterns of the scientific publications, the ver- P(k) for the net preferential attachment, two key features of tices being papers published in refereed jour- highly connect real networks. Using a model incorporating nals and the edges being links to the articles large chance o DISTRIBUTION FUNCTION connectivity. There are tw works that are n els. First, both with a fixed nu then randomly c connected (WS N. In contrast, m open and they f tion of new ver number of vert the lifetime of t actor network g actors to the sys ACTOR COLLABORATION WWW POWER GRID Fig. 1. The distribution function of connectivities for various large networks. (A) Actor collaboration nentially over t GAMMA= 2.3 graph with N GAMMA = 2.3 212,250 vertices and average connectivity k GAMMA= 4 28.78. (B) WWW, N Web pages (8) 325,729, k 5.46 (6). (C) Power grid data, N 4941, k 2.67. The dashed lines have constantly grow slopes (A) actor 2.3, (B) www 2.1 and (C) power 4. papers. Conseq510 15 OCTOBER 1999 VOL 286 SCIENCE www.sciencemag.org 20
  • GRAPH STRUCTURE OF THE WEB“GRAPH STRUCTURE IN THE WEB” ANDREJ BRODER, RAVI KUMAR, ET AL. 2000. 21
  • SCALE FREE NETWORKS 6 6 4 (a) 10 (b) (c) 10 4 4 10 10 2 10 2 2 10 10 0 0 0 10 10 10 0 2 4 0 2 4 0 2 4 10 10 10 10 10 10 10 10 10 word frequency citations web hits (d) (e) 4 (f) 10 6 100 10 3 10 3 10 10 2 10 0 1 10 6 7 0 2 4 6 10 10 10 10 10 10 2 3 4 5 6 7 books sold telephone calls received earthquake magnitude 2 (g) 4 10 (h) 100 (i) 10 3 0 10 10 2 10 -2 10 10 1 -4 10 10 1 2 3 4 5 0.01 0.1 1 10 10 10 10 1 10 100 crater diameter in km peak intensity intensity 4 10 (l) (j) 4 10 (k) 100 2 2 10 10 10 0 0 1 10 10 9 10 4 5 6 3 5 7 10 10 10 10 10 10 10 10 net worth in US dollars name frequency population of citySTEVEN H. STROGATZ, 2001 MARK E.J. NEWMAN, 2006 FIG. 4 Cumulative distributions or “rank/frequency plots” of twelve quantities reputed to follow power laws. The distributions were computed as described in Appendix A. Data in the shaded regions were excluded from the calculations of the exponents in Table I. Source references for the data are given in the text. (a) Numbers of occurrences of words in the novel Moby Dick by Hermann Melville. (b) Numbers of citations to scientific papers published in 1981, from time of publication until June 1997. (c) Numbers of hits on web sites by 60 000 users of the America Online Internet service for the day of 1 December 1997. 22 (d) Numbers of copies of bestselling books sold in the US between 1895 and 1965. (e) Number of calls received by AT&T telephone customers in the US for a single day. (f) Magnitude of earthquakes in California between January 1910 and May 1992.
