Lecture 4: Connect (4/4)
How the friendship we form connect us?
Why we are within a few clicks on Facebook?
COMS 4995-1: Introduction to Social Networks
Tuesday, September 18th
1
Some announcements
 This course is now officially
“sexy [kinda]”
congratulations!
 1st assign. due Thursday 4:10pm
 Part A+C on papers!
 Part B(+raw results of C) on
dropbox
 Sign the cover sheet
 1 late days: 5% (you have 3 free during semester)
2
Outline
 Milgram’s “small world” experiment
 It’s a “combinatorial small world”
 It’s a “complex small world”
 It’s an “algorithmic small world”
4
Small-world model
 Main idea: social networks follows a structure with
a random perturbation
 Formal construction:
1. Connect all nodes at distance in a regular lattice
2. Rewire each edge uniformly with probability p
(variant: connect each node to q neighbors, chosen
uniformly)
Collective dynamics of ‘small-world’ networks.
D. Watts, S. Strogatz, Nature (1998) 5
Small-world model
 Main idea: social networks follows a structure with
a random perturbation
Collective dynamics of ‘small-world’ networks.
D. Watts, S. Strogatz, Nature (1998) 6
Outline
 Milgram’s “small world” experiment
 It’s a “combinatorial small world”
 It’s a “complex small world”
 It’s an “algorithmic small world”
7
Where are we so far?
Analogy with a cosmological
principle
− Are you ready to accept a
cosmological theory that
does not predict life?
In other words, let’s perform
a simple sanity check
8
A thought experiment
1. Consider a randomly augmented lattice (N nodes)
9
A thought experiment
1. Consider a randomly augmented lattice (N nodes)
2. Perform “small world” Milgram experiment
Can you tell what will happen?
(a) The folder arrives in 6 hops
(b) The folder arrives in O(ln(N)) hops
(c) The folder never arrives
(d) I need more information
10
A thought experiment
(a) The folder arrives in 6 hops
NOT TRUE
 It actually does look like a naive answer
 More precisely:
 By previous result we know that shortest paths is of
the order of ln(N), which contradicts this statement.
11
A thought experiment
(b) The folder arrives in O(ln(N))
ACCORDING TO OUR PRINCIPLE, OUGHT TO BE TRUE
BECAUSE IT WAS OBSERVED BY MILGRAM
 A sufficient condition for this to be true is:
 Milgram’s procedure extract shortest path
 Answering this critical question boils down to an
algorithmic problem
12
A thought experiment
(c) The folder never arrives
SEEMS UNLIKELY
unless the procedure is badly designed (cycle)
or we model people dropping
or if the grid contains hole
13
A thought experiment
(d) I need more information
 In particular, how to model Milgram’s procedure
 “If you do not know a target, forward the folder to your
friend or acquaintance that is most likely to know her.”
14
What is Greedy Routing?
 A mathematical model of what Milgram measured
 Participants know where the target is located
 They use grid information + shortcuts “incidentally”
N.B.: Grid “dimensions” can describe geography or
other sociological property (occupation, language)
 Example:
15
st analyze a simple case, the randomly augmented lattice with
1.
How does greedy routing perform?
3 In a randomly augmented lattice of dimension k = 1 with N
routing uses at least Ω( (N )) steps.
: Let
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16
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17
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18
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it
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e, to reach before t before n st eps, it requires to reach t from the
of I l using l local edges.
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√
P
{ Xi ∈ Il } ≤
P [X i ∈ I l ] ≤ √
N steps are required. i =Hence
the expectedi = 1,...,n
number of st eps N
1,...,n
dy rout ing is lower bounded by a1 √const ant time the square root
A thought experiment
for n =
l =
 In a lineHence,
Milgram’s
uses
N ,steps
t his occurs wit h probability at m
conclude t hat , wit h probability at least 1/ 2 t he n first short cut
 square
root is not
{ X 1 , X 2 , . . . , X n } lie outside I l . In t hat case, if we assume t hat
satisfying
smallprocedure
world ofneeds
4 In a randomly
lattice
dimension
in I l , augmented
t hefor
greedy
eit her k ≥ 1 containing
2
 Not
better when
k>1 !
y routing
usesmuch
in
expectation
at least
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 even worse, the proof applies to any distributed alg.
s a good practice
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it requires
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eem that
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s k increases.
