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Homophily and AffiliationWeb Science (VU) (707.000)
Elisabeth Lex
KTI, TU Graz
April 18, 2016
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Outline
1 Repetition
2 Homophily
3 Homophily: Selection and Social Influence
4 Affiliation and Affiliation Networks
5 Link Formation in Social Affiliation Networks
6 The Schelling Model
7 Summary and Practical Examples
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Repetition
Repetition from last lecture
We discussed structural characteristics of social networks, and someprocesses that affect formation of links in a network
i.e. Triadic Closure
When individuals B and C have a common friend A, then there areincreased opportunities and sources of trust on which to base theirinteractions, and A will also have incentives to facilitate their friendship
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Repetition
Motivation (1/2)
However: without regard of context in which networks form!
These contexts have effects on network
Each person in a social network has a distinctive set of characteristics
Similarities or compatibilities among two people’s characteristics canstrongly influence whether a link forms between them
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Repetition
Motivation (2/2)
Network’s surrounding contexts: factors that exist outside thenodes and edges of a network and which affect how network evolves
Today, we will discuss how such effects operate, and what they implyabout the structure of social networks
We will find that surrounding contexts can be viewed in networkterms as well by expanding the network to represent contexts togetherwith node and edge properties
We will discuss different processes of network formation
Plus: A model for segregation in networks based on certain factors
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Homophily
Homophily
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Homophily
Homophily: Introduction (1/2)
Homophily is the principle that we tend to be similar to our friends
E.g. to places to live, occupations, interests, beliefs, and opinions
One of the most basic notions governing the structure of socialnetworks
Most of us have specific friendships that cross such boundaries
But in aggregate, many links in social networks tend to connectpeople who are similar to one another
“Birds of a feather flock together”
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Homophily
Homophily: Introduction (2/2)
Compare: friendship forms because two people are introducedthrough a common friend vs friendship forms because two peopleattend the same school:
First is intrinsic in the networkSecond can only be understood when investigating contextualinformation beyond the network
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Homophily
Homophily: Introduction (2/2)
Compare: friendship forms because two people are introducedthrough a common friend vs friendship forms because two peopleattend the same school:
First is intrinsic in the network (triadic closure)Second can only be understood when investigating contextualinformation beyond the network (social environment)
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Homophily
Example: Homophily depicted in a Social Network
Social network from a town’s middle school and high school (USA)
Two divisions: (i) left to right: based on race, (ii) top to bottom:based on age and school attendance (students in middle schoolseparated from students in high school)
Homophily can divide social network into densely connected, homogeneousparts that are weakly connected
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Homophily
Homophily and Triadic Closure
Social contexts provide a basis for triadic closure!
Why? Say A-B and A-C friendships exist, homophily suggests that Band C are each likely to be similar to A in a number of dimensions,and thus possibly also similar to each other
Higher chance that B-C friendship will form due to this similarity
Even if neither of them is aware that the other one knows A
Ergo: links may arise from a combination of several factors, e.g.influence of position of other nodes in the network, and influence ofsurrounding contexts such as similar interests, age,..
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Homophily
Measuring Homophily: Example Network with Male andFemale People
Suppose we have a network in which a p fraction of all individuals aremale, and a q fraction of all individuals are female.
Consider: random network R = (V ,E r ), each node is assigned malewith probability p, and female with probability q
Let G = (V ,E ) be a random sample of R with p fraction of male,and q fraction of female.
Consider any edge (i , j) ∈ E r of this random network R
Let random variable Xij = 1 if it is a cross-edge (i.e., first end of theedge is male and the second end is female, or vice versa), and Xij = 0otherwiseThen Xij is a Bernoulli random variable such that P(Xij = 1) = 2pq
If fraction of cross-edges is significantly less than 2pq, then there isevidence for homophily
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Homophily
Example: friendship network of an elementary-schoolclassroom
Is there homophily by gender?
Colored nodes are female, white nodes are male5 of the 18 edges are cross-gender, so male p = 2/3 and femaleq = 1/3Compare fraction of cross-gender edges to 2pq = 4/9 = 8/18.With no homophily, we expect 8 cross-gender edges and not 5 -evidence of homophily.In practice, working definition of “significantly less than.”, e.g.statistical significanceAlso possible for network to have fraction of cross-gender edges that issignificantly more than 2pq, then network exhibits inverse homophily
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Homophily: Selection and Social Influence
Homophily: Selection and Social Influence
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Homophily: Selection and Social Influence
Mechanisms Underlying Homophily: Selection and SocialInfluence (1/2)
Selection: the tendency of people to form friendships with others whoare like them in terms of immutable characteristics such as ethnicity
E.g. People choose friends who are most similar from a small groupof available persons
Globally, selection can be more implicit: For example, when peoplelive in neighborhoods, attend schools, or work for specific companies,the social environment is already favoring opportunities to formfriendships with others like oneself.
