Lecture Three

Lecture Three
Stories of Complex Sociotechnical Systems:
Measurement, Mechanisms, and Meaning
Lipari Summer School, Summer, 2012
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Prof. Peter Dodds
Social Contagion
Models
Granovetter’s model
Network version
Department of Mathematics & Statistics | Center for Complex Systems |
Vermont Advanced Computing Center | University of Vermont
Groups
Simple disease
spreading models
References
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
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Outline
A Very Dismal Science
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Contagion
Winning: it’s not for everyone
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Social Contagion Models
Granovetter’s model
Network version
Groups
Groups
Simple disease
spreading models
References
Simple disease spreading models
References
2 of 137
Complex
Sociotechnical
Systems
Economics, Schmeconomics
Alan Greenspan (September 18, 2007):
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
“I’ve been dealing with these big
mathematical models of forecasting the
economy ...
Social Contagion
Models
Granovetter’s model
Network version
Groups
If I could figure out a way to determine
whether or not people are more fearful
or changing to more euphoric,
Simple disease
spreading models
References
I don’t need any of this other stuff.
I could forecast the economy better than
any way I know.”
http://wikipedia.org
3 of 137
Complex
Sociotechnical
Systems
Economics, Schmeconomics
Greenspan continues:
“The trouble is that we can’t figure that out. I’ve been in
the forecasting business for 50 years. I’m no better than I
ever was, and nobody else is. Forecasting 50 years
ago was as good or as bad as it is today. And the reason
is that human nature hasn’t changed. We can’t improve
ourselves.”
A Very Dismal
Science
Jon Stewart:
Simple disease
spreading models
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
References
“You just bummed the @*!# out of me.”
wildbluffmedia.com
I
I
From the Daily Show () (September 18, 2007)
The full inteview is here ().
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Economics, Schmeconomics
James K. Galbraith:
NYT But there are at least 15,000 professional
economists in this country, and you’re saying only
two or three of them foresaw the mortgage crisis?
[JKG] Ten or 12 would be closer than two or three.
NYT What does that say about the field of economics,
which claims to be a science? [JKG] It’s an
enormous blot on the reputation of the profession.
There are thousands of economists. Most of them
teach. And most of them teach a theoretical
framework that has been shown to be fundamentally
useless.
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
From the New York Times, 11/02/2008 ()
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Collective Cooperation:
I
Standard frame:
Locally selfish behavior
→ collective cooperation.
I
Different frame:
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Locally moral/fair behaviour
→ collective bad actions.
Simple disease
spreading models
References
I
So why do we study frame 1 instead of frame 2?
I
Tragedy of the Commons is one example of frame 2.
I
Better question:
Who is it that studies frame 1 over frame 2. . . ?
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Homo Economicus
I
‘What makes people think like Economists?
Evidence on Economic Cognition from the “Survey of
Americans and Economists on the Economy” ’ [8]
Bryan Caplan, Journal of Law and Economics, 2001
People behave like Homo economicus:
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
1. if they are well educated,
2. if they are male,
3. if their real income rose over the last 5 years,
Groups
Simple disease
spreading models
References
4. if they expect their real income to rise over the next 5
years,
5. if they have a high degree of job security,
6. but not because of high income nor ideological
conservatism.
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Wealth distribution in the United States:
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Questions used in a recent study by Norton and
Ariely: [29]
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
I
What percentage of all wealth is owned by
individuals grouped into quintiles?
I
How do people believe wealth is distributed?
I
How do people believe wealth should be distributed?
Network version
Groups
Simple disease
spreading models
References
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Wealth distribution in the United States:
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
9 of 137
Wealth distribution in the United States:
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
10 of 137
This is a Collateralized Debt Obligation:
Contagion
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
A confusion of contagions:
I
Was Harry Potter some kind of virus?
I
What about Vampires?
I
Did Sudoku spread like a disease?
I
Language? The alphabet? [17]
I
Religion?
I
Democracy...?
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
12 of 137
Contagion
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Naturomorphisms
I
“The feeling was contagious.”
I
“The news spread like wildfire.”
I
“Freedom is the most contagious virus known to
man.”
—Hubert H. Humphrey, Johnson’s vice president
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
I
Groups
Simple disease
spreading models
References
“Nothing is so contagious as enthusiasm.”
—Samuel Taylor Coleridge
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Social contagion
Complex
Sociotechnical
Systems
A Very Dismal
Science
Eric Hoffer, 1902–1983
There is a grandeur in the uniformity of the mass. When
a fashion, a dance, a song, a slogan or a joke sweeps
like wildfire from one end of the continent to the other,
and a hundred million people roar with laughter, sway
their bodies in unison, hum one song or break forth in
anger and denunciation, there is the overpowering
feeling that in this country we have come nearer the
brotherhood of man than ever before.
I
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
Hoffer () was an interesting fellow...
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The spread of fanaticism
Hoffer’s acclaimed work: “The True Believer:
Thoughts On The Nature Of Mass Movements” (1951) [20]
Quotes-aplenty:
I
I
I
“We can be absolutely certain only about things we
do not understand.”
“Mass movements can rise and spread without belief
in a God, but never without belief in a devil.”
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
“Where freedom is real, equality is the passion of the
masses. Where equality is real, freedom is the
passion of a small minority.”
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Complex
Sociotechnical
Systems
Imitation
A Very Dismal
Science
Contagion
“When people are free
to do as they please,
they usually imitate
each other.”
—Eric Hoffer
“The Passionate State
of Mind” [21]
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
despair.com
16 of 137
Complex
Sociotechnical
Systems
The collective...
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
“Never Underestimate
the Power of Stupid
People in Large
Groups.”
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
despair.com
17 of 137
Contagion
Complex
Sociotechnical
Systems
A Very Dismal
Science
Definitions
I
I
(1) The spreading of a quality or quantity between
individuals in a population.
(2) A disease itself:
the plague, a blight, the dreaded lurgi, ...
I
from Latin: con = ‘together with’ + tangere ‘to touch.’
I
Contagion has unpleasant overtones...
I
Just Spreading might be a more neutral word
I
But contagion is kind of exciting...
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
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Examples of non-disease spreading:
Interesting infections:
I
Spreading of buildings in the US... ()
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
I
Viral get-out-the-vote video. ()
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Contagions
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Two main classes of contagion
1. Infectious diseases:
tuberculosis, HIV, ebola, SARS, influenza, ...
2. Social contagion:
fashion, word usage, rumors, riots, religion, ...
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
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Winning: it’s not for everyone
Complex
Sociotechnical
Systems
A Very Dismal
Science
Where do superstars come from?
