04198250 social networks 2015€¦ · · 2015-09-22jazz artcle jill. membership closure in...
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04198250 Social Networks 2015
Lecture 03: Networks in their surrounding contexts
The role of surrounding contexts
● Triadic closure: explains link formaton by propertes intrinsic in the network
● Today we will look at how factors outside the standard network model infuences link formaton
● In partcular: similarity between individuals● .. and the other way around
Homophily
• “Birds of a feather fock together.”• Your friends are more similar to you in age,
race, interests, opinions, etc. than a random collecton of individuals
• Homophily: principle that we tend to be similar to our friends
• One of the most basic notons governing the formaton of a social network
Moody (2001)
Social network from town’s middle and high schools
Moody (2001)
Social network from town’s middle and high schools
Clustering by race
Moody (2001)
Social network from town’s middle and high schools
Clustering by age
Homophily and Triadic Closure• Triadic closure: When two individuals share a
common friend, a friendship between them is more likely to occur
• Homophily suggests two individuals are more alike because of common friend, so link may occur even if neither is aware of mutual friend!
• Difcult to atribute formaton of link to any one factor
Can we develop a simple test for the
presence of homophily in a network?
Example Male/Female Network9 nodes: 6 males, 3 females
11 edges: 6 m-m, 2 f-f, 2 m-f
Does it exhibit homophily?
We have Homophily if few cross-edges
• Assume n nodes, with pn males and qn females (fractons p and q with 0<p,q<1 and p+q=1).
• If there is no homophily then edges are random.
If nodes randomly assigned a gender, then probability of an edge being male-male is p², female-female is q² and cross-gender is 2pq.
• Thus, if << 2pq cross-gender edges then the network shows homophily!
p²
q²2pq
Figure out p, q, n and decide if the network below exhibits homophily.
Notes on Homophily Test
• What if network had significantly more than 2pq cross-gender edges?
– Inverse homophily
– Example: Male-Female datng relatonships
Notes on Homophily Test
• Homophily test can be used to test any characteristc like race, age, natve language, preferences, etc.
• For characteristcs that have more than 2 values, perform same type of calculaton
• Compare number of heterogeneous edges to what randomly generated graph would look like using real data as probabilites
Why is homophilu ofen present in a social network?
• Two answers: Selecton and social infuence
• Selection: People tend to choose friends that are like themselves
• Can operate at different scales and levels of intentonality
– You actvely choose friends that are like yourself among a small group of people
– Your school’s populaton is relatvely homogeneous compared to overall populaton, so your environment compels you to choose friends like yourself
Mutable and Immutable Characteristcs
• Link formaton operates differently based on type of characteristc
• Immutable: Characteristcs that don’t change (gender, race) or change consistently with the populaton (age, generaton)
• Mutable: Characteristcs that can change over tme (behaviors, beliefs, interests, opinions)
Selecton and Social Infuence
• Research has shown that people are susceptble to social influence: they may change their behaviors to more closely resemble the behaviors of their friends (surprise!)
• “Bad company corrupts good character.” – 1 Corinthians 15:33
• Social infuence is reverse of selecton.
– Selecton: Individual characteristcs drive the formaton of links
– Social infuence: Existng links shape people’s mutable characteristcs
Longitudinal Studies
• Difcult to tell if selecton or social infuence at play with a single snapshot of a network
• Longitudinal studies tracking social connectons and individual behaviors over tme can help researchers uncover effect of social infuence
• Does behavior change afer changes to the network, or does the network change afer changes in behavior?
Example• Longitudinal studies have been used to determine if selecton
or social infuence has had more effect on teenagers’ scholastc achievement and drug use
• Understanding these effects can be helpful in developing interventons
• Assume drug use displays homophily in a network. Then, a
program to target social infuence (get friends to infuence other friends to stop) might work well, but...