With selection, the individual characteristics drive the formation oflinks
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Homophily: Selection and Social Influence
Mechanisms Underlying Homophily: Selection and SocialInfluence (2/2)
People may modify their behavior to align with behaviors of friends.
This is described as socialization and social influence - existing socialconnections in a network influence individual characteristics of nodes
With social influence, the existing links in the network serve to shapepeople’s (mutable) characteristics
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Homophily: Selection and Social Influence
Interplay between Selection and Social Influence
Track social connections and behaviors within a group over time
Behavioral changes that occur after an individual’s networkconnections changes as well as changes in the network that occurafter an individual changes her behavior
Hard to resolve how selection and social influence interact, andwhether one is more strongly at work than the other
E.g. study processes why friends have similar outcomes in terms ofscholastic achievement or drug usage
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Homophily: Selection and Social Influence
Example of Interplay between Selection and SocialInfluence (1/2)
Christakis and Fowler studied effect of social networks onhealth-related outcomes.
Using longitudinal data covering roughly 12 000 people, they trackedobesity status and social network structure over a 32-year period
They found: obese and non-obese people cluster in the networkconsistent with homophily: people tend to be more similar to theirnetwork neighbors wrt their obesity status than in a version of thesame network where obesity status is assigned randomly.
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Homophily: Selection and Social Influence
Example of Interplay between Selection and SocialInfluence (2/2)
Why?
Because of selection effects, in which people are choosing to formfriendships with others of similar obesity status?
Because of the effects of homophily according to other characteristics,in which the network structure indicates existing patterns of similarityin other dimensions that correlate with obesity status?
Because changes in the obesity status of a person’s friends wasexerting a (presumably behavioral) influence that affected his or herfuture obesity status?
Christakis and Fowler found that, even accounting for effects of types (i)and (ii), there is significant evidence for an effect of type (iii) as well: thatobesity is a health condition displaying a form of social influence wherechanges in your friends’ obesity status having an effect on you.
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Homophily: Selection and Social Influence
Summary
We have discussed contextual factors that affect formation of links ina network
Based on similarities in characteristics of nodes, behaviours, activitiesnodes are engaged in
Careful analysis needed to distinguish between factors that contributeto a conclusion
May be not clear why people tend to be similar to neighbours andhow this affects evolution of network and how behaviour of people innetwork is influenced
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Affiliation and Affiliation Networks
Affiliation and Affiliation Networks
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Affiliation and Affiliation Networks
Affiliation
Any context can be modeled into network itself even if it existsoutside the network: then, the network has both people and contextsas nodes
Today: person (“actors”) and activities (“events”) they participate inand how they affect formation of links
Activity: e.g. part of a company, organization, neighbourhood;pursuing a particular hobby or interest
Activities when shared between two people tend to increase likelihoodthat they will interact and form a link in the network
“Focal points of social interactions” - “foci”
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Affiliation and Affiliation Networks
Affiliation Networks
Represent the affiliation of people with foci
Affiliation networks are 2-mode networks: number of modes refers tothe number of distinct kinds of social entities in a network
Nodes of one type affiliate with nodes of the other types
Connections among members of one of the modes are based onlinkages established through the second
Affiliation networks allow to study the dual perspectives of the actorsand the events
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Affiliation and Affiliation Networks
Affiliation Networks: Example
Memberships of people on corporateboards of directors:
Two companies are implicitly linked byhaving the same person sit on boththeir boards. Higher chance forinformation and influence to flowbetween the two companies
Two people are implicitly linked byserving together on a board
Misses other contextual information:e.g. here, we see a former USVice-President
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Affiliation and Affiliation Networks
Other Examples for Affiliation Networks
Facebook.com users and groups/networks
XING.com users and groups
Netflix customers and movies
Amazon customers and books
Scientific network of authors and articles
etc
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Affiliation and Affiliation Networks
Properties of Affiliation Networks
Rates of Participation: the number of foci (e.g. events) with whicheach person is affiliated. Equal to degree of node representing theperson in the bipartite graph
Size of focus: number of people affiliated with a focus. Equal todegree of the node representing the event in the bipartite graph
Density (i.e., mean number of events to which pairs of personsbelong), reachability, connectedness, diameter (i.e., length of thelongest path between any pair of persons/events) defined on onemode networks that are extracted from affiliation networks
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Affiliation and Affiliation Networks
Affiliation Network Representation
Affiliation Matrix: rows index actors, columns events
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Affiliation and Affiliation Networks
Two Mode Networks and One Mode Networks
Folding: process of transforming two mode networks into one modenetworks
I.e., replacing paths of length two in the bipartite graph among actorsby an (undirected) edge.