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e
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aare
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 “Small
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results
that
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1
ance in the
lattice is
of the
order aofdaunting
O(N k ), algorithmic
so that the relative
… finding
them
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task
btained withP rshort
s augmentation
becomes
as k
op osicut
t ion
4 I n a randomlyactually
augmented
lattice ofworse
dimension
1
4N
1
k+ 1
nodes, greedy routing uses at least Ω(N
19
k
k+ 1
) steps.
The “ small world” model defined by K leinberg is a variat ion of
Outline
 Milgram’s “small world” experiment
 It’s a “combinatorial small world”
 It’s a “complex small world”
 It’s an “algorithmic small world”
 Beyond uniform random augmentation
20
Autopsy of “Small-world” failure
1.3.2
anandom
d om au
gm ent at
ation
i on wit
w i ht hbias
bi
1.3.2 R R
augment
entdom
at i onauwit
bi
as
an
gmhT
ent
atfailure
i on
w ioft of
hMMilgram’s
biilgram’s
as
he
experiment
under
The
failure
experiment under
√√ t
 In a uniformly
augmented
latticeas
shortcuts
do
exist NNsh
int
uit
ively
explained
follows:
about
intuitively
explained
as
follows:
about
experiment
under
the
uniform
augmentation
may
be
e of M ilgram’s√ experiment under
t he uniform√√augment at io
√
About
shorcuts
leads
toon
when
lead
t oto
tabout
he
int
erval
I Il average
l that
= Nwill
Non
T
heprob
pro
the
interval
when
. .The
ollows:
Nlead
shortcuts
exist
l when
explained
as follows:
N
short
cut
s
exist
average
√ about
√ s are uniformly dist ribut ed among a
short
cut
are
distribut
ed among
allt
en
= NI l. when
The problem
the fact
thatfrom
these
intl erval
lshortcuts
= Ncomesfrom
. T
heuniformly
problem
comes
t he fact
of
st
eps
t oto
find
one
of
tthese
hese
st
art
ing
from
istribut
edthey
among
all
N ed
nodes
and,
hence,
it takes
a lot
steps
find
one
starting
from
 But
areof
dispersed
among
are
uniformly
dist
ribut
among
all
Nof nodes
and,
hence,
itanta
t hey
uniformly
dist
ribut
ed,progress
no
algorit
hmc
they
are
uniformly
distributed,
no
algorithm
hese
arting
an
arbitrary
point.
Moreover,
since
findstone
of from
t hese
stare
art
ing
from
arbit
rary
point
. M oreo
Moreover,
previous
steps
doan
not
lead
to
following
otother
her
short
cuthope
s,progress
since
moving
any
but ed, nodist
algorithm
can
hope
to make
some
by some
following
shortcuts,
since
moving
anyw
niformly
ribut
ed,
no
algorit
hm
can
t
o
make
pr
 So need about N/√N = √N trials
osince
find
one.
s,t her
since
moving
does not
improvedoes
his chance
to
find
one.
short
cut s,tanywhere
moving
anywhere
not improve h
T he
following
spect acularillust
illu
The
followingresult
result is
is a spectacular
e.
 Isresult
there
augmentation?
s a spect
acularanother
illustrat
ion
thatt illust
properties
oft hat
algorithmic
can
allow
hem
tto
o ion
exploit
informat
ionin
can
allow
them
exploit
information
lowing
is
amic
spect
acular
rat
propert ies
ofi
xploit
information
in ainformat
surprising
ow t hem
t o exploit
ionmanner.
in a surprising manner.
ugment
latt ti ce
ice w
wit
A uAgm
ent i ning
g l at
i t h aa bias:
b i as: Kleinb
K lein
21
The 10 papers
that will make you a social expert
22
10 sociological must-reads
1.
2.
S.Milgram, “The small world problem,” Psychology today, 1967.
M. Granovetter, “The strength of weak ties: A network theory revisited,” Sociological theory, vol. 1, pp.
201–233, 1983.
3. M. McPherson, L. Smith-Lovin, and J. M. Cook, “Birds of a Feather: Homophily in Social Networks,”
Annual review of sociology, vol. 27, pp. 415–444, Jan. 2001.
