1. Topic 13 Network Models Credits: C. Faloutsos and J. Leskovec Tutorial
E. Kolaczyk Notes
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Social Networks
Network: A collection of inter-connected things
Also called “graph mining”
Data consisting of nodes and edges
Note: different than “graphical models” (graphical representation of dependence of random variables)
Edges represent:
Relationship between nodes
Behavior observed between nodes
High similarity between nodes
Edges typically weighted
Nodes and edges both can have attributes associated
Can be directed or undirected
Directed: phone calls, s
Undirected: collaboration, physical networks, friendship
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Examples
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4. Networks are everywhere!
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Layout
Layout matters!
Especially with directed graphs
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6. Facebook “Friend Wheel”
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7. LinkedIn LinkedIN community from LinkedIn labs
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8. Networks: A Matter of Scale
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9. Measurements on networks: Nodes and Edges
Node degree (node)
Number of edges coming in and out of a node is its degree
If directed, in-degree and out-degree are different
Degree centrality (node):
How ‘central’ is a given data point
How many times does it appear in a ‘shortest path’
Centrality = importance
Centrality (edge):
How central is an edge?
Similar ‘shortest path’ definition
Does removing it create more clusters?
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10. Measurements on networks (graph)
Degree Distribution
The distribution of all edge degrees characterizes the graph
Normal or highly skewed?
Clustering Coefficient (graph):
How “dense” is the graph?
Given n nodes, how many possible edges?
Density = #Edges/Possible edges
How likely is it that your friends are friends
Count: how many triangles
Diameter (graph)
Largest shortest path
Shortest paths (graph)
Histogram of shortest paths
Connectivity (graph)
Fully connected?
Connected components
For directed: strongly connected components
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Models on networks
Random (Erdos-Renyi)
All edges occur randomly w probability p
Degree distribution follows Poisson distribution
Exponential (p*) models
Statistical model: Extension of Erdos-Renyi
Defines a probability distribution over graph properties
Preferential attachment
Generative Model: New nodes create m links (based on Poisson)
attach to existing nodes proportional to degree of that node
Rich get richer
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Real-world networks
Degree distributions in real-world networks are heavily skewed to the right
preferential attachment fits this model
Long tail of values above the mean
Large mean, small median, small diameter
Leads to a “power law”
Let k = degree and pk = the number of nodes that have that degree
A plot of log k vs. log pk should be linear.
Many real world data sets follow a power law:
Online sales
Word length distributions
Number of friends on Facebook!
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More Power Law
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14. Erdos-Renyi vs. Power-law
From Leskovec & Faloutsos
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Small World
Real-world data sets tend to have power-law distributions
Also, tend to have a “small world” property
Everyone is reachable via a small number of edges
Small diameters
Stanley Milgram experiment 1967
People given letter, asked to forward to one friend
source: random residents of Omaha
target: stockbroker in Boston
Of completed chains, averaged 6 hops
hence,
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Small World Networks
Watts and Strogatz [1998] introduced small-world.
Navigable Social Networks [Kleinberg 2000]
Showed how small world networks are created
put n people on a k-dimensional grid
connect each to its immediate neighbors
add one long-range link per person
Everyone will be connected via a short path
This is the way the real world works!!!
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Small World Networks
Another look
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Sampling Networks
How do you sample from a massive network?
Simplest method – Induced Subgraph
Randomly sampled nodes and edges between them
Not so great!
Yellow nodes randomly sampled but don’t have the same graph properties!
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19. Sampling Networks Snowball Sampling:
Pick a random sample and then follow their ‘tree’ for a set number of ‘hops’
Still not perfect but better
Other ideas abound but little agreement
Great area for research!
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20. Network Problems of Interest
Link Prediction:
can we use existing network data to infer links where they don’t exist?
Links in the future?
Missing data
Simple methods
Look for many common neighbors
Complex methods
Stochastic Blockmodels
Similar to using SVD to ‘fill in’ a matrix
Agarwal and Pregibon ‘04
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21. Network Problems of Interest
Graph Matching / Similarity
Fraud (‘repetitive debtors’)
Citation de-noising
Need a metric to define difference between graphs
Collective Inference
What can you learn about someone from their network?
Fraud (‘guilt by association’)
Viral marketing
Following example courtesy of Sofus MacSkassy
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22. A Relational Neighbor Classifier (wvRN)?
Sofus A. Macskassy

23. A Relational Neighbor Classifier (wvRN)? ? ? ?
Sofus A. Macskassy

24. A Relational Neighbor Classifier (wvRN)? ? ? ?
Sofus A. Macskassy

25. Collective wvRN
Classify all entities in the network simultaneously, because (if done well) inferences about neighbors can reduce statistical bias (cf. Jensen et al. KDD-04) ? ? ? ? ? ? ? ? ? ?
Sofus A. Macskassy

26. Collective wvRN
Classify all entities in the network simultaneously, because (if done well) inferences about neighbors can reduce statistical bias (cf. Jensen et al. KDD-04) ? ? ? ? ? ? ? ? ? ?
Sofus A. Macskassy

27. Collective wvRN
Classify all entities in the network simultaneously, because (if done well) inferences about neighbors can reduce statistical bias (cf. Jensen et al. KDD-04) ? ? ? ? ? ? ? ? ? ?
Sofus A. Macskassy

28. Collective wvRN
Classify all entities in the network simultaneously, because (if done well) inferences about neighbors can reduce statistical bias (cf. Jensen et al. KDD-04) ? ? ? ? ? ? ? ? ? ?
Sofus A. Macskassy

29. Collective wvRN
Classify all entities in the network simultaneously, because (if done well) inferences about neighbors can reduce statistical bias (cf. Jensen et al. KDD-04) ? ? ? ? ? ? ? ? ? ?
Sofus A. Macskassy

30. Collective wvRN
Classify all entities in the network simultaneously, because (if done well) inferences about neighbors can reduce statistical bias (cf. Jensen et al. KDD-04) ? ? ? ?
Sofus A. Macskassy

31. Network Problems of Interest
Diffusion
Information or virus diffusion
Community Detection
Subgroups have a higher density within the subgroup
Can remove edges with high centrality to try and find communities
Understanding of Social Networks
Facebook
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References
Leskovec / Faloutsos Tutorial (mostly part 1)
Eric Kolacyzk Notes and book
Watts and Strogatz: “Collective dynamics of `small-world' networks”: Nature 393 p
Networks. MEJ Newman book.
Linked: How Everything Is Connected to Everything Else and What It Means : Albert Barabasi
Enron Data
Tools
Graphviz.org for visualization
Igraph (R package)
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