Each two mode network can be folded into 2 one mode networks
Thus, events can be described as collections of actors affiliated with itand actors can be described as collections of events with which theyare affiliated.
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Affiliation and Affiliation Networks
Example
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Affiliation and Affiliation Networks
Co-Evolution of Social and Affiliation Networks
Social networks and affiliation networks change over time: newfriendship links are formed, and people become associated with newfoci
Co-evolution that reflects the interplay between selection and socialinfluence
If two people participate in a shared focus, this provides them with anopportunity to become friends
If two people are friends, they can influence each other’s choice of foci
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Affiliation and Affiliation Networks
Social-Affiliation Network
Extends notion of affiliation network - two distinct kinds of edges:
Edge in a social network that connects two people, and indicatesfriendship (or alternatively some other social relation, like professionalcollaboration)Edge that connects a person to a focus, and indicates the participationof the person in the focus.
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Link Formation in Social Affiliation Networks
Link Formation in Social AffiliationNetworks
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Link Formation in Social Affiliation Networks
Link formation in Social Affiliation Networks
Suppose we have two nodes B and C with a common neighbor A in thenetwork, and suppose that an edge forms between B and C
Three types of link formation in social affiliation networks:
Triadic Closure: If A, B, and C each represent a person, then theformation of the link between B and C is triadic closureFocal Closure: If B and C represent people, but A represents a focus -tendency of two people to form a link when they have a focus incommonMembership Closure: If A and B are people, and C is a focus, then thisis social influence (B takes part in focus that friend A already involvedin) (social influence)
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Link Formation in Social Affiliation Networks
Example of a Social-Affiliation Network
Figure: Social-affiliation network containing both people and foci, edges can formunder the effect of several different kinds of closure processes: two people with afriend in common, two people with a focus in common, or a person joining afocus that a friend is already involved in
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Link Formation in Social Affiliation Networks
Example of a Social-Affiliation Network
Figure: Social-affiliation network containing both people and foci, edges can formunder the effect of several different kinds of closure processes: two people with afriend in common, two people with a focus in common, or a person joining afocus that a friend is already involved in
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Link Formation in Social Affiliation Networks
Closure Types and Social Influence vs Selection
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Link Formation in Social Affiliation Networks
Some thoughts on Link Formation
Described methods good examples for small group settings
Hard to analyse on a larger scale: track mechanisms as they operatein large populations
Accumulation of many small effects can produce somethingobservable in the aggregate
In reality: many link formation processes are “off the record”
Not easy to choose group of people (and social foci), and accuratelyquantify relative contributions that different mechanisms make toformation of real network links
Approach: Through online studies, caveat - never a priori clear howmuch one can extrapolate from digital interactions to interactionsthat are not computer-mediated, or even from onecomputer-mediated setting to another
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Link Formation in Social Affiliation Networks
Triadic Closure
Is it even more likely if nodes have more than 1 friend in common?
Figure: Anna and Esther have two friends in common, while Claire and Danielonly have one friend in common
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Link Formation in Social Affiliation Networks
Triadic Closure
Is it even more likely if nodes have more than 1 friend in common?
Figure: Anna and Esther have 2 friends in common, Claire and Daniel have 1common friend
Yes! Two have more common friends, more sources of opportunity andtrust for the interaction, more people with incentive to bring themtogether.
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Link Formation in Social Affiliation Networks
Empirical Assessment of Triadic Closure [Kossinets andWatts, 2006] (1/2)
We address this triadic closure empirically using network data as follows
Take two snapshots of the network at different times.
For each k, we identify all pairs of nodes who have exactly k friends incommon in the first snapshot, but who are not directly connected byan edge.
Define T(k) to be the fraction of these pairs that have formed anedge by the time of the second snapshot.
This is the empirical estimate for the probability that a link will formbetween two people with k friends in common.It reflects the power of triadic closure.
Plot T(k) as a function of k to illustrate the effect of common friendson the formation of links
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Link Formation in Social Affiliation Networks
Empirical Assessment of Triadic Closure [Kossinets andWatts, 2006] (2/2)
Dataset: history of e-mail communication among 22,000 US studentsover a one-year period
Network constructed that evolved over time: two people are linked ifthey had exchanged e-mail in each direction at some point in the past60 days.