I
Rosen (1981): “The Economics of Superstars”
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Examples:
Granovetter’s model
Network version
Groups
I
Full-time Comedians (≈ 200)
I
Soloists in Classical Music
I
Economic Textbooks (the usual myopic example)
I
Highly skewed distributions (again)...
Simple disease
spreading models
References
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Superstars
Complex
Sociotechnical
Systems
A Very Dismal
Science
Rosen’s theory:
I
Individual quality q maps to reward R(q)
I
R(q) is ‘convex’ (d2 R/dq 2 > 0)
Two reasons:
I
1. Imperfect substitution:
A very good surgeon is worth many mediocre ones
2. Technology:
Media spreads & technology reduces cost of
reproduction of books, songs, etc.
I
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
No social element—success follows ‘inherent quality’
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Superstars
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Adler (1985): “Stardom and Talent”
I
Assumes extreme case of equal ‘inherent quality’
I
Argues desire for coordination in knowledge and
culture leads to differential success
I
Success is then purely a social construction
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
23 of 137
Dominance hierarchies
Chase et al. (2002): “Individual differences versus social
dynamics in the formation of animal dominance
hierarchies” [11]
The aggressive female Metriaclima zebra ():
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
Pecking orders for fish...
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Dominance hierarchies
I
Fish forget—changing of dominance hierarchies:
Table 1. Percentage of groups with different numbers of fish
changing ranks between first and second hierarchies (n ! 22)
No. of fish changing ranks
0
2
3
4
I
Percentage of groups
27.3
36.4
18.2
18.2
rank on prior attributes of itself creates the linear structure of the
hierarchies. Although 50% of the fish changed ranks from one
hierarchy to the other, almost all the hierarchies were linear in
structure. Some factor other than differences in attributes seems
to have ensured high rates of linearity. In the next experiment,
we tested to determine whether that factor might be social
dynamics.
It might seem possible that ‘‘noise,’’ random fluctuations in
individuals’ attributes or behaviors, could account for the observed differences between the first and second hierarchies.
However, a careful consideration of the ways in which fluctuations might occur shows that this explanation is unlikely. For
example, what if the differences were assumed to have occurred
because some of the fish changed their ranks on attributes from
the 1.firstTransition
to the second
Fig.
patternshierarchies?
between ranksTo
of account
fish in thefor
firstour
andresults,
second
this assumption
would
require a mixture
of stability
instahierarchies.
Frequencies
of experimental
groups showing
each and
pattern
are
indicated
in parentheses.
arrowstimes
indicate
of right
rank.
bility in attribute
ranksOpen-headed
at just the right
andtransitions
in just the
Solid-headed
arrows
show
dominance
relationships
in
intransitive
triads;
all
proportion of groups. The rankings would have had to have been
the
fish for
in anall
intransitive
samefor
rank.
stable
the fish triad
in allshare
the the
groups
the day or two it took
them to form their first hierarchies (or we would not have seen
stable dominance
relationships
Then,
(one-sided
binomial
test: n ! by
22,our
P criterion).
" 0.001 and
P in
"three0.03,
quarters of the
(but 27%
not inofthe
one-quarter)
respectively).
Ingroups
this light,
theremaining
groups with
identical
various numbers
fish would have had to have swapped ranks
hierarchies
is veryofsmall.
on attributes in the 2-week period of separation so as to have
produced different second hierarchies. And finally, the rankings
Discussion. When we rewound the tape of the fish to form new
on attributeswe
for
all thedid
fish
inget
all the
the same
groups
would have
had
to
hierarchies,
usually
not
hierarchy
twice.
The
have become
oncepersisted
more forand
thethe
dayindividuals
or two it took
them
linearity
of thestable
structures
stayed
the
to form
hierarchies.
same,
buttheir
theirsecond
ranks did
not. Thus our results differ considerably
22 observations: about 3/4 of the time, hierarchy
changed
rank on prior attributes of itself creates th
hierarchies. Although 50% of the fish c
hierarchy to Complex
the other, almost all the hi
structure. Some
factor other than differe
Sociotechnical
to have ensured high rates of linearity.
Systems
we tested to determine whether that
dynamics.
It might seem possible that ‘‘noise,’’
individuals’ attributes or behaviors, cou
served differences between the first a
A Very
Dismal of the
However, a careful
consideration
Science
tions might occur
shows that this expla
example, what if the differences were as
because someContagion
of the fish changed their r
the first to the second hierarchies? To
this assumption would require a mixtur
Winning:
for ti
bility in attribute
ranks at it’s
just not
the right
proportion ofeveryone
groups. The rankings woul
stable for all the fish in all the groups fo
them to formSocial
their first
hierarchies (or w
Contagion
stable dominance relationships by our cr
Models
quarters of the
groups (but not in the
various numbers
of fish would
Granovetter’s
modelhave had
on attributes Network
in the 2-week
version period of se
produced different second hierarchies. A
Groups
on attributes for
all the fish in all the gr
have become stable once more for the d
disease
to form theirSimple
second hierarchies.
Alternatively,
instead ofmodels
attribute ra
spreading
nance rank as in the prior attribute mo
of fish might be considered to have been
References
at one meeting one might dominate, b
there was some chance that the other
problem with this model is that earlier
demonstrates that in situations in which
group has even a small chance of dom
probability of getting linear hierarchies
even in a more restrictive model in whic
are close in rank in the first hierarchies ha
of reversing their relationships, such
observed in this experiment, the probab
linear hierarchies as we observed is sti
available from the authors).
We know of only one other study (4
assembled groups to form initial hier
individuals for a period, and then reass
second hierarchy (but see Guhl, ref. 4
groups had pairwise encounters between
bly). Unfortunately, their techniques of
sible to compare results, because they
of 137acts
between the frequency of25
aggressive
in pairs toward one another in the two
Complex
Sociotechnical
Systems
Music Lab Experiment
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
48 songs
30,000 participants
multiple ‘worlds’
Inter-world variability
I
How probable is a social state?
I
Can we estimate variability?
Simple disease
spreading models
References
Salganik et al. (2006) “An experimental study of inequality and
unpredictability in an artificial cultural market” [33]
26 of 137
Music Lab Experiment
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
27 of 137
Complex
Sociotechnical
Systems
Music Lab Experiment
A Very Dismal
Science
Contagion
Experiment 1
Experiments 2–4
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
28 of 137
Complex
Sociotechnical
Systems
Music Lab Experiment
Experiment 1
1
12
24
36
48
48
36
24
12
1
Rank market share in indep. world
I
Rank market share in influence worlds
Rank market share in influence worlds
A Very Dismal
Science
Contagion
Experiment 2
1
Winning: it’s not for
everyone
12
Social Contagion
Models
24
Granovetter’s model
Network version
Groups
36
Simple disease
spreading models
48
48
36
24
12
1
Rank market share in indep. world
References
Variability in final rank.
