Analysis of Social Media MLD 10-802, LTI 11-772 William Cohen
1-25-010
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Recap: What are we trying to do? Like the normal curve: Fit
real-world data Find an underlying process that explains the data
Enable mathematical understandingl (closed- form?) Modelssome small
but interesting part of the data
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Graphs Some common properties of graphs: Distribution of node
degrees Distribution of cliques (e.g., triangles) Distribution of
paths Diameter (max shortest- path) Effective diameter (90 th
percentile) Connected components Some types of graphs to consider:
Real graphs (social & otherwise) Generated graphs: Erdos-Renyi
Bernoulli or Poisson Watts-Strogatz small world graphs
Barbosi-Albert preferential attachment
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Graphs Some types of graphs to consider: Real graphs (social
& otherwise) Generated graphs: Erdos-Renyi Bernoulli or Poisson
Watts-Strogatz small world graphs Barbosi-Albert preferential
attachment All pairs connected with probability p
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Graphs Some types of graphs to consider: Real graphs (social
& otherwise) Generated graphs: Erdos-Renyi Bernoulli or Poisson
Watts-Strogatz small world graphs Barbosi-Albert preferential
attachment Regular, high-homophily lattice Plus random shortcut
links
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Graphs Some types of graphs to consider: Real graphs (social
& otherwise) Generated graphs: Erdos-Renyi Bernoulli or Poisson
Watts-Strogatz small world graphs Barbosi-Albert preferential
attachment New nodes have m neighbors High-degree nodes are
preferred Rich get richer
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Graphs Some common properties of graphs: Distribution of node
degrees Distribution of cliques (e.g., triangles) Distribution of
paths Diameter (max shortest- path) Effective diameter (90 th
percentile) Connected components Some types of graphs to consider:
Real graphs (social & otherwise) Generated graphs: Erdos-Renyi
Bernoulli or Poisson Watts-Strogatz small world graphs
Barbosi-Albert preferential attachment
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Graphs Some common properties of graphs: Distribution of node
degrees Distribution of cliques (e.g., triangles) Distribution of
paths Diameter (max shortest- path) Effective diameter (90 th
percentile) Connected components
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Graphs Some common properties of graphs: Distribution of node
degrees Distribution of cliques (e.g., triangles) Distribution of
paths Diameter (max shortest- path) Effective diameter (90 th
percentile) Connected components In a big Erdos-Renyi graph this is
very small (1/n) In social graphs, not so much More later
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Graphs Some common properties of graphs: Distribution of node
degrees Distribution of cliques (e.g., triangles) Distribution of
paths Diameter (max shortest- path) Effective diameter (90 th
percentile) Mean diameter Connected components In a big Erdos-Renyi
graph this is small (logn/logz) In social graphs, it is also small
(6 degrees)
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Graphs Some common properties of graphs: Distribution of node
degrees Distribution of cliques (e.g., triangles) Distribution of
paths Diameter (max shortest- path) Effective diameter (90 th
percentile) Mean diameter Connected components In a big Erdos-Renyi
graph there is one giant connected component because two giant
connected components cannot co-exist for long.
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n/a Poor fit
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More terms Centrality and betweenness: how does your position
in a network affect what you do and how you do it? And how can we
define these precisely? High centrality: ringleaders? High
betweenness: go-between, conduit between different groups?
Structural hole Group cohesiveness: number of edges within a
(sub)group
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More terms
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Association network: bipartite network where nodes are people
or organizations
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A larger association network
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Triads and clustering coefficients In a random Erdos-Renyi
graph: In natural graphs two of your mutual friends might well be
friends: Like you they are both in the same class (club, field of
CS, ) You introduced them
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Watts-Strogatz model Start with a ring Connect each node to k
nearest neighbors homophily Add some random shortcuts from one
point to another small diameter Degree distribution not scale-free
Generalizes to d dimensions
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Even more terms Homophily: tendency for connected nodes to have
similar properties Social contagion: connected nodes become similar
over time Associative sorting: similar nodes tend to connect
Disassociative sorting: vice-versa Association network: bipartite
network where nodes are people or organizations
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A big question Homophily: similar nodes ~= connected nodes
Which is cause and which is effect? Do birds of a feather flock
together? Do you change your behavior based on the behavior of your
peers? Do both happen in different graphs? Can there be a
combination of associative sorting and social contagion in the same
graph?
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A big question about homophily Which is cause and which is
effect? Do birds of a feather flock together? Do you change your
behavior based on the behavior of your peers? How can you tell?
Look at when links are added and see what patterns emerge (triadic
closure): Pr(new link btwn u and v | #common friends)
Final example: spatial segregation How picky do people have to
be about their neighbors for homophily to arise? Imagine a grid
world where Agents are red or blue Agents move to a random location
if they are unhappy Agents are happy unless
