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Generating Networks with Arbitrary Properties

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Jérôme KUNEGIS
Jérôme KUNEGIS
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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 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
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 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
Basic Idea for Generating Networks: Random Graphs Each edge has probability p of existing Paul Erdős Jérôme Kunegis Generating Networks with Arbitrary Properties 6
Random Graphs Are Not Realistic Real network Random graph Jérôme Kunegis Generating Networks with Arbitrary Properties 7
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
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
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
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
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
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
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
Solution: Integral of Measure of Voronoi Cells Wanted distribution × × Random graphs × × x2 × × × × × × × × × × × × x1 Jérôme Kunegis Generating Networks with Arbitrary Properties 15
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
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 Properties 18
Example: Generate Network with Same Properties as Zachary's Karate Club Jérôme Kunegis Generating Networks with Arbitrary Properties 19

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Generating Networks with Arbitrary Properties

  • 2. Social Interaction “You’re my friend” Jérôme Kunegis Generating Networks with Arbitrary Properties 2
  • 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. Abstract: It's a Network Jérôme Kunegis Generating Networks with Arbitrary Properties 4
  • 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. 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. Random Graphs Are Not Realistic Real network Random graph Jérôme Kunegis Generating Networks with Arbitrary Properties 7
  • 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. 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. 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. 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. 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. 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. 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. Solution: Integral of Measure of Voronoi Cells Wanted distribution × × Random graphs × × x2 × × × × × × × × × × × × x1 Jérôme Kunegis Generating Networks with Arbitrary Properties 15
  • 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. Result: Close, But Not Exact Jérôme Kunegis Generating Networks with Arbitrary Properties 17
  • 18. Convergence Speed (σ = 3) Edge count 2-star count Triangle count Jérôme Kunegis Generating Networks with Arbitrary Properties 18
  • 19. Example: Generate Network with Same Properties as Zachary's Karate Club Jérôme Kunegis Generating Networks with Arbitrary Properties 19