29 of 137
Complex
Sociotechnical
Systems
Music Lab Experiment
0.6
Experiment 1
A Very Dismal
Science
Experiment 2
Gini coefficient G
Contagion
Winning: it’s not for
everyone
0.4
Social Contagion
Models
0.2
Granovetter’s model
Network version
Groups
0
Social Influence
I
Indep.
Social Influence
Indep.
Inequality as measured by Gini coefficient:
N
G=
Simple disease
spreading models
References
N
s X
s
X
1
|mi − mj |
(2Ns − 1)
i=1 j=1
30 of 137
Complex
Sociotechnical
Systems
Music Lab Experiment
0.015
Experiment 1
Experiment 2
A Very Dismal
Science
Unpredictability U
Contagion
Winning: it’s not for
everyone
0.01
Social Contagion
Models
0.005
Granovetter’s model
Network version
0
Social
Influence
I
Independent
Social
Influence
Independent
Unpredictability
U=
References
Ns X
Nw X
Nw
X
1
Ns
Groups
Simple disease
spreading models
Nw
2
|mi,j − mi,k |
i=1 j=1 k =j+1
31 of 137
Music Lab Experiment
Sensible result:
I
Stronger social signal leads to greater following and
greater inequality.
Peculiar result:
I
Stronger social signal leads to greater
unpredictability.
Very peculiar observation:
I
The most unequal distributions would suggest the
greatest variation in underlying ‘quality.’
I
But success may be due to social construction
through following.
I
‘Payola’ leads to poor system performance.
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
32 of 137
Music Lab Experiment—Sneakiness
Complex
Sociotechnical
Systems
A Very Dismal
Science
Exp. 3
Exp. 4
Unchanged world
Inverted worlds
500
Exp. 3
Song 1
300
Song 48
Song 48
200
Song 1
Song 1
100
Song 48
400 752
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
100
Network version
Song 2
Song 2
Song 2
Song 47
50
Song 47
Song 48
1200 1600 2000 2400 2800
Song 47
Song 47
150
Song 1
0
0
Song 2
200
Downloads
Downloads
400
Contagion
Exp. 4
Unchanged world
Inverted worlds
250
0
0
400
752
Subjects
I
Inversion of download count
I
The ‘pretend rich’ get richer ...
I
... but at a slower rate
1200 1600 2000 2400 2800
Subjects
Groups
Simple disease
spreading models
References
33 of 137
Social Contagion
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
http://xkcd.com/610/ ()
34 of 137
Social Contagion
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
35 of 137
Social Contagion
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
36 of 137
Social Contagion
Complex
Sociotechnical
Systems
Examples abound
A Very Dismal
Science
Contagion
I
fashion
I
striking
I
smoking () [13]
I
residential
segregation [34]
I
ipods
I
obesity () [12]
I
Harry Potter
Winning: it’s not for
everyone
I
voting
Social Contagion
Models
I
gossip
I
Rubik’s cube
I
religious beliefs
I
leaving lectures
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
SIR and SIRS contagion possible
I
Classes of behavior versus specific behavior: dieting
37 of 137
Social Contagion
Complex
Sociotechnical
Systems
Two focuses for us:
A Very Dismal
Science
I
I
Widespread media influence
Contagion
Word-of-mouth influence
Winning: it’s not for
everyone
We need to understand influence:
I
Who influences whom? Very hard to measure...
I
What kinds of influence response functions are
there?
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
(see Romero et al. [31] , Ugander et al. [39] )
I
Are some individuals super influencers?
Highly popularized by Gladwell [16] as ‘connectors’
I
The infectious idea of opinion leaders (Katz and
Lazarsfeld) [22]
38 of 137
The hypodermic model of influence
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
39 of 137
The two step model of influence [22]
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
40 of 137
The general model of influence
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
41 of 137
Social Contagion
Why do things spread?
Complex
Sociotechnical
Systems
A Very Dismal
Science
I
Because of special individuals?
Contagion
I
Or system level properties?
Winning: it’s not for
everyone
I
Is the match that lights the fire important?
I
Yes. But only because we are narrative-making
machines...
I
We like to think things happened for reasons...
I
Reasons for success are usually ascribed to intrinsic
properties (e.g., Mona Lisa)
I
System/group properties harder to understand—-no
natural frame/metaphor
I
Always good to examine what is said before and
after the fact...
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
42 of 137
Complex
Sociotechnical
Systems
From Pratchett’s “Lords and Ladies”:
A Very Dismal
Science
Granny Weatherwax () on trying to borrow the mind of a
swarm of bees—
Winning: it’s not for
everyone
“But a swarm, a mind made up of thousands of mobile
parts, was beyond her. It was the toughest test of all.
She’d tried over and over again to ride on one, to see the
world through ten thousand pairs of multifaceted eyes all
at once, and all she’d ever got was a migraine and an
inclination to make love to flowers.”
Contagion
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
(p. 42). Harper Collins, Inc. Kindle Edition.
43 of 137
The Mona Lisa
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
I
“Becoming Mona Lisa: The Making of a Global
Icon”—David Sassoon
I
Not the world’s greatest painting from the start...
I
Escalation through theft, vandalism, parody, ...
44 of 137
The completely unpredicted fall of Eastern
Europe
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
Timur Kuran: [26, 27] “Now Out of Never: The Element of
Surprise in the East European Revolution of 1989”
45 of 137
The dismal predictive powers of editors...
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
46 of 137
Getting others to do things for you
Complex
Sociotechnical
Systems
From ‘Influence’ [14] by Robert Cialdini ()
Six modes of influence:
1. Reciprocation: The Old Give and Take... and Take;
e.g., Free samples, Hare Krishnas.
2. Commitment and Consistency: Hobgoblins of the
Mind; e.g., Hazing.
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
3. Social Proof: Truths Are Us;
e.g., Jonestown (),
Kitty Genovese () (contested).
Simple disease
spreading models
References
4. Liking: The Friendly Thief ; e.g., Separation into
groups is enough to cause problems.
5. Authority: Directed Deference;
e.g., Milgram’s obedience to authority
experiment. ()
6. Scarcity: The Rule of the Few; e.g., Prohibition.
47 of 137
Social Contagion
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
I
Cialdini’s modes are heuristics that help up us get
through life.
Winning: it’s not for
everyone
I
Very useful but can be leveraged...
