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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 - - PowerPoint PPT Presentation

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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

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SLIDE 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

  • f 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 5

Precepts of similarity measures

a vertex is similar to itself neighborhoods are similar two vertices are similar if their

http://www-personal.umich.edu/˜pholme/ – p.3/47

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SLIDE 6

Precepts of similarity measures

a vertex is similar to itself neighborhoods are similar two vertices are similar if their

http://www-personal.umich.edu/˜pholme/ – p.4/47

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SLIDE 7

Precepts of similarity measures

a vertex is similar to itself neighborhoods are similar two vertices are similar if their

structural equivalence / structural similarity

http://www-personal.umich.edu/˜pholme/ – p.5/47

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SLIDE 8

Precepts of similarity measures

a vertex is similar to itself neighborhoods are similar two vertices are similar if their

structural equivalence / structural similarity

i j

http://www-personal.umich.edu/˜pholme/ – p.6/47

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SLIDE 9

Structural similarity measures

i j

http://www-personal.umich.edu/˜pholme/ – p.7/47

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SLIDE 10

Structural similarity measures

i j Γi

http://www-personal.umich.edu/˜pholme/ – p.8/47

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SLIDE 11

Structural similarity measures

i j Γi Γj

http://www-personal.umich.edu/˜pholme/ – p.9/47

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SLIDE 12

Structural similarity measures

i j Γi Γj |Γi ∩ Γj| = 2

http://www-personal.umich.edu/˜pholme/ – p.10/47

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SLIDE 13

Structural similarity measures

i j Γi Γj |Γi ∩ Γj| = 2 |Γi ∩ Γj|/|Γi ∪ Γj| Jaccard (1901)

http://www-personal.umich.edu/˜pholme/ – p.11/47

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SLIDE 14

Structural similarity measures

i j Γi Γj |Γi ∩ Γj|/|Γi ∪ Γj| |Γi ∩ Γj| = 2 |Γi ∩ Γj|/ |Γi||Γj| Salton (1989)

http://www-personal.umich.edu/˜pholme/ – p.12/47

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SLIDE 15

Structural similarity measures

i j Γi Γj |Γi ∩ Γj|/|Γi ∪ Γj| |Γi ∩ Γj|/ |Γi||Γj| |Γi ∩ Γj| = 2 Ravasz et al. (2002) |Γi ∩ Γj|/ min(|Γi|, |Γj|)

http://www-personal.umich.edu/˜pholme/ – p.13/47

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SLIDE 16

Precepts of similarity measures

a vertex is similar to itself neighborhoods are similar two vertices are similar if their

http://www-personal.umich.edu/˜pholme/ – p.14/47

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SLIDE 17

Precepts of similarity measures

a vertex is similar to itself neighborhoods are similar two vertices are similar if their

regular equivalence / regular similarity

i j

http://www-personal.umich.edu/˜pholme/ – p.15/47

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SLIDE 18

Precepts of similarity measures

a vertex is similar to itself a vertex is similar to another if the its neighborhood is similar to the other vertex

http://www-personal.umich.edu/˜pholme/ – p.16/47

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SLIDE 19

Precepts of similarity measures

a vertex is similar to itself a vertex is similar to another if the its neighborhood is similar to the other vertex

  • ur similarity

j

  • r

i j i

http://www-personal.umich.edu/˜pholme/ – p.17/47

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SLIDE 20

Our similarity measure

j

  • r

i j i

A starting point...

Sij = φ

  • v

AivSvj + ψδij ⇒ (setting ψ = 1) S = (I − φA)−1 = I + φA + φ2A2 + · · ·

http://www-personal.umich.edu/˜pholme/ – p.18/47

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SLIDE 21

Our similarity measure

We replace φl by individual factors C ij

l representing

1 / expected # of paths of length l between i and j ... Sij =

∞

  • l=0

Cij

l (Al)ij

We obtain Cij

l ≈

       (2m/kikj)λ1−l

1

l 1 δij l = 0 Unfortunately Cij

l (Al)ij ∈ O(1), so...

http://www-personal.umich.edu/˜pholme/ – p.19/47

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SLIDE 22

Our similarity measure

...we scale down each term by a factor αl, 0 < α < 1: Sij = δij + 2m kikj

∞

  • l=1

αlλ−l+1

1

Al

ij

=

  • 1 − 2mλ1

kikj

  • δij + 2mλ1

kikj

  • I − α

λ1 A −1

ij

...and omit the first term only contributing to the diagonal ...

http://www-personal.umich.edu/˜pholme/ – p.20/47

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SLIDE 23

Our similarity measure

Sij = 2mλ1 kikj

  • I − α

λ1 A −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

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SLIDE 24

Evaluation: Model

Stratified network model:

▽http://www-personal.umich.edu/˜pholme/ – p.22/47

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SLIDE 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

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SLIDE 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) = p0 exp(−a∆t). (We choose p0 = 0.12 and a = 2.0.)

