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Reading Group “Networks,
Crowds and Markets”
Session 1: Graph Theory and Social Networks
Overview
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Introduction Reading Group
Ch. 2 Graphs, Paths and Small Worlds
Ch. 3 Strength of Weak Ties
Ch. 4 Homophily
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Schelling model
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Introduction to the Reading Group
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Book: Networks, Crowds and Markets
Why this book?
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Multidisciplinary and Comprehensive
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Networks: Jon Kleinberg, Computer Scientist
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Crowds and Markets: David Easley: Economist
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Up to date (2010)
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Good Reputation
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Introduction to the Reading Group
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Additional comments
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Treated chapters are in Syllabus
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Chapters are online:
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http://www.cs.cornell.edu/home/kleinber/networksbook/
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Book is at Undergraduate level

Consider Advanced Material and additional papers
when presenting
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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
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A network is mathematically represented by a
graph, G=<V,E>, a set of vertices (nodes) V
and the edges (ties, links) between them
A graph can be directed or undirected
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Graphs, Paths and Distances
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A path is a sequence of (distinct) nodes, v1, v2,
…, vk, such that for each i in {1,…,k-1} there is
an edge between vi and vi+1
GJHML is a
path
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Graphs, Paths and Distances
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The distance between two nodes v1 and v2 is
the length of the shortest path between them
The shortest path
between G and L
is (among others)
GJHL and its length
is 3
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Small-World Phenomenon
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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
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Random participants in Nebraska and Kansas were
asked to send a chain letter to Boston through firstname based acquaintances
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Distribution of Chain Lengths
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Small Worlds
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Milgram found that average lengths of the
chains in the experiment was around six
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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
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Suppose everyone has on average 100
acquaintances and there is little overlap between
acquaintanceships
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Me: 1
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Acquaintances: 100
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Acquaintances at distance 2: 100^2=10,000
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Acquaintances at distance 3: 100^3=1,000,000
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Acquaintances at distance 4: 100^4=100,000,000
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Acquaintances at distance 5: 100^5=10,000,000,000
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Chapter 3
STRENGTH OF WEAK TIES
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Strength of Weak Ties
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Links differ in terms of strength
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Friends vs. Acquaintance
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Amount of contact time, affection, trust
Mark Granovetter (1974): Getting a Job
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Jobseekers obtain useful job info through social
network
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
More often from acquaintances than from close friends
Why?
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Strength of Weak Ties
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Granovetter (1973): The Strength of Weak Ties
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Link between local network property and
global network structure
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Local: Triadic closure of triads with strong ties
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Local-Global: Strong ties cannot be bridges
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Global: Bridges more important for information
transmission
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Conclusion: Weak ties are more important
for information transmission
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Strength of Weak Ties
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Triadic closure of triads with strong ties
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A satisfies strong triadic closure property:
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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
B
A
C
C
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Strength of Weak Ties
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A bridge is a tie that connects two otherwise
unconnected components
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Information within group is often same
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Information between groups is different
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Bridge provides link to different information source,
and is therefore more important
E
B
C
D
A
F
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Strength of Weak Ties
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Tie AB is a local bridge if A and B have no
friends in common
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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
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Strength of Weak Ties
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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
C
A
E
D
F
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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)
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Chapter 4
HOMOPHILY
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Homophily

Agents in a social network have other
characteristics apart from their links
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Non-mutable: race, gender, age
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Mutable: place to live, occupation, activities,
opinions, beliefs

Links and mutable characteristics co-evolve
over time
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Homophily
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When we take a snapshot in time, we observe
that these node characteristics are correlated
across links
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E.g. Academics have often academic
friends, etc.
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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
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Mechanisms underlying Homophily
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Selection
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A and B have similar characteristics -> A and B form a
link AB
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Social Influence
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A and B have a link -> B chooses the same (mutable)
characteristic as A
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E.g. A starts smoking, and B follows (peer pressure)
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Social-Affiliation Network
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Network of persons and social foci (activities)
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Triadic Closure
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Focal Closure
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Selection: Karate introduces Anna to Daniel
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Membership Closure
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Social Influence: Anna introduces Bob to
Karate
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Homophily
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Both Selection and Social Influence drive
homophily
How important is each mechanism?
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Important question: Different mechanism
implies different policy,
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e.g. Policy to prevent teenagers from smoking
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Social Influence. Target “key players” and let them
positively influence rest

Selection. Target on characteristics (e.g. family
background) alone
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Homophily
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Both Selection and Social Influence drive
homophily
How important is each mechanism?
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Difficult question:

Requires longitudinal data
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Requires observation of (almost) all characteristics

If a characteristic is not observed, then social
influence effect is overestimated
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Homophily
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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
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Sacerdote: Social influence among students after
randomized dorm assignment
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Homophily and Segregation
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Neighborhoods tend to be segregated
according to race or culture
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Ghetto formation
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What is the mechanism behind that?
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Segregation in Chicago
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Homophily and Segregation
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Segregation model of Thomas Schelling
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Agent-based model
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Two different agents: X and O types
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Agents live on a grid
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weak satisficing preferences for homophily
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At least k of the 8 neighbors of same type
Each period, agents who are not satisfied move to a
location where they are
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Schelling’s model (k=3)
X
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Schelling’s model (k=3)
X
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Schelling’s model online
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http://cs.gmu.edu/~eclab/projects/mason/project
s/schelling/
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Schelling’s model
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Surprising relation between micro-behavior
and macro-outcomes

Weak satisficing preferences for homophily
sufficient to create complete segregation
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Segregation arises due to miscoordination
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There exists an allocation involving complete
integration satisfying all agents, but individual
decisionmaking does not lead Typ
to hier
that
outcome
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Overview

Introduction Reading Group

Ch. 2 Graphs, Paths and Small Worlds

Ch. 3 Strength of Weak Ties

Ch. 4 Homophily


Schelling model
Planning
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Next week: 6 March 13:00


Natasa Golo and Dan Braha
Next Reading Group: 13 March 13:30 h

Maurice Koster: Ch. 8 and Ch. Typ
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