Question: What is the probability T(k) that a person becomes friendswith another person as a function of the number of common friends(k)?Calculation of T(k): Determation of “average” version of by takingmultiple pairs of snapshots
Building a curve for T(k) on each pair of snapshots, and thenaveraging all obtained curvesObservations in each snapshot were one day apart, so computationgives avg. probability that two people form a link per day, as a functionof the number of common friends.
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Link Formation in Social Affiliation Networks
Results
Black: curve determined from thedata. Dotted: comparison toprobabilities computed according tobaseline (common friends provideindependent probabilities of linkformation)
Clear evidence for triadic closure:
Probability of link formationincreases steadily as number ofcommon friends increasesRoughly linear increase, with 2common friends more than twice theeffect on link formation!
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Link Formation in Social Affiliation Networks
Focal Closure
Using the same approach, (Kossinets and Watts, 2006) computeprobabilities for the focal closure
What is the probability that two people form a link as a function ofthe number of foci they are jointly affiliated with?
Example: Combining university e-mail dataset with information aboutthe class schedules for each student
Each class became a focus, and two students shared a focus if theyhad taken a class together
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Link Formation in Social Affiliation Networks
Empirical Results
Single shared class has similar effect on link formation as a singleshared friendBut: curve for focal closure behaves quite differently from triadicclosure curve
Turns downward and appears to approximately level off, rather thanturning slightly upwardThus, subsequent shared classes after the first produce a ’diminishingreturns’ effect
Interesting open question to understand how this generalizes to othertypes of shared foci, and to other domains
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Link Formation in Social Affiliation Networks
Membership Closure
What is the probability T(k) that a person becomes involved with aparticular focus as a function of the number of friends (k) who arealready involved in it?
(Kossinets and Watts, 2006) did not provide any analysis on this
Therefore, a study on membership closure in Wikipedia is presentedhere
Social network of Wikipedia editors who maintain talk pages
Edge between two editors if they have communicated via the talkpages
Editor’s behavior corresponds to set of articles she edited - associationto focus
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Link Formation in Social Affiliation Networks
Observations
Initial increasing effect similar as with triadic closure: the probabilityof editing a Wikipedia article more than twice as great when you have2 connections into the focus (“editing the same articles”) rather than1
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Link Formation in Social Affiliation Networks
Interplay Between Selection and Social Influence
Question: is the homophily arising because:
Editors are forming connections with those who have edited the samearticles they have - selection?
Editors are led to the articles of those they talk to - social influence?
Let’s look at a measure of their behavioral similarity
Similarity = number of articles edited by both editor A and editor Bnumber of articles edited by at least editor A or editor B
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Link Formation in Social Affiliation Networks
Interplay Between Selection and Social Influence
Question: is the homophily arising because:
Editors are forming connections with those who have edited the samearticles they have - selection?
Editors are led to the articles of those they talk to - social influence?
Let’s look at a measure of their behavioral similarity
Similarity = number of articles edited by both editor A and editor Bnumber of articles edited by at least editor A or editor B
Corresponds to neighborhood overlap of two editors in affiliation networkof editors and articles!
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Link Formation in Social Affiliation Networks
Analysis on the Wikipedia
For each pair of editors A and B who have ever communicated, recordtheir similarity over time (actors in Wikipedia are timestamped)
Time moves in discrete units, advancing by one “tick” whenevereither A or B performs an action on Wikipedia Time
Next, declare time 0 for the pair A and B to be the point at whichthey first communicated
Results show similarity as function of time, one for each pair ofeditors who ever communicated
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Link Formation in Social Affiliation Networks
Results
Similarity increases both before and after moment of first interaction(both selection and social influence)
More steep before 0 (particular role on similarity)
Rise in similarity before two editors meet
Note that similarity much higher for interacting users than for pairs ofeditors who have not interacted
Blue line shows similarity of a random pair (non-interacting)
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Link Formation in Social Affiliation Networks
Homophily and Segregation
Homophily can explain segregation e.g. formation of ethnically and raciallyhomogeneous neighborhoods in cities.
Figure: Map by Moebius and Rosenblatt: percentage of African-Americans percity block in Chicago in 1940 and 1960. In blocks colored yellow and orange,percentage is below 25, in brown and black above 75.
