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Generating Networks with Arbitrary Properties
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3
4. Abstract: It's a Network
Jérôme Kunegis
Generating Networks with Arbitrary Properties
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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)
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For testing algorithms: Generate a larger
network then those currently known
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Jérôme Kunegis
Generating Networks with Arbitrary Properties
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6. Basic Idea for Generating Networks: Random Graphs
Each edge has
probability p of existing
Paul Erdős
Jérôme Kunegis
Generating Networks with Arbitrary Properties
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7. Random Graphs Are Not Realistic
Real network
Random graph
Jérôme Kunegis
Generating Networks with Arbitrary Properties
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8. Real Networks Have Special Properties
Many triangles
(“clustering”)
Many 2-stars
(“preferential attachment”)
Short paths (“small world”)
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Assortativity
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Power-law-like degree distributions
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Connectivity
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Reciprocity
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Global structure
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Subgraph patterns
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etc., etc., etc., etc., etc.
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Jérôme Kunegis
Generating Networks with Arbitrary Properties
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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 G
m = 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
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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
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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 )
P(x)
We get
Mode[x] = 0
!!
0
Jérôme Kunegis
0
0.3
Generating Networks with Arbitrary Properties
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x
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12. Idea
Require not that E[x] = c, but that x follow a normal distribution
P(x)
0
0
0.3
1
x
P(G) = Pnorm (x1, x2, …; μ1, μ2, …, σ1, σ2, …)
Jérôme Kunegis
Generating Networks with Arbitrary Properties
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13. Real Networks Have a Distribution of Values Anyway
P(G) = Pnorm (x1, x2, …)
Data from konect.uni-koblenz.de
Jérôme Kunegis
Generating Networks with Arbitrary Properties
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14. Monte Carlo Markov Chain Methods
+ Current graphs
× Possible next steps
Wanted distribution
×
Random graphs
+
×
x2
×
×
×
P = high
×
Sampling will be bias
towards the distribution
of random graphs
P = low
×
×
×
×
×
×
×
×
×
×
x1
Jérôme Kunegis
Generating Networks with Arbitrary Properties
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15. Solution: Integral of Measure of Voronoi Cells
Wanted distribution
×
×
Random graphs
×
×
x2
×
×
×
×
×
×
×
×
×
×
×
×
x1
Jérôme Kunegis
Generating Networks with Arbitrary Properties
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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
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Find nearest possible network (i.e., nearest “×”)
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Claim: This distribution at each step is similar to the
underlying measure, giving an unbiased sampling.
Jérôme Kunegis
Generating Networks with Arbitrary Properties
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17. Result: Close, But Not Exact
Jérôme Kunegis
Generating Networks with Arbitrary Properties
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