Social Networks

SOCIAL NETWORKS
FIVE SHORT STORIES

LEONID ZHUKOV

NATIONAL RESEARCH UNIVERSITY
HIGHER SCHOOL OF ECONOMICS

LZHUKOV@HSE.RU

1

FIVE SHORT STORIES

SCIENTISTS 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 BEGINNING
1736: LEONARD EULER. KOENIGSBERG BRIDGES

1929: FRIGYES KARINTHY “CHAINS - LANCSZEMEK”

4

60TH AND 70TH

1959: PAUL ERDOS, RANDOM NETWORKS

1967: STANLEY MILGRAM, SMALL WORLD

1973: MARK GRANOVETER, STRENGTH OF
WEAK TIES

5

LAST 10 YEARS

ALBERT-LÁSZLÓ BARABÁSI, NORHEASTERN, PHYSICS.

DUNKAN WATTS, COLUMBIA, SOCIOLOGY

PAUL NEWMAN, UNIV OF MICHIGAN, PHYSICS

JOHN KLEINBERG, CORNELL, COMPUTER SCIENCE

MATTHEW JACKSON, STANFORD, ECONOMICS

6

SUBJECTS

COMPUTER SCIENCE: ALGORITHMS, GRAPH THEORY,
SEARCH ON GRAPH, PATHS LENGTH, CONNECTED
COMPONENTS, CLIQUES, GRAPH COLORING ETC

SOCIOLOGY: SOCIAL ROLES, STATUS, IDENTITY,
COMMUNITIES, INFLUENCE, COHESIVENESS

PHYSICS: STATISTICS, PHASE TRANSITIONS, EVOLUTION
MODELS, DYNAMICAL SYSTEM

ECONOMICS: NETWORK GAMES, OPTIMALITY,
EQUILIBRIUM

7

COMPLEX NETWORKS

NETWORK(GRAPH) : NODES
AND CONNECTIONS (EDGES)

COMPLEX

NOT REGULAR NOR RANDOM

VARIOUS

UNIVERSAL

CLASS C NETWORKS IMAGE BY BARRETT LYON

8

COMPLEX NETWORKS

PROTEIN - 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 STUDY
296 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 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

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 D
AVE 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 Network

WEB 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 Boston
scribed by a continuously decreasing func- searchers have uncovered scale-free struc" University and Stockholm University, for
tion. When plotted on a double-logarith- tures in a stunning range of systems. instance, has shown that a netWork of
mic 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. Conseq

510 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 city

STEVEN 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
ki
s- 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 to
es 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) i
hnologies 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 2t
s-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 result
n- They feature THEthe distribution. In model B,
TO of
ystems. 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 networks

al by so- start with underlying assumption edges. At
opted we is, the BIRTHOFASCALE-FREE NETWORK giving m. Although
ence. It had
is 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 system's 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 the
h 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 these
is That scaling, P(k) is not stationary: because N is
mental ques- systems in more detail. For example, in the
om?
e
is,
constant and the number No. 1,edges increases
society func-
molecules, or
r?
Rev. Mod. Phys., Vol. 74,
of January 2002
y,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 many
ding 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 lion's 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 MODELS

RANDOM GRAPH BARABASI-ALBERT MODEL

25

THE THIRD STORY

26

THE STRENGTH OF WEAK TIES

“THE STRENGTH OF WEAK TIES”. MARK GRANOVETTER. 1973

27

TIE STRENGTH OF TIES
STRENGTH OF A TIE:

AMOUNT OF TIME

EMOTIONAL INTENSITY

INTIMACY

RECIPROCITY

TRIADIC CLOSURE

IF A AND B AND A AND C ARE STRONGLY
LINKED, THEN THE TIE BETWEEN B AND
C IS ALWAYS PRESENT

CLUSTERING COEFFICIENT

28

BRIDGES
BRIDGE - A LINE IN A NETWORK WHICH
PROVIDES THE ONLY PATH BETWEEN TWO
POINTS

BRIDGE IS LOCAL BRIDGE IF ITS REMOVAL
INCREASES DISTANCE BETWEEN TWO
POINTS

NO STRONG TIE IS A BRIDGE

ROLE IN DIFFUSION

29


WHERE 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% RARELY

WEAK TIES = ”LONG TIES”,
CONNECT PEOPLE FROM
DIFFERENT COMMUNITIES

30

FACEBOOK
All Friends Maintained Relationships

One-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 DETECTION

COMMUNITY - A SET OF NODES CONNECTED AMONG THEMSELVES MORE THAN WITH THE
REST OF THE NETWORK

33

GRAPH THEORY METHODS

GRAPH 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 11
BETWEENNESS”. LINTONC.FREEMAN, 1977.
3 10

(c)

“FINDING AND EVALUATING COMMUNITY STRUCTURE IN 2
6
9

NETWORKS” . MARK E. J. NEWMAN AND MICHELLE
1 4 5 7 8 11
GIRVAN. 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 THEORY