  • to the Limitdeviates from a Poisson distribution. We have seen in (1) Growth: Starting with a sma Secs. III.D and VI.B.3 that random-graph theory and nodes, at every time step, we add k m( m k i edges that link the new i n property of many complex networks” (7), it was the WS model cannot reproduce thisasfeature. While it is more of a prediction than a fact, because nature 0) could have chosen many different architec- m k mN 1 nodes already presenti in the system.etworks: A Decade distribution (Sec. V), of modernconstruc- straightforward to constructasrandom graphs that have a tures there are networks. Yet, probably the t PREFERENTIAL ATTACHMENT power-law degree most surprising discovery these topology: theory is the universality of the network tions only postpone an important question: towhat is the network Many real networks, from the cell the Internet, independent of their age, function, and scope, (2) Preferential attachment: When to which the new node connects, w probability that a new node jwill be 1 k mechanism responsible for the emergence of isscale-free converge to similar architectures. It this uni- networks? We shall see versality that allowed researchers from different in this section that answering i depends on the degree k i of node at the components of such complex systems as the cell, the thisdecade, an avalanche Orequiredisciplines to from network theory asnetwork question will research Smon shift embrace modeling a com- R E P of R T a paradigm. The sum in the denominato ki ly wired together. In the past s, independent of theirBARABASIto modeling the network assembly networks of topology ALBERT MODEL Today, the scale-free nature of and evolu- age, function, and scope, converge to that allowed researchers from different disciplines to embrace key scientific interest, from protein interactions to 2 system except. the newly int kis- The decade-oldadiscovery of scale-free networks was one of social networks andequal networkm 2inter- not 1 j k jm 22mt km, 0 ). The prob- igm. that new While atis connected with from the tion. vertex this point these two approaches do ) of t/k t/k 2 (t j m leading toes appear to be NETWORK: ON EVERYwe shall NEW that there particularlythe system [thatfind ability density P(k) can be obtained from distinct, STEP A ze the emergence of network science, a new research field with linked documents that make up the WWW to the probability to any vertex in interconnected hardware behind the Internet, has GROWING j ccomplishments. is a fundamental THAT difference between the modeling ap- is, (k) NODEwe ADDEDin might been establishedfromSuch small-world P[k iAfterkti time k, which procedure r k]/ ksteps this over long Downloaded from www.sciencemag.org on July 24, 2009 y are suspected that the scale-freetook1/(m 0 LINKS TO EXISTING TheP(k) IS 1)]. better maps t not only and the data sets const property (6) random graphs beyond doubt. and evidence (t) ihnologies proach NODES comes a model (Fig. 2B) one required to reproduce the network periodstions indicated stationarynetwork ev leads to P(k) time . with N t m 0 nodes and mt edges leads to the that this solution etworks that not be unique to the WWW. The main purpose of but also from the agreement between empirical t to fail but the 1999 Science paper wasthereport this data and analytical models that predictthe power- models, and to t 2ts-proved like unexpected similarity show thatdistribution. the negative side goal prompting some not o exp( law and indicating ATTACHMENT: PROBABILITYthewas re- too k), degree networks of quite structureabsence earlyof PREFERENTIAL that without between While the the of euphoria former (10, 11). Yet, invariant state with the probability models, different nature to two mechanisms, effects, thematicians growth models is to construct a searchers tothe many