Generating networks
with arbitrary properties
Social Interaction

“You’re my friend”

Jérôme Kunegis

Generating Networks with Arbitrary Properties

2
Many Social Interactions
”
y friend
m
You’re
“

Jérôme Kunegis

’re m
y frie
nd”

”
nd
rie

y friend”

“You

y

’re
my
fri...
Abstract: It's a Network

Jérôme Kunegis

Generating Networks with Arbitrary Properties

4
Problem: Generate Realistic Graphs

Why generate graphs?
To visualize an existing network: generate a
smaller graph with s...
Basic Idea for Generating Networks: Random Graphs

Each edge has
probability p of existing

Paul Erdős
Jérôme Kunegis

Gen...
Random Graphs Are Not Realistic

Real network

Random graph

Jérôme Kunegis

Generating Networks with Arbitrary Properties...
Real Networks Have Special Properties

Many triangles
(“clustering”)

Many 2-stars
(“preferential attachment”)

Short path...
Solution: Exponential Random Graph Models
Example with three statistics:
P(G) = exp( a1 m + a2 t + a3 s + b )
m, t, s: Pro...
Problems of Exponential Random Graph Models

P(G) = exp( a1 x1 + a2 x2 + … + ak xk + b )

Many exponential random graph mo...
Explanation of Degeneracy

Consider a variable x between 0 and 1
with expected value 0.3.
An exponential random model for ...
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...
Real Networks Have a Distribution of Values Anyway

P(G) = Pnorm (x1, x2, …)

Data from konect.uni-koblenz.de

Jérôme Kune...
Monte Carlo Markov Chain Methods
+ Current graphs
× Possible next steps
Wanted distribution
×
Random graphs
+
×

x2

×

×
...
Solution: Integral of Measure of Voronoi Cells

Wanted distribution
×
×

Random graphs
×

×

x2

×

×

×

×

×
×

×

×
×

...
How To Compute The Integral over Voronoi Cells
Answer: We don't have to.
Sampling strategy:
Sample point in statistic-spac...
Result: Close, But Not Exact

Jérôme Kunegis

Generating Networks with Arbitrary Properties

17
Convergence Speed (σ = 3)

Edge count

2-star count

Triangle count

Jérôme Kunegis

Generating Networks with Arbitrary Pr...
Example: Generate Network with Same Properties as Zachary's Karate Club

Jérôme Kunegis

Generating Networks with Arbitrar...
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Generating Networks with Arbitrary Properties

  1. 1. Generating networks with arbitrary properties
  2. 2. Social Interaction “You’re my friend” Jérôme Kunegis Generating Networks with Arbitrary Properties 2
  3. 3. Many Social Interactions ” y friend m You’re “ Jérôme Kunegis ’re m y frie nd” ” nd rie y friend” “You y ’re my frie nd ” f my ’re ou “Y “You’re m ’re ou “Y m d” n ie fr “Yo u y friend m “You’re Generating Networks with Arbitrary Properties ” 3
  4. 4. Abstract: It's a Network Jérôme Kunegis Generating Networks with Arbitrary Properties 4
  5. 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 5
  6. 6. Basic Idea for Generating Networks: Random Graphs Each edge has probability p of existing Paul Erdős Jérôme Kunegis Generating Networks with Arbitrary Properties 6
  7. 7. Random Graphs Are Not Realistic Real network Random graph Jérôme Kunegis Generating Networks with Arbitrary Properties 7
  8. 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 8
  9. 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 9
  10. 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 10
  11. 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 1 x 11
  12. 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 12
  13. 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 13
  14. 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 14
  15. 15. Solution: Integral of Measure of Voronoi Cells Wanted distribution × × Random graphs × × x2 × × × × × × × × × × × × x1 Jérôme Kunegis Generating Networks with Arbitrary Properties 15
  16. 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. Jérôme Kunegis Generating Networks with Arbitrary Properties 16
  17. 17. Result: Close, But Not Exact Jérôme Kunegis Generating Networks with Arbitrary Properties 17
  18. 18. Convergence Speed (σ = 3) Edge count 2-star count Triangle count Jérôme Kunegis Generating Networks with Arbitrary Properties 18
  19. 19. Example: Generate Network with Same Properties as Zachary's Karate Club Jérôme Kunegis Generating Networks with Arbitrary Properties 19

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