Social Contagion
Models
Granovetter’s model
Network version
Messing with social connections
I
Ads based on message content
(e.g., Google and email)
I
BzzAgent ()
I
Facebook’s advertising: Beacon ()
Groups
Simple disease
spreading models
References
48 of 137
Complex
Sociotechnical
Systems
Thomas Schelling () (Economist/Nobelist):
A Very Dismal
Science
Contagion
I
Tipping models—Schelling
(1971) [34, 35, 36]
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
I
I
I
I
Threshold models—Granovetter
(1978) [19]
Herding models—Bikhchandani,
Hirschleifer, Welch (1992) [4, 5]
I
[youtube] ()
Simulation on checker boards
Idea of thresholds
Network version
Groups
Simple disease
spreading models
References
Social learning theory,
Informational cascades,...
49 of 137
Social contagion models
Thresholds
I
Basic idea: individuals adopt a behavior when a
certain fraction of others have adopted
I
‘Others’ may be everyone in a population, an
individual’s close friends, any reference group.
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
I
Response can be probabilistic or deterministic.
I
Individual thresholds can vary
I
Assumption: order of others’ adoption does not
matter... (unrealistic).
I
Assumption: level of influence per person is uniform
(unrealistic).
Simple disease
spreading models
References
50 of 137
Social Contagion
Some possible origins of thresholds:
I
Inherent, evolution-devised inclination to coordinate,
to conform, to imitate. [3]
I
Lack of information: impute the worth of a good or
behavior based on degree of adoption (social proof)
Economics: Network effects or network externalities
I
I
I
I
Externalities = Effects on others not directly involved
in a transaction
Examples: telephones, fax machine, Facebook,
operating systems
An individual’s utility increases with the adoption level
among peers and the population in general
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
51 of 137
Complex
Sociotechnical
Systems
Action based on perceived behavior of others:
1
2.5
1
0.6
1.5
0.4
0.2
C
φt+1 = F (φt)
B
2
f (φ∗)
Pr(a
i,t+1
=1)
A
0.8
1
0.5
0
0
φ∗i
φi,t
1
0
0
0.5
∗
1
0.8
A Very Dismal
Science
0.6
0.4
Contagion
0.2
Winning: it’s not for
everyone
0
0
φ
0.5
φt
1
Social Contagion
Models
Granovetter’s model
I
Two states: Susceptible and Infected.
I
φ = fraction of contacts ‘on’ (e.g., rioting)
Simple disease
spreading models
I
Discrete time update (strong assumption!)
References
I
This is a Critical mass model
I
Many other kinds of dynamics are possible.
Network version
Groups
Implications for collective action theory:
1. Collective uniformity 6→ individual uniformity
2. Small individual changes → large global changes
53 of 137
Threshold model on a network
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Many years after Granovetter and Soong’s work:
Winning: it’s not for
everyone
Social Contagion
Models
“A simple model of global cascades on random networks”
D. J. Watts. Proc. Natl. Acad. Sci., 2002 [40]
Granovetter’s model
Network version
Groups
Simple disease
spreading models
I
Mean field model → network model
I
Individuals now have a limited view of the world
References
55 of 137
Threshold model on a network
Complex
Sociotechnical
Systems
A Very Dismal
Science
Interactions between individuals now represented by
a network
Contagion
I
Network is sparse
Winning: it’s not for
everyone
I
Individual i has ki contacts
Social Contagion
Models
I
Influence on each link is reciprocal and of unit weight
I
Each individual i has a fixed threshold φi
I
Individuals repeatedly poll contacts on network
I
Synchronous, discrete time updating
I
Individual i becomes active when
fraction of active contacts akii ≥ φi
I
Individuals remain active when switched (no
recovery = SI model)
I
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
56 of 137
Complex
Sociotechnical
Systems
Threshold model on a network
A Very Dismal
Science
Contagion
t=1
e
t=2
e
a
Winning: it’s not for
everyone
t=3
e
a
a
Social Contagion
Models
Granovetter’s model
d
b
c
d
b
c
d
b
Network version
Groups
c
Simple disease
spreading models
References
I
All nodes have threshold φ = 0.2.
57 of 137
Snowballing
The Cascade Condition:
1. If one individual is initially activated, what is the
probability that an activation will spread over a
network?
2. What features of a network determine whether a
cascade will occur or not?
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
First study random networks:
I
Start with N nodes with a degree distribution pk
I
Nodes are randomly connected (carefully so)
I
Aim: Figure out when activation will propagate
I
Determine a cascade condition
References
58 of 137
Snowballing
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Follow active links
I
I
I
An active link is a link connected to an activated
node.
If an infected link leads to at least 1 more infected
link, then activation spreads.
We need to understand which nodes can be
activated when only one of their neigbors becomes
active.
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
59 of 137
Complex
Sociotechnical
Systems
The most gullible
A Very Dismal
Science
Vulnerables:
Contagion
I
We call individuals who can be activated by just one
contact being active vulnerables
Winning: it’s not for
everyone
I
The vulnerability condition for node i:
Social Contagion
Models
Granovetter’s model
Network version
1/ki ≥ φi
I
Which means # contacts ki ≤ b1/φi c
I
For global cascades on random networks, must have
a global cluster of vulnerables [40]
I
Cluster of vulnerables = critical mass
I
Network story: 1 node → critical mass → everyone.
Groups
Simple disease
spreading models
References
60 of 137
Complex
Sociotechnical
Systems
Cascade condition
A Very Dismal
Science
Back to following a link:
I
I
I
A randomly chosen link, traversed in a random
direction, leads to a degree k node with probability
∝ kPk .
Follows from there being k ways to connect to a
node with degree k .
Normalization:
∞
X
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
kPk = hk i
k =0
I
So
P(linked node has degree k ) =
kPk
hk i
61 of 137
Complex
Sociotechnical
Systems
Cascade condition
A Very Dismal
Science
Next: Vulnerability of linked node
I
Linked node is vulnerable with probability
Z
βk =
1/k
φ0∗ =0
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
f (φ0∗ )dφ0∗
I
If linked node is vulnerable, it produces k − 1 new
outgoing active links
I
If linked node is not vulnerable, it produces no active
links.
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
62 of 137
Complex
Sociotechnical
Systems
Cascade condition
A Very Dismal
Science
Putting things together:
I
Expected number of active edges produced by an
active edge:
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
∞
X
kPk
R=
(k − 1) · βk ·
hk i
k =1 |
{z
}
+
success
=
∞
X
(k − 1) · βk ·
k =1
kPk
0 · (1 − βk ) ·
hk i
|
{z
}
failure
Network version
Groups
Simple disease
spreading models
References
kPk
hk i
63 of 137
Complex
Sociotechnical
Systems
Cascade condition
A Very Dismal
Science
Contagion
So... for random networks with fixed degree distributions,
cacades take off when:
R=
∞
X
(k − 1) · βk ·
k =1
kPk
≥ 1.
hk i
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
I
βk = probability a degree k node is vulnerable.