systems scale-free, even and OF CONNECTION TO A graph with correct topological ThePsolutiona 2m 2of this equatio e much preferential (Fig. 1). attachment, are the NODE IS label was scarce at best. However, attachment eliminates evidence scale- preferential PROPORTIONAL edges following 3 power law with a k ove features, inNODE DEGREE of netthe was to force us to better understand put of underlying causes the modeling when resultn- They feature THEthe distribution. In model B, TO ofystems. free When we concluded 1999 that we “expect the scale-free networks will k that every node i at its intr (see Fig. 21). The scaling exponent is red randomly that the the invariant state […] is a generic the factors that shape network structure. For ex- scale emphasis on capturing the network dynamics. That N vertices and no behind evolving or dy- 3, independent of in the model. it the only parameter 72 ´ R. Albert and A.-L. Barabasi: Statistical mechanics of complex networksal by so- start with underlying assumption edges. At opted we is, the BIRTHOFASCALE-FREE NETWORK giving m. Although ence. It hadis for ex- ining each time step, we randomly select a vertex namic networks is that if we capture correctly the pro- A SCALE-FREE NETWORK grows incrementally reproduces theTheoretical approaches distribu- B. observed scale-free No. 1, Ja from two to 11 nodes in this example. When deciding where to establish a link, a new node (green) prefers to attach to an existing node (red) that already has many other connections. Rev. Mod. Phys., Vol. 74, These two basic mechanisms-growth ehandshakes omenon ob- and connect cesses it with probability networks that /we see today, proposed model cannot be expected and preferential that assembled the attachment-will eventually (k i ) ki tion, the lead to the systems being dominated by hubs, nodes having an enormous number of links.c- Watts and j k j tothen we will obtainsystem. Although at as to account forThe aspects ofproperties of the s ch resonated an vertex i in the their topology correctly well. Dy- .---- -1 ~ ~ all dynamical the studied net- ~ namics takes the driving role, topology being only a by-w of theh beyond so- ccess early times the this modeling philosophy. product of model exhibits power-law works. For be addressed usingproposed analyti that, we theory to various by Ba continuum need model theseis That scaling, P(k) is not stationary: because N is mental ques- systems in more detail. For example, in theom? e is, constant and the number No. 1,edges increases society func-molecules, or r? Rev. Mod. Phys., Vol. 74, of January 2002y,This ques- time, after T N 2 time steps the system with connected actors are more likely to be ~~ why scale-free networks are so ubiquitous model we assumed linear preferential attach- ment; that is, (k) k. However, although an existing node that has twice as manyding 10 years y propertyreaches a state in which all vertices are con- in general (k) could have an arbitrary non- chosen for new roles. On the Internet the in the real world. connections), one hub will tend to run ree more connected routers, which typically Growth and preferential attachment away with the lions share of connections. have greater bandwidth, are more desir- can even help explicate the presence of In such "winner take all" scenarios, the FIG. 21. Numerical simulations of network evolution: (a) Degree distribution of the Barabasi-Al ´c-may show s nected. The failure of models A and B indi- able for new users. In the U.S. biotech in- dustry, well-established companies such as scale-free networks in biological systems. network eventually assumes a star topol- linear form (k) k , simulations indicate 300 000 and , m 0 m 1; , m 0 m 3; , m 0 m 5; and , m 0 m 7. The slope of the das 23 Andreas Wagner of the University of ogy with a central hub. the best fit to the data. The inset shows the rescaled distribution (see text) P(k)/2m 2 for the same v Genzyme tend to attract more alliances, New Mexico and David A. Fell of Oxford ame 10 years which further increases their desirability Brookes University in England have AnAchilles Heel dashed line being 3; (b) P(k) for m 0 m 5 and various system sizes, , N 100 000; , N 1