• But if homophily is due to selecton, former drug users may choose new friends and drug-using behavior of others is not strongly affected
Social Infuence of Obesity
• Christakis and Fowlertracked obesity statusand social networkof 12,000 people over32 years
• Found homophily based on obesity status• They wanted to know why
Christakis & Fowler (2007)htp://www.ted.com/talks/nicholas_christakis_the_hidden_infuence_of_social_networks.html
Social Infuence of Obesity
• Christakis and Fowlertracked obesity statusand social networkof 12,000 people over32 years
• Found homophily based on obesity status• They wanted to know why
Christakis & Fowler (2007)htp://www.ted.com/talks/nicholas_christakis_the_hidden_infuence_of_social_networks.html
Why Obesity Homophily?
1. Selecton effects – people choose to befriend others of similar obesity status?
2. Confounding effects of homophily – other factors that correlate with obesity status?
3. Social infuence – if friends changed their obesity status, did it infuence person’s future obesity status?
• Discovered significant evidence for hypothesis 3: Obesity is a
type of “contagion” that can spread through social infuence!
Afliaton Network• Putng context into the network by showing connectons to
actvites, companies, organizatons, neighborhoods, etc.
• Bipartite graph: every edge joins two nodes belonging to different sets
Sue
Bill
Chess Club
Band
Social-Afliaton Network
• Two types of edges:– Person to person: Friendship or other social relatonship
– Person to foci: Partcipaton in the focus
Sue
Bill
Chess Club
Band
Gary
Alice
Closure Processes
SueGary
Alice
Triadic closure
Band Sue
Bill
Sue
Gary
Band
Focal closure: closure due to selecton
Membership closure: closure due to social infuence
Research Questons About Closure• Would Sue be more likely to become friends with Alice if they
shared more than one friend?
• In other words, is triadic closure dependent on the number of shared friends?
• More formally: What is the probability that two people form a link as a functon of the number of mutual friends they share?
SueGary
Alice
SueGary
Alice Moe
Research Questons About Closure
• For focal closure, what is the probability that two people form a link as a functon of the number of foci they are jointly afliated with?
• For membership closure, what is the probability that a person becomes involved in a partcular focus as a functon of the number of friends who are already involved in it?
Band Sue
Bill Ping Pong
Sue
Gary
Band
Jill
Research Methodology: Measuring Triadic Closure
1. Take two snapshots of network at tmes t1 and t2
2. For each k, identfy all pairs of nodes who have exactly k friends in common at t1 but who are not directly connected by an edge
3. Define T(k) to be fracton of these pairs that form an edge by
t2. This is the probability that a link will form between two people with k friends in common
4. Plot T(k) as functon of k to illustrate effect of common friends on link formaton
Email Social Network
• Kossinets and Wats (2006) examined email communicaton of 22,000 students over one year at a large US university
• Made link between two people if an email was sent between the two in the last 60 days
• Each snapshot is one day apart
• T(k) averaged over multple pairs of snapshots
Triadic Closure in Email Data Set
Almost no emails
exchanged when no friends in common
Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf
Triadic Closure in Email Data Set
Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf
Significant increase
going from 1 to 2 friends
Significant increase but
on much smaller
subpopulation
Triadic Closure in Email Data Set
Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf
Baseline model: probability p of
forming link when k friends in commonTbase(k) = 1 – (1 –
p)^k
Tbase(k) = 1 – (1 –p)^(k-1)
Email Social Network• To evaluate focal closure, Kossinets and Wats
obtained class schedules for 22,000 students• Created social-afliaton network where
classes are foci• Determined probability of link formaton as
functon of number of shared foci
ENG 101 Sue
Bill HIST 202
Focal Closure in Email Data Set
Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf
Sharing a class has
nearly same effect as sharing a
friend
Diminishing returns
Tbase(k) = 1 – (1 – p)^k
Measuring Membership Closure
• Backstrom et al. (2006) created social-afliaton network for blogging site LiveJournal
– Friendships designated by users in their profile– Foci are membership in user-defined communites
Sue
Gary
Jazz Comm
Jill
Membership Closure in LiveJournal Data Set
Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf
Significant increase
going from 1 to 2
Diminishing returns
Measuring Membership Closure
• Crandall et al. (2008) created social-afliaton network for Wikipedia
– Node for editors maintaining a user talk page
– Link if two editors have communicated using talk page
– Foci are artcles edited by editors
Sue
Gary
Jazz Artcle
Jill
Membership Closure in Wikipedia Data Set
Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf
Significant increase
going from 1 to 2
Diminishing returns
Selecton and Social Infuence
• Further examinaton of Wikipedia data set to see evidence of homophily produced by selecton and social infuence
• How do we measure similarity of two editors?