I
Pk = probability a node has degree k .
64 of 137
Complex
Sociotechnical
Systems
Cascade condition
A Very Dismal
Science
Two special cases:
I
Contagion
(1) Simple disease-like spreading succeeds: βk = β
β·
∞
X
kPk
(k − 1) ·
≥ 1.
hk i
k =1
I
(2) Giant component exists: β = 1
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
1·
∞
X
(k − 1) ·
k =1
kPk
≥ 1.
hk i
65 of 137
Complex
Sociotechnical
Systems
Cascades on random networks
A Very Dismal
Science
1
Contagion
Final
cascade size
〈S〉
0.8
I
0.6
Fraction of
Vulnerables
0.4
0.2
No
Cascades
0
1
Cascades
Possible
2
3
Low influence
4
z
No
Cascades
5
6
7
High influence
I
I
Cascades occur
only if size of max
vulnerable cluster
> 0.
System may be
‘robust-yet-fragile’.
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
‘Ignorance’
facilitates
spreading.
Example networks
66 of 137
Complex
Sociotechnical
Systems
Cascade window for random networks
A Very Dismal
Science
30
Contagion
25
1
Winning: it’s not for
everyone
0.8
〈S〉
no cascades
20
0.6
Social Contagion
Models
0.4
influence z
0.2
15
0
1
2
3
4
5
6
7
z
Granovetter’s model
Network version
Groups
10
5
0
0.05
Simple disease
spreading models
cascades
0.1
References
0.15
0.2
0.25
φ = uniform individual threshold
I
‘Cascade window’ widens as threshold φ decreases.
I
Lower thresholds enable spreading.
67 of 137
Cascade window for random networks
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
68 of 137
Complex
Sociotechnical
Systems
Early adopters are not well connected:
I
Degree distributions of nodes adopting at time t:
t =0
t =1
0.8
0.2
t=0
0.15
0.1
10
15
20
0
0
t =4
20
t=4
0.3
0.2
0.2
0.1
0.1
10
15
20
0
0
t = 12
t=6
5
10
15
20
t = 12
t = 14
10
15
20
0
0
t=8
0.2
0.1
0.1
5
10
15
20
10
15
20
0
0
15
20
0
0
Social Contagion
Models
Granovetter’s model
Network version
Groups
t = 10
Simple disease
spreading models
5
10
15
20
References
t = 18
0.2
t = 16
t = 18
0.15
0.1
0.05
5
10
0.3
0.2
0
0
5
t = 10
0.1
0.05
5
0
0
0.4
0.15
0.1
0.05
20
0.2
0.15
0.1
15
t = 16
0.2
0.15
10
0.3
t = 14
0.2
0.2
5
0.4
0.4
0.3
5
0
0
t =8
0.5
0.4
0
0
15
t =6
0.5
0
0
10
Winning: it’s not for
everyone
0.4
0.2
5
Contagion
t=3
0.6
0.4
0.2
5
0.8
t=2
0.6
0.4
A Very Dismal
Science
t =3
0.8
t=1
0.6
0.05
0
0
t =2
0.05
5
10
15
20
0
0
5
10
15
20
Pk ,t versus k
69 of 137
Complex
Sociotechnical
Systems
The multiplier effect:
“Influentials, Networks, and Public Opinion Formation” [41]
Journal of Consumer Research, Watts and Dodds, 2007.
Cascade size ratio
1
B
Cascade size Savg
0.8
Social Contagion
Models
4
Degree ratio
Granovetter’s model
Network version
3
Groups
0.6
Simple disease
spreading models
2
0.4
Average
individuals
0.2
0
1
2
3
4
Influence navg
5
References
1
6
0
Contagion
Winning: it’s not for
everyone
Top 10% individuals
A
A Very Dismal
Science
1
2
3
4
5
6
Gain
Influence navg
I
Fairly uniform levels of individual influence.
I
Multiplier effect is mostly below 1.
70 of 137
Complex
Sociotechnical
Systems
The multiplier effect:
A Very Dismal
Science
Top 10% individuals
A
1
B
Cascade size
Savg
0.8
I
Cascade size ratio
Contagion
Winning: it’s not for
everyone
12
9
Social Contagion
Models
Degree ratio
0.6
Granovetter’s model
Network version
6
Groups
0.4
Simple disease
spreading models
3
0.2
References
0
1
2
3
4
Influence navg
5
6
0
1
2
3
n
Average
Individuals
4
5
6
avg
Gain
Skewed influence distribution example.
71 of 137
Special subnetworks can act as triggers
Complex
Sociotechnical
Systems
A
i0
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
B
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
I
φ = 1/3 for all nodes
72 of 137
Complex
Sociotechnical
Systems
The power of groups...
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
“A few harmless flakes
working together can
unleash an avalanche
of destruction.”
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
despair.com
74 of 137
Incorporating social context:
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
I
Assumption of sparse interactions is good
I
Degree distribution is (generally) key to a network’s
function
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
I
Still, random networks don’t represent all networks
I
Major element missing: group structure
I
“Threshold Models of Social Influence” [42]
Watts and Dodds, 2009.
Oxford Handbook of Analytic Sociology.
Eds. Hedström and Bearman.
Network version
Groups
Simple disease
spreading models
References
75 of 137
Group structure—Ramified random networks
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
p = intergroup connection probability
q = intragroup connection probability.
76 of 137
Complex
Sociotechnical
Systems
Bipartite networks
1
2
3
contexts
4
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
individuals
a
b
c
d
e
Network version
Groups
Simple disease
spreading models
References
b
d
unipartite
network
a
c
e
77 of 137
Complex
Sociotechnical
Systems
Context distance
A Very Dismal
Science
occupation
Contagion
Winning: it’s not for
everyone
education
health care
Social Contagion
Models
Granovetter’s model
Network version
kindergarten
teacher
high school
teacher
nurse
doctor
Groups
Simple disease
spreading models
References
a
b
c
d
e
78 of 137
Complex
Sociotechnical
Systems
Generalized affiliation model
A Very Dismal
Science
geography
occupation
Contagion
age
Winning: it’s not for
everyone
0
100
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
a
b
c
d
e
(Blau & Schwartz, Simmel, Breiger)
79 of 137
Generalized affiliation model networks with
triadic closure
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
I
exp−αd
Connect nodes with probability ∝
where
α = homophily parameter
and
d = distance between nodes (height of lowest
common ancestor)
I
τ1 = intergroup probability of friend-of-friend
connection
I
τ2 = intragroup probability of friend-of-friend
connection
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
80 of 137
Cascade windows for group-based networks
Complex
Sociotechnical
Systems
A Very Dismal
Science
Single seed
B
Coherent group seed
C
Contagion
Winning: it’s not for
everyone
Random
Group networks
A
Random set seed
Social Contagion
Models
Granovetter’s model
Network version
Generalized Affiliation
Model networks
Groups
D
E
F
Simple disease
spreading models
References
81 of 137
Multiplier effect for group-based networks:
A Very Dismal
Science
Degree ratio
A
1
B
3
Cascade
size ratio
Savg
0.8
2
0.6
0.4
1
Gain
8
12
16
0
4
20
8
navg
C
12
16
D
Groups
Simple disease
spreading models
3
0.8
Savg
Granovetter’s model
Network version
20
navg
1
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
0.2
0
4
Complex
Sociotechnical
Systems
References
2
0.6
0.4
Cascade
size ratio < 1!
