Date post: | 10-May-2015 |
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Generating networks with arbitrary properties
Jérôme Kunegis Generating Networks with Arbitrary Properties 2
Social Interaction
“You’re my friend”
Jérôme Kunegis Generating Networks with Arbitrary Properties 3
Many Social Interactions
“You’re my friend”“You’re m
y friend”
“You’re my friend”“You’re my friend”
“You’re
my f
riend”“You’re m
y friend”
“You’re my friend”
Jérôme Kunegis Generating Networks with Arbitrary Properties 4
Abstract: It's a Network
Jérôme Kunegis Generating Networks with Arbitrary Properties 5
Problem: Generate Realistic Graphs
Why generate graphs?
● To visualize an existing network: generate a smaller graph with same properties as a large real (note: sampling a subset will skew the properties)
● For testing algorithms: Generate a larger network then those currently known
Jérôme Kunegis Generating Networks with Arbitrary Properties 6
Basic Idea for Generating Networks: Random Graphs
Each edge has probability p of existing
Paul Erdős
Jérôme Kunegis Generating Networks with Arbitrary Properties 7
Random Graphs Are Not Realistic
Random graph
Real network
Jérôme Kunegis Generating Networks with Arbitrary Properties 8
Real Networks Have Special Properties
Many triangles (“clustering”)
Many 2-stars(“preferential attachment”)
● Short paths (“small world”)● Assortativity● Power-law-like degree distributions● Connectivity● Reciprocity● Global structure● Subgraph patterns● etc., etc., etc., etc., etc.
Jérôme Kunegis Generating Networks with Arbitrary Properties 9
Solution: Exponential Random Graph Models
Example with three statistics:
P(G) = exp( a1 m + a2 t + a3 s + b )
m, t, s: Properties of Gm = Number of edges; t = Number of triangles; s = Number of 2-stars
a1, a2, a3, b: Parameters of the model
Jérôme Kunegis Generating Networks with Arbitrary Properties 10
Problems of Exponential Random Graph Models
P(G) = exp( a1 x1 + a2 x2 + … + ak xk + b )
Many exponential random graph models are degenerate: They contain mostly almost-empty or almost-full graphs
But on average, they produce the correct statistics!
Jérôme Kunegis Generating Networks with Arbitrary Properties 11
Explanation of Degeneracy
Consider a variable x between 0 and 1 with expected value 0.3.
An exponential random model for it is given by:
P(x) = exp( ax + b )
We getMode[x] = 0 !!
x
P(x)
0 10.30
Jérôme Kunegis Generating Networks with Arbitrary Properties 12
Idea
Require not that E[x] = c, but that x follow a normal distribution
x
P(x)
0 10.30
P(G) = Pnorm (x1, x2, …; μ1, μ2, …, σ1, σ2, …)
Jérôme Kunegis Generating Networks with Arbitrary Properties 13
Real Networks Have a Distribution of Values Anyway
P(G) = Pnorm (x1, x2, …)
Data from konect.uni-koblenz.de
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Monte Carlo Markov Chain Methods
x1
x2
Wanted distribution
Random graphs
×
×
×
×
×
×
×
×
×
×××
×
×
×
×
Sampling will be bias towards the distribution of random graphs
P = high
P = low
+
+ Current graphs× Possible next steps
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Solution: Integral of Measure of Voronoi Cells
x1
x2
Wanted distribution
Random graphs
×
×
×
×
×
×
×
×
×
×××
×
×
×
×
Jérôme Kunegis Generating Networks with Arbitrary Properties 16
How To Compute The Integral over Voronoi Cells
Answer: We don't have to.
Sampling strategy:
● Sample point in statistic-space according to our wanted distribution● Find nearest possible network (i.e., nearest “×”)
Claim: This distribution at each step is similar to the underlying measure, giving an unbiased sampling.
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Result: Close, But Not Exact
Jérôme Kunegis Generating Networks with Arbitrary Properties 18
Convergence Speed (σ = 3)
Edge count
2-star count
Triangle count
Jérôme Kunegis Generating Networks with Arbitrary Properties 19
Example: Generate Network with Same Properties as Zachary's Karate Club