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V E R T E X S I M I L A R I T Y A N D I T S A P P L I C A T I O N T O F U N C T I O N A L P R E D I C T I O N Petter Holme University of Michigan, Ann Arbor, U.S.A. with Elizabeth Leicht and Mark Newman (University of Michigan) and Mikael

  1. V E R T E X S I M I L A R I T Y A N D I T S A P P L I C A T I O N T O F U N C T I O N A L P R E D I C T I O N Petter Holme University of Michigan, Ann Arbor, U.S.A. with Elizabeth Leicht and Mark Newman (University of Michigan) and Mikael Huss (Royal Institute of Technology, Stockholm, Sweden) http://www-personal.umich.edu/ ∼ pholme/ http://www-personal.umich.edu/˜pholme/ – p.1/47

  2. Vertex equivalence / similarity In complex networks the nodes have different functions. These functions are reflected in their position in the networks. ▽ http://www-personal.umich.edu/˜pholme/ – p.2/47

  3. Vertex equivalence / similarity In complex networks the nodes have different functions. These functions are reflected in their position in the networks. Can we, from the network structure, guess if two vertices have similar function? ▽ http://www-personal.umich.edu/˜pholme/ – p.2/47

  4. Vertex equivalence / similarity In complex networks the nodes have different functions. These functions are reflected in their position in the networks. Can we, from the network structure, guess if two vertices have similar function? How can we use this information to classify the vertices / predict their functions? http://www-personal.umich.edu/˜pholme/ – p.2/47

  5. Precepts of similarity measures two vertices are similar if their a vertex is similar to itself neighborhoods are similar http://www-personal.umich.edu/˜pholme/ – p.3/47

  6. Precepts of similarity measures two vertices are similar if their a vertex is similar to itself neighborhoods are similar http://www-personal.umich.edu/˜pholme/ – p.4/47

  7. Precepts of similarity measures two vertices are similar if their a vertex is similar to itself neighborhoods are similar structural equivalence / structural similarity http://www-personal.umich.edu/˜pholme/ – p.5/47

  8. Precepts of similarity measures two vertices are similar if their a vertex is similar to itself neighborhoods are similar structural equivalence / structural similarity i j http://www-personal.umich.edu/˜pholme/ – p.6/47

  9. Structural similarity measures i j http://www-personal.umich.edu/˜pholme/ – p.7/47

  10. Structural similarity measures Γ i i j http://www-personal.umich.edu/˜pholme/ – p.8/47

  11. Structural similarity measures Γ i i Γ j j http://www-personal.umich.edu/˜pholme/ – p.9/47

  12. Structural similarity measures Γ i i | Γ i ∩ Γ j | = 2 Γ j j http://www-personal.umich.edu/˜pholme/ – p.10/47

  13. Structural similarity measures | Γ i ∩ Γ j | / | Γ i ∪ Γ j | Γ i i | Γ i ∩ Γ j | = 2 Γ j j Jaccard (1901) http://www-personal.umich.edu/˜pholme/ – p.11/47

  14. Structural similarity measures | Γ i ∩ Γ j | / | Γ i ∪ Γ j | Γ i | Γ i ∩ Γ j | / � | Γ i || Γ j | i | Γ i ∩ Γ j | = 2 Γ j j Salton (1989) http://www-personal.umich.edu/˜pholme/ – p.12/47

  15. Structural similarity measures | Γ i ∩ Γ j | / | Γ i ∪ Γ j | Γ i | Γ i ∩ Γ j | / � | Γ i || Γ j | i | Γ i ∩ Γ j | = 2 Γ j j | Γ i ∩ Γ j | / min( | Γ i | , | Γ j | ) Ravasz et al. (2002) http://www-personal.umich.edu/˜pholme/ – p.13/47

  16. Precepts of similarity measures two vertices are similar if their a vertex is similar to itself neighborhoods are similar http://www-personal.umich.edu/˜pholme/ – p.14/47

  17. Precepts of similarity measures two vertices are similar if their a vertex is similar to itself neighborhoods are similar regular equivalence / regular similarity i j http://www-personal.umich.edu/˜pholme/ – p.15/47

  18. Precepts of similarity measures a vertex is similar to another a vertex is similar to itself if the its neighborhood is similar to the other vertex http://www-personal.umich.edu/˜pholme/ – p.16/47

  19. Precepts of similarity measures a vertex is similar to another a vertex is similar to itself if the its neighborhood is similar to the other vertex our similarity or i j i j http://www-personal.umich.edu/˜pholme/ – p.17/47

  20. Our similarity measure or i j i j A starting point... � A iv S vj + ψδ ij ⇒ (setting ψ = 1) S ij = φ v S = ( I − φ A ) − 1 = I + φ A + φ 2 A 2 + · · · http://www-personal.umich.edu/˜pholme/ – p.18/47

  21. Our similarity measure We replace φ l by individual factors C ij l representing 1 / expected # of paths of length l between i and j ... ∞ � C ij l ( A l ) ij S ij = l= 0 We obtain  (2 m/k i k j ) λ 1 − l l � 1  C ij  1 l ≈  l = 0  δ ij   Unfortunately C ij l ( A l ) ij ∈ O (1), so... http://www-personal.umich.edu/˜pholme/ – p.19/47