  • PREFERENTIAL ATTACHMENT 24
  • TWO MODELSRANDOM GRAPH BARABASI-ALBERT MODEL 25
  • STRENGTH OF WEAK TIES THE THIRD STORY 26
  • THE STRENGTH OF WEAK TIES“THE STRENGTH OF WEAK TIES”. MARK GRANOVETTER. 1973 27
  • TIE STRENGTH OF TIESSTRENGTH OF A TIE: AMOUNT OF TIME EMOTIONAL INTENSITY INTIMACY RECIPROCITYTRIADIC CLOSURE IF A AND B AND A AND C ARE STRONGLY LINKED, THEN THE TIE BETWEEN B AND C IS ALWAYS PRESENTCLUSTERING COEFFICIENT 28
  • BRIDGESBRIDGE - A LINE IN A NETWORK WHICHPROVIDES THE ONLY PATH BETWEEN TWOPOINTS BRIDGE IS LOCAL BRIDGE IF ITS REMOVALINCREASES DISTANCE BETWEEN TWOPOINTSNO STRONG TIE IS A BRIDGEROLE IN DIFFUSION 29
  • STRENGTH OF WEAK TIESWHERE IS THE STRENGTH?JOB CHANGES: 16.7% FRIENDS (1-2 CONTACTS A WEEK) 55.6% ACQUAINTANCES (OCCASIONAL CONTACTS, MORE THEN ONCE A YEAR ) 27.8% RARELYWEAK TIES = ”LONG TIES”,CONNECT PEOPLE FROMDIFFERENT COMMUNITIES 30
  • FACEBOOK All Friends Maintained RelationshipsOne-way Communication Mutual Communication DECLARED FRIENDSHIP MAINTAINED RELATIONSHIPS ONE WAY COMMUNICATIONS CAMERON MARLOW ET. AL , 2009 31
  • IN THE CIRCLE OF FRIENDS THE FOURTH STORY 32
  • COMMUNITY DETECTIONCOMMUNITY - A SET OF NODES CONNECTED AMONG THEMSELVES MORE THAN WITH THEREST OF THE NETWORK 33
  • GRAPH THEORY METHODSGRAPH CUTS: FLOW METHODS MIN CUT, NORMALIZED CUTS GREEDY ALGORITHMS SPECTRAL METHODS MULTI RESOLUTION METHODS 34
  • BETWEENNESS CENTRALITY advanced material 2 9 6 NODE BETWEENNESS CENTRALITY IS 1 4 5 7 8 11 PROPORTIONAL TO THE NUMBER OF SHORTEST PATHS GOING THROUGH THE NODE 3 10 (a) 2 9 EDGE BETWEENNESS 6 1 4 5 7 8 11 ITERATIVELY REMOVING THE WEAKEST 3 10 (b) 2 9 6“A SET OF MEASURE OF CENTRALITY BASED ON 1 4 5 7 8 11BETWEENNESS”. LINTONC.FREEMAN, 1977. 3 10 (c)“FINDING AND EVALUATING COMMUNITY STRUCTURE IN 2 6 9NETWORKS” . MARK E. J. NEWMAN AND MICHELLE 1 4 5 7 8 11GIRVAN. 2004. 3 10 (d) Figure 3.17. The four steps (a)–(d) of the Girvan–Newman method applied to the network f Figure 3.15. 35
  • ECONOMICS OF FRIENDSHIP THE FIFTH, BUT NOT THE LAST STORY 36
  • GAME THEORYGAME THEORY IS THE STUDY OF THE WAYS IN WHICH STRATEGIC INTERACTIONS AMONGECONOMIC AGENTS PRODUCE OUTCOMES WITH RESPECT UTILITIES OF THOSE AGENTS,NOTION OF PAYOFFPAYOFF TABLERATIONAL PLAYERS, ACTING IN THEIR SELF INTERESTSNETWORK FORMATION GAME 37
  • UTILITARIAN RELATIONSHIPS LINKS - SOCIAL RELATIONSHIPS, FRIENDSHIP CONNECTIONS OFFER BENEFITS: FAVORS, SUPPORT, INFORMATION (0<D<1) 1.2. A SETBASED UTILITY FUNCTION DISTANCE OF EXAMPLES: 27 t t t PAY1 2 COSTS FOR DIRECT RELATIONSHIPS (0<C<1) 3 4  +  2 +  3  c FROM INDIRECT 2  2c PLAYERS BENEFIT 2 +  2 DETERIORATES WITHDISTANCE 2 +  RELATIONSHIPS, BENEFITS 2c  + 2 + 3  c (B^D) Figure 1.2.3 The utilities to the players in a three-link four-player network in1.2. A SET OF EXAMPLES: TOTAL BENEFITS - COSTS model. RELATIONSHIPS UTILITY = symmetric connections the 27 t t t 1 2 3 4 Given a network g,12 write the net utility or payo§ ui (g) that player i receives from  +  2 +  3g c a network as 2 +  2  2c 2 +  2  2c  + 2 + 3  c XFigure 1.2.3 The utilities to the players in a three-link four-player network in ui (g) =  `ij (g)  di (g)c; the j6symmetric pathconnected in model. =i: i and j are connections g where `ij (g) is the number of links in the shortest path between i and j, di (g) is the 38
  • STABILITY AND EFFICIENCYPAIRWISE STABILITY: NO PLAYER WANTS TO REMOVE A LINK NO TWO PLAYERS WANT TO BOTH ADD A LINKEFFICIENCY: STRONG EFFICIENCY (MAXIMIZES TOTAL UTILITY) PARETO EFFICIENCYTENSION BETWEEN STABILITY AND EFFICIENCY JACKSON, M.O. AND WOLINSKY, A. (1996) 39
