Transcript slides
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Reading Group “Networks, Crowds and Markets”
Session 1: Graph Theory and Social Networks
Overview
Introduction Reading Group Ch. 2 Graphs, Paths and Small Worlds Ch. 3 Strength of Weak Ties Ch. 4 Homophily Schelling model Typ hier de footer 2
Introduction to the Reading Group
Book:
Networks, Crowds and Markets
Why this book?
Multidisciplinary and Comprehensive
Networks
: Jon Kleinberg, Computer Scientist
Crowds and Markets
: David Easley: Economist Up to date (2010) Good Reputation Typ hier de footer 3
Introduction to the Reading Group
Additional comments Treated chapters are in Syllabus Chapters are online: http://www.cs.cornell.edu/home/kleinber/networks book/ Book is at Undergraduate level Consider Advanced Material and additional papers when presenting Typ hier de footer 4
Chapter 2
GRAPHS, PATHS AND SMALL WORLDS
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A social network
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A financial network
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A technological network: ARPANET
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Graphs, Paths and Distances
A network is mathematically represented by a graph, G=
Graphs, Paths and Distances
A path is a sequence of (distinct) nodes,
v 1 , v 2 , …, v k ,
such that for each
i
in {1,…,k-1} there is an edge between
v i
and
v i+1
GJHML is a path Typ hier de footer 10
Graphs, Paths and Distances
The
distance
between two nodes
v 1
and
v 2
is the length of the shortest path between them Typ hier de footer The shortest path between G and L is (among others) GJHL and its length is 3 11
Small-World Phenomenon
When we look at large social network with thousands of nodes, we find that distances are generally quite short, often less than 10. This is called the Small-World phenomenon Stanley Milgram e.a. in 1960s: Small World Experiment Random participants in Nebraska and Kansas were asked to send a chain letter to Boston through first name based acquaintances Typ hier de footer 12
Distribution of Chain Lengths
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Small Worlds
Milgram found that average lengths of the chains in the experiment was around six Six degrees of separation This number has been replicated in other studies, e.g. Leskovec & Horvitz in Microsoft Instant Messenger network Why is this?
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Small-World Phenomenon
Suppose everyone has on average 100 acquaintances and there is little overlap between acquaintanceships Me: 1 Acquaintances: 100 Acquaintances at distance 2: 100^2=10,000 Acquaintances at distance 3: 100^3=1,000,000 Acquaintances at distance 4: 100^4=100,000,000 Acquaintances at distance 5: 100^5=10,000,000,000 Typ hier de footer 15
Chapter 3
STRENGTH OF WEAK TIES
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Strength of Weak Ties
Links differ in terms of strength Friends vs. Acquaintance Amount of contact time, affection, trust Mark Granovetter (1974):
Getting a Job
Jobseekers obtain useful job info through social network More often from acquaintances than from close friends Why?
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Strength of Weak Ties
Granovetter (1973):
The Strength of Weak Ties
Link between
local network property
and
global network structure
Local:
Triadic closure
of triads with strong ties Local-Global: Strong ties cannot be
bridges
Global: Bridges more important for information transmission Conclusion: Weak ties are more important for information transmission Typ hier de footer 18
Strength of Weak Ties
Triadic closure of triads with strong ties A satisfies
strong triadic closure property:
for all B and C for which there is a strong tie AB and AC, there is also a (strong or weak) tie BC B A C B A Typ hier de footer C 19
Strength of Weak Ties
A
bridge
is a tie that connects two otherwise unconnected components Information within group is often same Information between groups is different Bridge provides link to different information source, and is therefore more important E B C D A Typ hier de footer F 20
Strength of Weak Ties
Tie AB is a
local bridge
if A and B have no friends in common The span of a local bridge AB is the distance between A and B after removal of AB itself AB is a
local bridge
of span 4 A B Typ hier de footer 21
Strength of Weak Ties
Claim: if a node A satisfies the Strong Triadic Closure and is involved in at least two strong ties, then
any local bridge
it is involved in
must be a weak tie
Proof by contradiction: suppose C satisfies STC and CD is a strong bridge, then there is a triple BCD with BC and CD strong. But then, BD should be linked.
B E C D A Typ hier de footer 22 F
Strength of Weak Ties
Empirical support
for
Strength of Weak Ties Theory Onnela et al. (2007) Empirical support
against
Strength of Weak Ties Theory Van der Leij & Goyal (2011) Typ hier de footer 23
Chapter 4
HOMOPHILY
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Homophily
Agents in a social network have
other characteristics
apart from their links Non-mutable: race, gender, age Mutable: place to live, occupation, activities, opinions, beliefs Links and mutable characteristics co-evolve over time Typ hier de footer 25
Homophily
When we take a snapshot in time, we observe that these node characteristics are correlated across links E.g. Academics have often academic friends, etc.
This phenomenon that people are linked to similar others is called
homophily
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Homophily at a U.S. High School
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Homophily
Mechanisms underlying Homophily Selection A and B have similar characteristics -> A and B form a link AB Social Influence A and B have a link -> B chooses the same (mutable) characteristic as A E.g. A starts smoking, and B follows (peer pressure) Typ hier de footer 28
Social-Affiliation Network
Network of persons and social
foci
(activities) Typ hier de footer 29
Triadic Closure
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Focal Closure
Selection
:
Karate
introduces
Anna
to
Daniel
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Membership Closure
Social Influence
:
Anna
introduces
Bob
to
Karate
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Homophily
Both Selection and Social Influence drive homophily How important is each mechanism?
Important question: Different mechanism implies different policy, e.g. Policy to prevent teenagers from smoking Social Influence. Target “key players” and let them positively influence rest Selection. Target on characteristics (e.g. family background) alone Typ hier de footer 33
Homophily
Both Selection and Social Influence drive homophily How important is each mechanism?
Difficult question: Requires longitudinal data Requires observation of (almost) all characteristics If a characteristic is not observed, then social influence effect is overestimated Typ hier de footer 34
Homophily
Measuring the mechanisms behind homophily is a hot topic Kossinets & Watts (2006): Detailed course and e-mail interaction data from university Centola (2010, 2011): Experimental data on social influence controlling network structure Sacerdote: Social influence among students after randomized dorm assignment Typ hier de footer 35
Homophily and Segregation
Neighborhoods tend to be segregated according to race or culture Ghetto formation What is the mechanism behind that?
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Segregation in Chicago
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Homophily and Segregation
Segregation model of Thomas Schelling Agent-based model Two different agents: X and O types Agents live on a grid
weak satisficing preferences for homophily
At least
k
of the 8 neighbors of same type Each period, agents who are not satisfied move to a location where they are Typ hier de footer 38
Schelling’s model (k=3)
X Typ hier de footer 39
Schelling’s model (k=3)
X Typ hier de footer 40
Schelling’s model online
http://cs.gmu.edu/~eclab/projects/mason/project s/schelling/ Typ hier de footer 41
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Schelling’s model
Surprising relation between
micro-behavior
and
macro-outcomes
Weak satisficing preferences for homophily sufficient to create complete segregation Segregation arises due to miscoordination There exists an allocation involving
complete integration satisfying all agents
, but individual decisionmaking does not lead to that outcome Typ hier de footer 43
Overview
Introduction Reading Group Ch. 2 Graphs, Paths and Small Worlds Ch. 3 Strength of Weak Ties Ch. 4 Homophily Schelling model Planning Next week: 6 March 13:00 Natasa Golo and Dan Braha Next Reading Group: 13 March 13:30 h 44