  22. Our similarity measure ...we scale down each term by a factor α l , 0 < α < 1: ∞ δ ij + 2 m � α l λ − l+ 1 � A l � S ij = ij 1 k i k j l= 1 � − 1 � 1 − 2 m λ 1 δ ij + 2 m λ 1 I − α � � �� = A k i k j k i k j λ 1 ij ...and omit the first term only contributing to the diagonal ... http://www-personal.umich.edu/˜pholme/ – p.20/47

  23. Our similarity measure � − 1 � S ij = 2 m λ 1 I − α �� A k i k j λ 1 ij E. A. Leicht, P . Holme & M. E. J. Newman, Vertex similarity in networks, e-print physics/0510143 http://www-personal.umich.edu/˜pholme/ – p.21/47

  24. Evaluation: Model Stratified network model: ▽ http://www-personal.umich.edu/˜pholme/ – p.22/47

  25. Evaluation: Model Stratified network model: Assign an “age” t = 1 , · · · , 10 to N vertices with uniform randomness. ▽ http://www-personal.umich.edu/˜pholme/ – p.22/47

  26. Evaluation: Model Stratified network model: Assign an “age” t = 1 , · · · , 10 to N vertices with uniform randomness. Let there be a link between i and j with probability P ( ∆ t ) = p 0 exp( − a ∆ t ). (We choose p 0 = 0 . 12 and a = 2 . 0.) ▽ http://www-personal.umich.edu/˜pholme/ – p.22/47

  27. Evaluation: Model Stratified network model: Assign an “age” t = 1 , · · · , 10 to N vertices with uniform randomness. Let there be a link between i and j with probability P ( ∆ t ) = p 0 exp( − a ∆ t ). (We choose p 0 = 0 . 12 and a = 2 . 0.) The probability of a link drops by a factor of e a for every additional year separating their ages. http://www-personal.umich.edu/˜pholme/ – p.22/47

  28. Evaluation: Model http://www-personal.umich.edu/˜pholme/ – p.23/47

  29. Evaluation: Model 100 10 � S ij � 1 0.1 0 0 1 2 3 4 5 6 7 8 9 age difference, σ age ( i, j ) http://www-personal.umich.edu/˜pholme/ – p.24/47

  30. Evaluation: Model 100 10 � S ij � 1 0.1 0 0 1 2 3 4 5 6 7 8 9 age difference, σ age ( i, j ) http://www-personal.umich.edu/˜pholme/ – p.25/47

  31. Evaluation: Roget’ s Thesaurus Classes (Divisions) Sections Subsections Words http://www-personal.umich.edu/˜pholme/ – p.26/47

  32. Evaluation: Roget’ s Thesaurus Words Expressing Abstract Relations Words Relating to the Sentient and Moral Powers (Divisions) Words Relating to the Intellectual Faculties Sections Words Relating to Space . . . Subsections Words http://www-personal.umich.edu/˜pholme/ – p.27/47

  33. Evaluation: Roget’ s Thesaurus Words Expressing Abstract Relations Words Relating to the Sentient and Moral Powers Words Relating to the Intellectual Faculties Affections in General Words Relating to Space . Personal Affections . . Sympathetic Affections Religious Affections Subsections . . . Words http://www-personal.umich.edu/˜pholme/ – p.28/47

  34. Evaluation: Roget’ s Thesaurus Words Expressing Abstract Relations Words Relating to the Sentient and Moral Powers Words Relating to the Intellectual Faculties Affections in General Words Relating to Space . Personal Affections . . Sympathetic Affections Religious Affections . Religious doctrines . . Superhuman beings and regions Words Religious Sentiments . . . http://www-personal.umich.edu/˜pholme/ – p.29/47

  35. Evaluation: Roget’ s Thesaurus Words Expressing Abstract Relations Words Relating to the Sentient and Moral Powers Words Relating to the Intellectual Faculties Affections in General Words Relating to Space . Personal Affections . . Sympathetic Affections Religious Affections . Religious doctrines . . Superhuman beings and regions Deity Religious Sentiments Angel . . Satan . Heaven Hell . . . http://www-personal.umich.edu/˜pholme/ – p.30/47

  36. Evaluation: Roget’ s Thesaurus word our measure cosine similarity warning 32.0 omen 0.516 alarm danger 25.8 threat 0.471 omen 18.8 prediction 0.348 heaven 63.4 pleasure 0.408 hell pain 28.9 inferiority 0.222 discontent 7.0 weariness 0.267 plunge 33.6 dryness 0.447 water air 25.3 wind 0.316 moisture 25.3 ocean 0.316 http://www-personal.umich.edu/˜pholme/ – p.31/47

  37. Evaluation: AddHealth Friendship network of school children. ▽ http://www-personal.umich.edu/˜pholme/ – p.32/47

  38. Evaluation: AddHealth Friendship network of school children. 90 118 students at 168 schools. ▽ http://www-personal.umich.edu/˜pholme/ – p.32/47

  39. Evaluation: AddHealth Friendship network of school children. 90 118 students at 168 schools. Information about grade, race and gender http://www-personal.umich.edu/˜pholme/ – p.32/47

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