  • OPTIMAL NETWORK STRUCTURE IN SOME RANGE OF PARAMETERS, THESE NETWORKS ARE BOTH STABLE AND EFFICIENT COMPLETE NETWORK STAR NETWORK 40
  • FOLLOWING THE CROWD THE LAST STORY 41
  • INFORMATION CASCADERESTAURANT CHOICE: YOUR OWN INFORMATION (PRIVATE SIGNAL) INFORMATION ABOUT CHOICE MADE BY OTHERS (EXTERNAL SIGNAL)SEQUENTIAL DECISION MAKINGONLY INFORMATION BASEDRATIONAL CHOICE IN BEST INTEREST 42
  • NETWORK EFFECTLOCAL LEVEL OF INTERACTION, FRIENDS INFLUENCE (NOT INTERESTED IN ENTIREPOPULATION OPINION)INFORMATION EFFECT: OBSERVE THE CHOICE OF OTHERSDIRECT BENEFIT EFFECT: ADVANTAGE OF COPYING DECISIONS OF OTHERS (MATCHINGTECHNOLOGY ETC) 43
  • This describes what happens on a single edge of the network, but the point is thateach node v is playing a copy of this game with each of its neighbors, and its payoff isthe sum of its payoffs in the games played on each edge. Hence, v’s choice of strategywill be based on the choices made by all of its neighbors, taken together. The basic question faced by v is the following: suppose that some of its neighbors COORDINATION GAMEadopt A, and some adopt B; what should v do in order to maximize its payoff? Thisclearly depends on the relative number of neighbors doing each, and on the relationbetween the payoff values a and b. With a little bit of algebra, we can make up adecision rule for v quite easily, as follows. Suppose that a p fraction of v’s neighborshave behavior A, and a (1 − p) fraction have behavior B; that is, if v has d neighbors,then pd adopt A and (1 − p)d adopt B, as shown in Figure 19.2. So if v chooses A, it PAYOFF MATRIX A B A B A B modeling diffusion through a network 501 v B A a,a 0,0 gets a payoff of pda, and if it chooses B, it gets a payoff of (1 − p)db. Thus, b,b the A B 0,0 A is better choice if (1-p)d B pd neighbors use A neighbors use B pda ≥ (1 − p)db,Figure 19.2. Node v must choose between behavior A and behavior B, based on what itsneighbors are doing. or, rearranging terms, if b THRESHOLD p≥ . a+b We’ll use q to denote this expression on the right-hand side. This inequality describes a very simple threshold rule: it says that if a fraction of at least q = b/(a + b) of your neighbors follow behavior A, then you should, too. And it makes sense intuitively: when q is small, then A is the much more enticing behavior, and it only takes a small fraction of your neighbors engaging in A for44 to do so as well. However, if q is you
  • NETWORK CASCADESCASCADE IS A “CHAIN REACTION” OF SWITCHING FROM ONE TYPE OF BEHAVIOR TOANOTHER T 45
  • CASCADE SIZECOMPLETE CASCADE PARTIAL CASCADE q=0.4 NATURAL COMMUNITIES/ BOUNDARIES CLUSTERS RELATIVE WEAK TIES ADVANTAGES 46
  • CASCADE MAXIMIZATION A -SEED SET K - SIZE OF A - CASCADE FROM A 47
  • MARKETING STRATEGY 48
  • COMPLEX NETWORKSFEATURES: POWER LAW SMALL AVERAGE DISTANCE HIGH CLUSTERING BUILD BY INDEPENDENT INTERACTING AGENTS 49
  • FACEBOOK WORLD PAUL BUTLER, FACEBOOK 50
  • TEXTBOOKS 51
  • EASY READ 52
  • REFERENCESERDOS, P. AND A. RÈNYI. “ON RANDOM GRAPHS”. PUBLICATIONES MATHEMATICAEDEBRECEN 6: 290-297, 1959JEFFREY TRAVERS AND STANLEY MILGRAM. “AN EXPERIMENTAL STUDY OF THESMALL WORLD PROBLEM.” SOCIOMETRY, 32(4):425–443, 1969.DUNCAN J. WATTS AND STEVEN H. STROGATZ. “COLLECTIVE DYNAMICS OF SMALL-WORLD NETWORKS”. NATURE, 393:440–442, 1998.MARK GRANOVETTER. “THE STRENGTH OF WEAK TIES” AMERICAN JOURNAL OFSOCIOLOGY, 78:1360–1380, 1973.C. MARLOW, L. BYRON, T. LENTO, AND I. ROSENN. “MAINTAINED RELATIONSHIPS ONFACEBOOK 2009”. ONLINE AT HTTP://OVERSTATED.NET/2009/03/09/MAINTAINED-RELATIONSHIPS-ON-FACEBOOK.ALBERT-LA ́SZLO ́ BARABA ́SI AND RE ́KA ALBERT. “EMERGENCE OF SCALING INRANDOM NETWORKS.” SCIENCE, 286:509–512, 1999. 53
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