1
0.2
0
0
4
8
navg
I
12
16
0
0
4
8
12
16
navg
Multiplier almost always below 1.
82 of 137
Assortativity in group-based networks
0.8
0.6
1
Average
Cascade size
A Very Dismal
Science
Contagion
0.5
0
Winning: it’s not for
everyone
0
4
8
12
Social Contagion
Models
k
0.4
Complex
Sociotechnical
Systems
Granovetter’s model
Network version
Groups
Degree distribution
for initially infected node
0.2
0
0
5
Local influence
10
15
Simple disease
spreading models
References
20
k
I
The most connected nodes aren’t always the most
‘influential.’
I
Degree assortativity is the reason.
83 of 137
Social contagion
Summary
I
‘Influential vulnerables’ are key to spread.
I
Early adopters are mostly vulnerables.
I
Vulnerable nodes important but not necessary.
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
I
Vulnerable groups may greatly facilitate spread.
I
Seems that cascade condition is a global one.
I
Most extreme/unexpected cascades occur in highly
connected networks.
I
‘Influentials’ are posterior constructs.
I
Many potential ‘influentials’ exist.
Groups
Simple disease
spreading models
References
84 of 137
Social contagion
Implications
I
I
I
Focus on the influential vulnerables.
Create entities that can be transmitted successfully
through many individuals rather than broadcast from
one ‘influential.’
Only simple ideas can spread by word-of-mouth.
(Idea of opinion leaders spreads well...)
I
Want enough individuals who will adopt and display.
I
Displaying can be passive = free (yo-yo’s, fashion),
or active = harder to achieve (political messages).
I
Entities can be novel or designed to combine with
others, e.g. block another one.
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
85 of 137
Mathematical Epidemiology
The standard SIR model
I
I
[28]
= basic model of disease contagion
Three states:
1. S = Susceptible
2. I = Infective/Infectious
3. R = Recovered or Removed or Refractory
I
S(t) + I(t) + R(t) = 1
I
Presumes random interactions (mass-action
principle)
I
Interactions are independent (no memory)
I
Discrete and continuous time versions
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
86 of 137
Mathematical Epidemiology
Complex
Sociotechnical
Systems
A Very Dismal
Science
Discrete time automata example:
Contagion
Winning: it’s not for
everyone
1 − βI
S
Transition Probabilities:
βI
Social Contagion
Models
Granovetter’s model
Network version
Groups
ρ
I
r
R
1−r
β for being infected given
contact with infected
r for recovery
ρ for loss of immunity
Simple disease
spreading models
References
1−ρ
87 of 137
Mathematical Epidemiology
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Original models attributed to
I
1920’s: Reed and Frost
Social Contagion
Models
Granovetter’s model
Network version
I
I
1920’s/1930’s: Kermack and McKendrick [23, 25, 24]
Coupled differential equations with a mass-action
principle
Groups
Simple disease
spreading models
References
88 of 137
Independent Interaction models
Differential equations for continuous model
d
S = −βIS + ρR
dt
d
I = βIS − rI
dt
d
R = rI − ρR
dt
β, r , and ρ are now rates.
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
Reproduction Number R0 :
I
R0 = expected number of infected individuals
resulting from a single initial infective
I
Epidemic threshold: If R0 > 1, ‘epidemic’ occurs.
89 of 137
Reproduction Number R0
Complex
Sociotechnical
Systems
A Very Dismal
Science
Discrete version:
I
Set up: One Infective in a randomly mixing
population of Susceptibles
I
At time t = 0, single infective random bumps into a
Susceptible
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
I
Probability of transmission = β
Simple disease
spreading models
I
At time t = 1, single Infective remains infected with
probability 1 − r
References
I
At time t = k , single Infective remains infected with
probability (1 − r )k
90 of 137
Complex
Sociotechnical
Systems
Reproduction Number R0
Discrete version:
I
Expected number infected by original Infective:
A Very Dismal
Science
Contagion
2
3
R0 = β + (1 − r )β + (1 − r ) β + (1 − r ) β + . . .
Winning: it’s not for
everyone
Social Contagion
Models
= β 1 + (1 − r ) + (1 − r )2 + (1 − r )3 + . . .
=β
1
= β/r
1 − (1 − r )
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
For S0 initial infectives (1 − S0 = R0 immune):
R0 = S0 β/r
91 of 137
Independent Interaction models
For the continuous version
I
Second equation:
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
d
I = βSI − rI
dt
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
d
I = (βS − r )I
dt
Network version
Groups
Simple disease
spreading models
References
I
Number of infectives grows initially if
βS(0) − r > 0 : βS(0) > r : βS(0)/r > 1
I
Same story as for discrete model.
92 of 137
Complex
Sociotechnical
Systems
Independent Interaction models
A Very Dismal
Science
Example of epidemic threshold:
Fraction infected
1
Contagion
0.8
Winning: it’s not for
everyone
0.6
Social Contagion
Models
Granovetter’s model
0.4
Network version
Groups
Simple disease
spreading models
0.2
0
0
References
1
2
3
4
R0
I
Continuous phase transition.
I
Fine idea from a simple model.