4. M. O. Lorenz, “Methods of measuring the concentration of wealth,” Publications of the American
Statistical Association, vol. 9, no. 70, pp. 209–219, 1905.
+ H. Simon, “On a Class of Skew Distribution Functions,” Biometrika, vol. 42, no. 3, pp. 425–440, 1955.
5. R. I. M. Dunbar, “Coevolution of Neocortical Size, Group-Size and Language in Humans,” Behav Brain
Sci, vol. 16, no. 4, pp. 681–694, 1993.
6. D. Cartwright and F. Harary, “Structural balance: a generalization of Heider's theory.,” Psychological
Review, vol. 63, no. 5, pp. 277–293, 1956.
7. M. Granovetter, “Threshold Models of Collective Behavior,” The American Journal of Sociology, vol. 83,
no. 6, pp. 1420–1443, May 1978.
8. B. Ryan and N. C. Gross, “The diffusion of hybrid seed corn in two Iowa communities,” Rural sociology,
vol. 8, no. 1, pp. 15–24, 1943.
+ S. Asch, “Opinions and social pressure,” Scientific American, 1955.
9. R. S. Burt, Structural Holes: The Social Structure of Competition. Harvard University Press, 1992.
10. F. Galton, “Vox Populi,” Nature, vol. 75, no. 1949, pp. 450–451, Mar. 1907.
23
Homophily
 People “love those who are like themselves”,
“Similarity begets friendship”
 Nichomachean Ethics, Aristotle & Phaedrus, Plato
 Do you think homophily
produces or hinder
small world?
Homophily in Online Dating:
When Do You Like Someone Like Yourself?
Andrew T. Fiore and Judith S. Donath
MIT Media Laboratory
20 Ames St., Cambridge, Mass., USA
{fiore, judith}@media.mit.edu
ABSTRACT
Psychologists have found that actual and perceived similarity
between potential romantic partners in demographics,
attitudes, values, and attractiveness correlate positively with
attraction and, later, relationship satisfaction. Online dating
systems provide a new way for users to identify and
communicate with potential partners, but the information they
provide differs dramatically from what a person might glean
from face-to-face interaction. An analysis of dyadic
interactions of approximately 65,000 heterosexual users of an
online dating system in the U.S. showed that, despite these
differences, users of the system sought people like them much
more often than chance would predict, just as in the offline
world. The users’ preferences were most strongly sameseeking for attributes related to the life course, like marital
history and whether one wants children, but they also
demonstrated significant homophily in self-reported physical
build, physical attractiveness, and smoking habits.
Author Keywords
24 Online
personals,
attraction,
computer-mediated
communication, online dating, relationships
ACM Classification Keywords
H5.3. Group and Organization Interfaces; Asynchronous
psychological and sociological perspectives (Lea & Spears
1995, Walther 1996, McKenna et al. 2002), and they have
examined the personals ads that appear in print publications
(Bolig et al. 1984, Ahuvia & Adelman 1992). This paper
describes a quantitative examination of the characteristics for
which online dating users seek others like them.
NATURE OF ONLINE PERSONALS DATA
We analyzed data from one online dating system in
particular. Through an agreement brokered by the Media
Laboratory with an online dating Web site (the “Site”), we
obtained access to a snapshot of activity on the Site over an
eight-month period, from June 2002 through February
2003. The data included users’ personal profile information,
their self-reported preferences for a mate, and their
communications via the site’s private message system with
other users. Anonymous ID numbers distinguished unique
users.
Table 1 indicates which profile characteristics users could
specify about themselves and about the partners they would
like to meet.
Data about private messages exchanged by the users
included the sender, recipient, subject, text, date and time of
delivery, and whether the recipient had read the message.
Augmenting lattice with a bias
 What if the augmentation exhibits a bias
 Most of the people you know are near,
 Occasionally, you know someone outside
 Does this break the lower bound proof?
Does finding a neighborhood of t becomes easier?
25
How to model augmentation bias
 Formal construction:
1. Connect nodes at distance p in a regular lattice
2. Connect each node to q other nodes, chosen with a
biased probability
3. p=q=1 to simplify
The small-world phenomenon: An algorithmic perspective.