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The Schelling Model
A Model for Spatial Segregation - TheSchelling Model
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The Schelling Model
The Schelling Model
Thomas Crombie Schelling (born 14 April 1921)
An American economist, and Professor of foreign affairs, nationalsecurity, nuclear strategy, and arms control at the School of PublicPolicy at University of Maryland, College Park.
Awarded the 2005 Nobel Memorial Prize in Economic Sciences(shared with Robert Aumann) for “having enhanced ourunderstanding of conflict and cooperation through game-theoryanalysis”
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The Schelling Model
The Schelling Model
Schelling showed that a small preference for one’s neighbors to be ofthe same color could lead to total segregation
Shows how global patterns of spatial segregation can arise fromhomophily at local level
Focuses on intentionally simplified mechanism to show how forcesthat lead to segregation are highly robust - they can operate evenwhen no one explicitely aims for segregated outcome
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The Schelling Model
How does the Schelling Model work? (1/2)
Assume a population of individuals (aka agents) of type X or OTypes represent immutable characteristics (e.g., age)Two populations of the two agent types are initially placed intorandom locations of a neighborhood represented by a gridAfter placing all agents, each cell either occupied by an agent oremptyThe neighbor relationships among the cells can be represented verysimply as a graph: cells are the nodes, edges are inserted between twocells that are neighbors on the grid
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The Schelling Model
How does the Schelling Model work? (2/2)
Now, determine if each agent is satisfied with its current location
Agent is satisfied if is surrounded by at least t percent of its own typeof neighboring agents
Threshold t applies to all agents in the model (in reality everyonemight have a different threshold they are satisfied with)
The higher t, the higher the likelihood that agents will not besatisfied with their current locationExample:
For example, if t = 3, agent X is satisfied if at least 3 of its neighborsare also XIf fewer than 3 are X, then the agent is not satisfied, and it will want tochange its location in the gridWhen an agent is not satisfied, it can be moved to any vacant locationin the gridAny algorithm can be used to choose new location (e.g., randomselection, nearest available location, 1 row at a time)
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The Schelling Model
Example
(a) Initial stage (b) After one round
Figure: Left image: all dissatisfied agents have an asterisk next to them. Rightimage: shows new configuration after all dissatisfied agents have been moved tounoccupied cells (1 row at a time) where they are satisfied. May cause otheragents to become unsatisfied, then new round of movement begins
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The Schelling Model
Netlogo Example of the Schelling Model
http://ccl.northwestern.edu/netlogo/
Go to File/Model Library/Social Science/Segregation
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The Schelling Model
Observations from Schelling’s Model
Spatial segregation takes place even though no individual agentactively seeks it
Segregation doesn’t happen due to built-in model agents that arewilling to be in the minority
Ideally, all agents are carefully arranged in an integrated pattern
However, from random start hard for agents to find such integratedpatterns
At more general level, Schelling model is an example of how fixedcharacteristics (e.g., ethnicity) can become highly correlated withmutable characteristics
E.g. decision where to live, which over time conforms to similarities inagents immutable types, producing segregation
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Summary and Practical Examples
Summary and Practical Examples
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Summary and Practical Examples
Summary
We have learned about:
Homophily
Affiliation, Affiliation Networks, Social Affiliation Networks
Link Formation in Social Affiliation Networks, 3 types of closure
The Schelling Model as spatial model for segregation
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Summary and Practical Examples
Sompe Practical Examples
Targeted marketing:
User recommendations in group settings. Authors found that social tiesare indicative of similar preferences. 1
Exploited this to detect communities of users with similar taste(homophily) and to exploit this to improve Collaborative Filtering (CF)
Investigating effects in society, e.g., the glass ceiling effect 2
Investigating political communications in social networks 3
1Narasimhan, et al., 2014. Exposing commercial value in social networks: matchingonline communities and businesses
2Chen Avin, Barbara Keller, Zvi Lotker, Claire Mathieu, David Peleg, andYvonne-Anne Pignolet. 2015. Homophily and the Glass Ceiling Effect in SocialNetworks. In Proceedings of the 2015 Conference on Innovations in TheoreticalComputer Science (ITCS ’15)
3Halberstamm & Knight, 2014. Homophily, Group Size, and the Diffusion of PoliticalInformation in Social Networks: Evidence from Twitter. NBER Working Paper No.20681
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Summary and Practical Examples
Thanks for your attention - Questions?
Slides use figures from Chapter 4 of Networks, Crowds and Markets byEasley and Kleinberg (2010)http://www.cs.cornell.edu/home/kleinberg/networks-book/Elisabeth Lex (KTI, TU Graz) Networks April 18, 2016 63 / 63