93 of 137
Independent Interaction models
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Many variants of the SIR model:
I
SIS: susceptible-infective-susceptible
I
SIRS: susceptible-infective-recovered-susceptible
I
compartment models (age or gender partitions)
I
more categories such as ‘exposed’ (SEIRS)
I
recruitment (migration, birth)
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
94 of 137
Disease spreading models
Complex
Sociotechnical
Systems
A Very Dismal
Science
For novel diseases:
Contagion
1. Can we predict the size of an epidemic?
Winning: it’s not for
everyone
2. How important is the reproduction number R0 ?
Social Contagion
Models
Granovetter’s model
Network version
R0 approximately same for all of the following:
I
1918-19 “Spanish Flu” ∼ 500,000 deaths in US
I
1957-58 “Asian Flu” ∼ 70,000 deaths in US
I
1968-69 “Hong Kong Flu” ∼ 34,000 deaths in US
I
2003 “SARS Epidemic” ∼ 800 deaths world-wide
Groups
Simple disease
spreading models
References
95 of 137
Size distributions
Size distributions are important elsewhere:
I
earthquakes (Gutenberg-Richter law)
I
city sizes, forest fires, war fatalities
I
wealth distributions
I
‘popularity’ (books, music, websites, ideas)
I
Epidemics?
Power laws distributions are common but not obligatory...
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
Really, what about epidemics?
I
Simply hasn’t attracted much attention.
I
Data not as clean as for other phenomena.
96 of 137
Feeling Ill in Iceland
Caseload recorded monthly for range of diseases in
Iceland, 1888-1990
Frequency
0.03
0.02
Iceland: measles
normalized count
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
0.01
0
1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990
Date
I
Simple disease
spreading models
References
Treat outbreaks separated in time as ‘novel’
diseases.
97 of 137
Complex
Sociotechnical
Systems
Really not so good at all in Iceland
A Very Dismal
Science
Contagion
Epidemic size distributions N(S) for
Measles, Rubella, and Whooping Cough.
75
105
N(S)
A
B
5
4
4
4
3
3
3
2
2
2
1
1
0.05
S
0.075
0.1
0
0
Granovetter’s model
C
5
0.025
Social Contagion
Models
75
5
0
0
Winning: it’s not for
everyone
Network version
Groups
Simple disease
spreading models
References
1
0.02
0.04
0.06
0
0
S
0.025
0.05
0.075
S
Spike near S = 0, relatively flat otherwise.
98 of 137
Complex
Sociotechnical
Systems
Measles & Pertussis
75
75
N (ψ)
4
A
1
5
0
4
10
10
−5
10
B
1
10
Contagion
0
−4
10
−3
10
3
−2
10
−1
10
10
−5
10
−4
10
−3
10
−2
10
3
ψ
2
2
1
1
0
0
N>(ψ)
N>(ψ)
5
A Very Dismal
Science
2
10
2
10
−1
Winning: it’s not for
everyone
10
ψ
Social Contagion
Models
Granovetter’s model
0.025
0.05
0.075
0.1
0
0
0.025
0.05
0.075
ψ
ψ
Insert plots:
Complementary cumulative frequency distributions:
Network version
Groups
Simple disease
spreading models
References
N(Ψ0 > Ψ) ∝ Ψ−γ+1
Limited scaling with a possible break.
99 of 137
Power law distributions
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Measured values of γ:
I
measles: 1.40 (low Ψ) and 1.13 (high Ψ)
I
pertussis: 1.39 (low Ψ) and 1.16 (high Ψ)
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
I
Expect 2 ≤ γ < 3 (finite mean, infinite variance)
Simple disease
spreading models
I
When γ < 1, can’t normalize
References
I
Distribution is quite flat.
100 of 137
Complex
Sociotechnical
Systems
Resurgence—example of SARS
A Very Dismal
Science
160
# New cases
D
Contagion
120
Winning: it’s not for
everyone
80
Social Contagion
Models
40
0
Nov 16, ’02
Granovetter’s model
Dec 16, ’02
Jan 15, ’03
Feb 14, ’03
Mar 16, ’03
Apr 15, ’03
May 15, ’03
Date of onset
I
Epidemic slows...
then an infective moves to a new context.
I
Epidemic discovers new ‘pools’ of susceptibles:
Resurgence.
I
Importance of rare, stochastic events.
Jun 14, ’03
Network version
Groups
Simple disease
spreading models
References
101 of 137
The challenge
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
So... can a simple model produce
1. broad epidemic distributions
and
2. resurgence ?
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
102 of 137
Complex
Sociotechnical
Systems
Size distributions
2000
A
1500
N(ψ)
A Very Dismal
Science
R0=3
Contagion
Simple models
typically produce
bimodal or unimodal
size distributions.
1000
500
0
0
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
0.25
0.5
0.75
1
ψ
Simple disease
spreading models
References
I
I
This includes network models:
random, small-world, scale-free, ...
Exceptions:
1. Forest fire models
2. Sophisticated metapopulation models
103 of 137
Burning through the population
Forest fire models:
[30]
I
Rhodes & Anderson, 1996
I
The physicist’s approach:
“if it works for magnets, it’ll work for people...”
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
A bit of a stretch:
1. Epidemics ≡ forest fires
spreading on 3-d and 5-d lattices.
Simple disease
spreading models
References
2. Claim Iceland and Faroe Islands exhibit power law
distributions for outbreaks.
3. Original forest fire model not completely understood.
104 of 137
Size distributions
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
From Rhodes and Anderson, 1996.
105 of 137
Sophisticated metapopulation models
Complex
Sociotechnical
Systems
A Very Dismal
Science
I
Community based mixing: Longini (two scales).
I
Eubank et al.’s EpiSims/TRANSIMS—city
simulations.
I
Spreading through countries—Airlines: Germann et
al., Corlizza et al.
I
Vital work but perhaps hard to generalize from...
I
: Create a simple model involving multiscale travel
I
Multiscale models suggested by others but not
formalized (Bailey, Cliff and Haggett, Ferguson et al.)
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
106 of 137
Size distributions
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
I
Very big question: What is N?
I
Should we model SARS in Hong Kong as spreading
in a neighborhood, in Hong Kong, Asia, or the world?
I
For simple models, we need to know the final size
beforehand...
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
107 of 137
Complex
Sociotechnical
Systems
Improving simple models
Contexts and Identities—Bipartite networks
A Very Dismal
Science
1
2
3
contexts
4
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
individuals
a
b
c
d
e
Granovetter’s model
Network version
Groups
Simple disease
spreading models
b
d
References
unipartite
network
a
c
e
I
boards of directors
I
movies
I
transportation modes (subway)
108 of 137
Improving simple models
Idea for social networks: incorporate identity.
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Identity is formed from attributes such as:
I
Geographic location
I
Type of employment
I
I
Age
Recreational activities
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
Groups are crucial...
I
formed by people with at least one similar attribute
I
Attributes ⇔ Contexts ⇔ Interactions ⇔
Networks. [43]
109 of 137
Complex
Sociotechnical
Systems
Infer interactions/network from identities
occupation
A Very Dismal
Science
Contagion
education
Winning: it’s not for
everyone
health care
Social Contagion
Models
kindergarten
teacher
high school
teacher
Granovetter’s model
nurse
doctor
Network version
Groups
Simple disease
spreading models
References
a
b
c
d
e
Distance makes sense in identity/context space.