number of artcles edited by both editorsnumber of artcles edited by at least one editor
• Similar to neighborhood overlap of two editors in an afliaton network of editors and artcles
Selecton and Social Infuence
• We have homophily! Pairs of Wikipedia editors who have communicated are significantly more similar in behavior than editors who have never communicated
• Does homophily arise because…– editors are talking with editors who have edited
the same artcle? (selecton)– editors are led to the artcles by those they talk
to? (social infuence)
Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf
Segregaton
• Cites ofen divided into homogeneous neighborhoods based on ethnic and racial lines
Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf
High density of African-
Americans
Low density of African-
Americans
Schelling Model of Segregaton
• Thomas Schelling introduced simple model in 1970s that shows how global paterns of self-chosen segregaton can emerge due to the effects of homophily at the local level, even if no individual wants segregated outcome.
• Populaton formed by two types of individuals called agents (X and O)
• The two types of agents represent immutable characteristc like race, ethnicity, country of origin, or natve language
X X OX O O OX X X XX O X X
O O
Agents placed randomly in grid
Agents are satisfied with their locaton in the grid if at least t of their neighbors are agents of the same type.
t = threshold
In each round, first identfy all unsatsfied agents
X X OX O O OX X X XX O X X
O O
Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type
Unsatsfied because only 2 neighbors
are X
X X OX O O OX X X XX O X X
O O
Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type
Unsatsfied because only 2 neighbors are X
X* X OX O O OX X X XX O X X
O O
Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type
Satsfied because 3 neighbors are O
X* X* OX O O OX X X XX O X X
O O
Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type
Satsfied because 4
neighbors are X
X* X* OX O O OX X X XX O X X
O O
Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type
Unsatsfied
X* X* OX O O OX X X XX O X X
O O
Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type
Satsfied
X* X* OX O* O OX X X XX O X X
O O
Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type
Unsatsfied
X* X* OX O* O OX X X XX O X X
O O
Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type
Etc… Here’s the final result where unsatsfied agents marked with *
X* X* OX O* O O*X X X X
X* O* X XO* O*
Now all unsatsfied agents move, one by one, to a locaton where they are satsfied
O O OX O O OX X X X X
X X X XO*
Previously satisfied agent no
longer satisfied!
Note that the grid is now more segregated than before
Sometimes an agent can’t find a new location that will satisfy – leave alone or move to random location
Model Variatons
• Schedule agents to move in random order or in sweeping moton
• Agents move to nearest locaton that satsfies or random one
• Threshold could be percentage or vary among or within populaton groups
• World could wrap (lef edge meets up with right edge)
• Many other variatons, but results usually end up the same:
self-imposed segregation!
Schelling Simulators
• Check out a simulator:• Uri Wilensky’s NetLogo Segregaton model
htp://ccl.northwestern.edu/netlogo/models/Segregaton
Luke’s Schelling Model using t = 3
Randomly placed Afer 20 rounds
Conclusions
• Spatal segregaton takes place even though no single agent is actvely seeking it
• Schelling model is an example of how mutable characteristcs can become highly correlated with immutable characteristcs
• Choice of where to live (mutable) over tme conforms to agents’ type (immutable) producing segregaton (homophily)