J. Kleinberg, Proc. of ACM STOC (2000) 26
We assume V =
(i 1 , . . . , i k ) ∈ { 1, 2, . . . , L } k
, (not e t hat
How to model augmentation bias
odes are connect ed t o all ot her nodes whose dist ance in
t most p (i.e. v = (i 1 , . . . , i k ) and v = (i 1 , . . . , i k ) are
i 1Formal
| + . . . +construction:
|i k − i k | ≤ p).
1 − 
1. Connect nodes at distance p in a regular lattice
n addit ion,
node
connect
ed t onodes,
q ot hers
nodes
i
2. each
Connect
eachisnode
to q other
chosen
withchosen
a
uch t hat biased probability
P [u
v] =
1
u− v
v= u
r
1
u− v
.
r
ly called t he clustering coefficient, is crit ical as it
r may
be called
thegrid.
clustering
coefficient
augment
attion
of t he
A node
t hatcut
is stwice
sthe
ance
p and
he
number
of random
short
q are two p
r times less

If
a
node
is
twice
further,
probability
is
st
ill
be
chosen
but
only
wit
h
a
probability
2
del, which have lit t le effect on t he performancet imes
of dist ribut
hatThe
, in
t he probability
describing
t he chance t o connect
small-world
phenomenon: An
algorithmic perspective.
J. Kleinberg,
of ACM
(2000)
nat
or onlyProc.
plays
t heSTOC
role
of a normalizing const ant .
rat io between t hese probability approaches 1. When
27
Impact of clustering coefficient
Small values of r
Approaches uniform augmentation
Large values of r
Approaches original lattice
28
Can we break the lower bound?
(a) Yes, finding a neighborhood of t becomes easier
A PRIORI NOT TRUE
 It is easier only if you are already near the target
 In general, it can take a larger number of steps
29
Can we break the lower bound?
(b) Yes, for another reason
 All positions are not equal, hence progress is possible
 As shortcut are used recursively, probability increases
 So we need to study the sequence of progress
30
The critical case
 Assume r=k (dimension of the grid)
 A neighborhood of t of radius d/2
 Contains (d/2)k nodes
 Each may be chosen with
probability roughly 1/(3d/2)k
 Growth of ball compensates
probability decreases!
 Harmonic distribution.
The small-world phenomenon: An algorithmic perspective.
J. Kleinberg, Proc. of ACM STOC (2000) 31
Augmented lattice
Navigable small world
dist. alg need O(log2(N)) steps
Combinatorial Small world
(Short paths exist)
dist. alg. need N(k-r)/(k+1)
steps
Not a small world
(Short paths do not exist)
alg. need N(r-k)/(r-(k-1)) steps
r
0
r=k
The small-world phenomenon: An algorithmic perspective.
J. Kleinberg, Proc. of ACM STOC (2000) 32
Theoretical follow ups
 Is the analysis of greedy routing tight?
 Yes, greedy routing performs in Ω(log2 n)
 Can we find path as short as log(n) (shortest path)?
 Yes, with extra information on neighboring nodes
 Or another augmentation
 Can we build augmentation for an infinite lattice?
 See homework exercice (check tomorrow night)
33
Theoretical follow ups (cont’d)
 Can we augment other graphs?
 G=(V,E) (i.e. a lattice) with distance known
 Random augmentation adds one shortcut per node
Is routing on G + shortcuts used incidentally efficient?
 Indeed all these graphs are polylog augmentable:
 Bounded ball growth, Doubling dimensions
 Bounded “width” (Trees, bounded treewidth graphs)
 What about all graphs? Lower Bound O(n1/√ln(n))
34
Practical follow up
Can we observe harmonic
distribution?
• Yes, using closeness
rank instead of distance
Can we prove it emerge?
• Recent results
• Through rewiring,
mobility
Geographic routing in social networks.
D. Liben-Nowell et. al. PNAS (2005)
35
Summary
 Milgram’s experiment prove that social networks
are navigable
 individuals can take advantage of short paths
 with basic information
 This is at odds with uniform random graphs
 The key ingredients to explain navigability
 A space easy to route (e.g. grid, trees, etc.).
 A subtle harmonic augmentation (e.g. ball radius).
36