110 of 137
Complex
Sociotechnical
Systems
Generalized context space
A Very Dismal
Science
geography
occupation
age
0
Contagion
100
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
a
b
c
d
e
(Blau & Schwartz [6] , Simmel [37] , Breiger [7] )
111 of 137
A toy agent-based model
Complex
Sociotechnical
Systems
A Very Dismal
Science
Geography—allow people to move between
contexts:
I
Locally: standard SIR model with random mixing
I
discrete time simulation
I
β = infection probability
I
γ = recovery probability
I
P = probability of travel
I
Movement distance: Pr(d) ∝ exp(−d/ξ)
I
ξ = typical travel distance
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
112 of 137
A toy agent-based model
Complex
Sociotechnical
Systems
Schematic:
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
7
〈k
initiator
〉
6
5
4
3
2
113 of 137
Model output
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
I
Define P0 = Expected number of infected individuals
leaving initially infected context.
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
I
I
Need P0 > 1 for disease to spread (independent of
R0 ).
Limit epidemic size by restricting frequency of travel
and/or range
Network version
Groups
Simple disease
spreading models
References
114 of 137
Model output
Varying ξ:
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
I
Transition in expected final size based on typical
movement distance (sensible)
115 of 137
Model output
Complex
Sociotechnical
Systems
Varying P0 :
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
I
Transition in expected final size based on typical
number of infectives leaving first group (also
sensible)
I
Travel advisories: ξ has larger effect than P0 .
116 of 137
Complex
Sociotechnical
Systems
Example model output: size distributions
R0=3
R =12
400
N(ψ)
N(ψ)
400
300
200
Contagion
0
300
Winning: it’s not for
everyone
200
Social Contagion
Models
Granovetter’s model
100
100
0
0
A Very Dismal
Science
683
1942
0.25
0.5
ψ
0.75
1
0
0
Network version
Groups
0.25
0.5
0.75
1
ψ
I
Flat distributions are possible for certain ξ and P.
I
Different R0 ’s may produce similar distributions
I
Same epidemic sizes may arise from different R0 ’s
Simple disease
spreading models
References
117 of 137
Complex
Sociotechnical
Systems
Model output—resurgence
A Very Dismal
Science
Contagion
# New cases
Standard model:
6000
Winning: it’s not for
everyone
D
R0=3
4000
Social Contagion
Models
Granovetter’s model
Network version
2000
0
0
Groups
500
1000
t
1500
Simple disease
spreading models
References
118 of 137
Complex
Sociotechnical
Systems
Model output—resurgence
A Very Dismal
Science
Standard model with transport:
# New cases
Contagion
200
E
Winning: it’s not for
everyone
R0=3
Social Contagion
Models
100
Granovetter’s model
0
0
Network version
500
1000
1500
# New cases
t
400
Groups
Simple disease
spreading models
G
References
R0=3
200
0
0
500
1000
1500
t
119 of 137
The upshot
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Simple multiscale population structure
+
stochasticity
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
leads to
Groups
Simple disease
spreading models
resurgence
+
broad epidemic size distributions
References
120 of 137
Conclusions
I
For this model, epidemic size is highly unpredictable
I
Model is more complicated than SIR but still simple
I
We haven’t even included normal social responses
such as travel bans and self-quarantine.
I
I
The reproduction number R0 is not terribly useful.
R0 , however measured, is not informative about
1. how likely the observed epidemic size was,
2. and how likely future epidemics will be.
I
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
Problem: R0 summarises one epidemic after the fact
and enfolds movement, the price of bananas,
everything.
121 of 137
Conclusions
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
I
I
I
Disease spread highly sensitive to population
structure
Rare events may matter enormously
(e.g., an infected individual taking an international
flight)
More support for controlling population movement
(e.g., travel advisories, quarantine)
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
122 of 137
Conclusions
Complex
Sociotechnical
Systems
A Very Dismal
Science
What to do:
Contagion
I
Need to separate movement from disease
Winning: it’s not for
everyone
I
R0 needs a friend or two.
Social Contagion
Models
I
Need R0 > 1 and P0 > 1 and ξ sufficiently large
for disease to have a chance of spreading
Granovetter’s model
Network version
Groups
Simple disease
spreading models
More wondering:
I
Exactly how important are rare events in disease
spreading?
I
Again, what is N?
References
123 of 137
Simple disease spreading models
Complex
Sociotechnical
Systems
A Very Dismal
Science
Valiant attempts to use SIR and co. elsewhere:
I
Adoption of ideas/beliefs (Goffman & Newell,
1964) [18]
I
Spread of rumors (Daley & Kendall, 1965) [15]
I
Diffusion of innovations (Bass, 1969) [2]
I
Spread of fanatical behavior (Castillo-Chávez &
Song, 2003)
I
Spread of Feynmann diagrams (Bettencourt et al.,
2006)
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
124 of 137
References I
[1]
[2]
[3]
[4]
M. Adler.
Stardom and talent.
American Economic Review, pages 208–212, 1985.
pdf ()
F. Bass.
A new product growth model for consumer durables.
Manage. Sci., 15:215–227, 1969. pdf ()
A. Bentley, M. Earls, and M. J. O’Brien.
I’ll Have What She’s Having: Mapping Social
Behavior.
MIT Press, Cambridge, MA, 2011.
Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
S. Bikhchandani, D. Hirshleifer, and I. Welch.
A theory of fads, fashion, custom, and cultural
change as informational cascades.
J. Polit. Econ., 100:992–1026, 1992.
125 of 137
References II
Complex
Sociotechnical
Systems
A Very Dismal
Science
[5]
S. Bikhchandani, D. Hirshleifer, and I. Welch.
Learning from the behavior of others: Conformity,
fads, and informational cascades.
J. Econ. Perspect., 12(3):151–170, 1998. pdf ()
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
[6]
[7]
P. M. Blau and J. E. Schwartz.
Crosscutting Social Circles.
Academic Press, Orlando, FL, 1984.
Groups
Simple disease
spreading models
References
R. L. Breiger.
The duality of persons and groups.
Social Forces, 53(2):181–190, 1974. pdf ()
126 of 137
References III
[8]
B. Caplan.
What makes people think like economists? evidence
on economic cognition from the “survey of
americans and economists on the economy”.
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Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
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Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
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A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
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Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
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Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
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Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
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Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
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Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
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Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
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Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
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Complex
Sociotechnical
Systems
A Very Dismal
Science
Contagion
Winning: it’s not for
everyone
Social Contagion
Models
Granovetter’s model
Network version
Groups
Simple disease
spreading models
References
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