GAME THEORY IS THE STUDY OF THE WAYS IN WHICH STRATEGIC INTERACTIONS AMONG
ECONOMIC AGENTS PRODUCE OUTCOMES WITH RESPECT UTILITIES OF THOSE AGENTS,

NOTION OF PAYOFF

PAYOFF TABLE

RATIONAL PLAYERS, ACTING IN THEIR SELF INTERESTS

NETWORK 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 in
1.2. 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

X
Figure 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 EFFICIENCY

PAIRWISE STABILITY:

NO PLAYER WANTS TO REMOVE A LINK

NO TWO PLAYERS WANT TO BOTH ADD A LINK

EFFICIENCY:

STRONG EFFICIENCY (MAXIMIZES TOTAL UTILITY)

PARETO EFFICIENCY

TENSION 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 CASCADE

RESTAURANT CHOICE:

YOUR OWN INFORMATION (PRIVATE SIGNAL)

INFORMATION ABOUT CHOICE MADE BY OTHERS (EXTERNAL SIGNAL)

SEQUENTIAL DECISION MAKING

ONLY INFORMATION BASED

RATIONAL CHOICE IN BEST INTEREST

42

NETWORK EFFECT

LOCAL LEVEL OF INTERACTION, FRIENDS INFLUENCE (NOT INTERESTED IN ENTIRE
POPULATION OPINION)

INFORMATION EFFECT: OBSERVE THE CHOICE OF OTHERS

DIRECT BENEFIT EFFECT: ADVANTAGE OF COPYING DECISIONS OF OTHERS (MATCHING
TECHNOLOGY ETC)

43

This describes what happens on a single edge of the network, but the point is that
each node v is playing a copy of this game with each of its neighbors, and its payoff is
the sum of its payoffs in the games played on each edge. Hence, v’s choice of strategy
will 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 GAME
adopt A, and some adopt B; what should v do in order to maximize its payoff? This
clearly depends on the relative number of neighbors doing each, and on the relation
between the payoff values a and b. With a little bit of algebra, we can make up a
decision rule for v quite easily, as follows. Suppose that a p fraction of v’s neighbors
have 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 its
neighbors 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 CASCADES

CASCADE IS A “CHAIN REACTION” OF SWITCHING FROM ONE TYPE OF BEHAVIOR TO
ANOTHER

T

45

CASCADE SIZE

COMPLETE 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 NETWORKS

FEATURES:

POWER LAW

SMALL AVERAGE DISTANCE

HIGH CLUSTERING

BUILD BY INDEPENDENT INTERACTING AGENTS

49

FACEBOOK WORLD

PAUL BUTLER, FACEBOOK

50

REFERENCES

ERDOS, P. AND A. RÈNYI. “ON RANDOM GRAPHS”. PUBLICATIONES MATHEMATICAE
DEBRECEN 6: 290-297, 1959

JEFFREY TRAVERS AND STANLEY MILGRAM. “AN EXPERIMENTAL STUDY OF THE
SMALL 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 OF
SOCIOLOGY, 78:1360–1380, 1973.

C. MARLOW, L. BYRON, T. LENTO, AND I. ROSENN. “MAINTAINED RELATIONSHIPS ON
FACEBOOK 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 IN
RANDOM NETWORKS.” SCIENCE, 286:509–512, 1999.

53

REFERENCES
ANDREJ BRODER, RAVI KUMAR, ET AL. “GRAPH STRUCTURE IN THE WEB” . IN PROC. 9TH
INTERNATIONAL WORLD WIDE WEB CONFERENCE, PAGES 309–320, 2000.

LINTON FREEMAN. “A SET OF MEASURE OF CENTRALITY BASED ON BETWEENNESS”.
40(1):35– 41, 1977.

MARK E. J. NEWMAN AND MICHELLE GIRVAN. “FINDING AND EVALUATING COMMUNITY
STRUCTURE IN NETWORKS”. PHYSICAL REVIEW E, 69(2):026113, 2004.

JACKSON, M.O. AND WOLINSKY, A. (1996) “A STRATEGIC MODEL OF SOCIAL AND ECO-
NOMIC NETWORKS,” JOURNAL OF ECONOMIC THEORY, VOL 71, NO. 1, PP 44 Ñ74.

SUSHIL BIKHCHANDANI, DAVID HIRSHLEIFER, AND IVO WELCH. “A THEORY OF FADS,
FASHION, CUSTOM AND CULTURAL CHANGE AS INFORMATION CASCADES.” JOURNAL OF
POLITICAL ECONOMY, 100:992–1026, 1992.

STEPHEN MORRIS. “CONTAGION”. REVIEW OF ECONOMIC STUDIES, 67:57–78, 2000.

DAVID KEMPE, JON KLEINBERG, AND EVA TARDOS. “MAXIMIZING THE SPREAD OF
INFLUENCE IN A SOCIAL NETWORK.” IN PROC. 9TH ACM SIGKDD INT. CONF. ON
KNOWLEDGE DISCOVERY AND DATA MINING, PAGES 137–146, 2003.

54

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