▽http://www-personal.umich.edu/˜pholme/ – p.22/47

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SLIDE 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) = p0 exp(−a∆t). (We choose p0 = 0.12 and a = 2.0.) The probability of a link drops by a factor of ea for every additional year separating their ages.

http://www-personal.umich.edu/˜pholme/ – p.22/47

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SLIDE 28

Evaluation: Model

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

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SLIDE 29

Evaluation: Model

age difference, σage(i, j) 100 10 1 0.1 1 2 3 4 5 6 7 8 9 Sij

http://www-personal.umich.edu/˜pholme/ – p.24/47

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SLIDE 30

Evaluation: Model

age difference, σage(i, j) 100 10 1 0.1 1 2 3 4 5 6 7 8 9 Sij

http://www-personal.umich.edu/˜pholme/ – p.25/47

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SLIDE 31

Evaluation: Roget’ s Thesaurus

Words Subsections Sections (Divisions) Classes

http://www-personal.umich.edu/˜pholme/ – p.26/47

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SLIDE 32

Evaluation: Roget’ s Thesaurus

Words Relating to the Sentient and Moral Powers Words Expressing Abstract Relations Words Relating to the Intellectual Faculties Words Relating to Space (Divisions) . . . Sections Subsections Words

http://www-personal.umich.edu/˜pholme/ – p.27/47

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SLIDE 33

Evaluation: Roget’ s Thesaurus

Words Relating to the Sentient and Moral Powers . . . Words Subsections . . . Words Expressing Abstract Relations Words Relating to the Intellectual Faculties Words Relating to Space Affections in General Personal Affections Sympathetic Affections Religious Affections

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

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SLIDE 34

Evaluation: Roget’ s Thesaurus

Words Relating to the Sentient and Moral Powers . . . . . . . . . Words Words Expressing Abstract Relations Words Relating to the Intellectual Faculties Words Relating to Space Religious Affections Affections in General Personal Affections Sympathetic Affections Religious doctrines Superhuman beings and regions Religious Sentiments

http://www-personal.umich.edu/˜pholme/ – p.29/47

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SLIDE 35

Evaluation: Roget’ s Thesaurus

Words Relating to the Sentient and Moral Powers . . . . . . . . . . . . Words Expressing Abstract Relations Words Relating to the Intellectual Faculties Words Relating to Space Religious Affections Affections in General Personal Affections Sympathetic Affections Superhuman beings and regions Religious doctrines Religious Sentiments Deity Hell Heaven Angel Satan

http://www-personal.umich.edu/˜pholme/ – p.30/47

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SLIDE 36

Evaluation: Roget’ s Thesaurus

word

  • ur measure

cosine similarity warning 32.0

  • men

0.516 alarm danger 25.8 threat 0.471

  • men

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

  • cean

0.316

http://www-personal.umich.edu/˜pholme/ – p.31/47

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SLIDE 37

Evaluation: AddHealth

Friendship network of school children.

▽http://www-personal.umich.edu/˜pholme/ – p.32/47

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SLIDE 38

Evaluation: AddHealth

Friendship network of school children. 90 118 students at 168 schools.

▽http://www-personal.umich.edu/˜pholme/ – p.32/47

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SLIDE 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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SLIDE 40

Evaluation: AddHealth α

0.9 0.92 0.94 0.96 0.98 1 0.1 0.15 0.2 0.25

  • corr. coeff., r(S, σage)

http://www-personal.umich.edu/˜pholme/ – p.33/47

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SLIDE 41

Similarity > > > prediction

? ? ? ? ?

Imagine a system with:

▽http://www-personal.umich.edu/˜pholme/ – p.34/47

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SLIDE 42

Similarity > > > prediction

? ? ? ? ?

Imagine a system with: Its network structure known.

▽http://www-personal.umich.edu/˜pholme/ – p.34/47

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SLIDE 43

Similarity > > > prediction

? ? ? ? ?

Imagine a system with: Its network structure known. The function of a fraction r ∈ (0, 1) of the vertices classified.

▽http://www-personal.umich.edu/˜pholme/ – p.34/47

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SLIDE 44

Similarity > > > prediction

? ? ? ? ?

Imagine a system with: Its network structure known. The function of a fraction r ∈ (0, 1) of the vertices classified. How can we assess the function of unclassified vertices? P . Holme & M. Huss, Role-similarity based functional prediction in networked systems: application to the yeast proteome, J. Roy.

  • Soc. Interface 2, (2005) pp. 327-333.

http://www-personal.umich.edu/˜pholme/ – p.34/47

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SLIDE 45

Prediction algorithm

? ? ? ? ?

Assign a functional similarity between classified vertices. (If vertices can have more than one function, we can use the Jaccard index |Fi ∩ Fj|/|Fi ∪ Fj|.)

▽http://www-personal.umich.edu/˜pholme/ – p.35/47

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SLIDE 46

Prediction algorithm

? ? ? ? ?

Assign a functional similarity between classified vertices. (If vertices can have more than one function, we can use the Jaccard index |Fi ∩ Fj|/|Fi ∪ Fj|.) Set Sij = δij if i or j is unclassificed.

▽http://www-personal.umich.edu/˜pholme/ – p.35/47

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SLIDE 47

Prediction algorithm

? ? ? ? ?

Assign a functional similarity between classified vertices. (If vertices can have more than one function, we can use the Jaccard index |Fi ∩ Fj|/|Fi ∪ Fj|.) Set Sij = δij if i or j is unclassificed. Update Sij (i or j is unclassificed) iteratively.

▽http://www-personal.umich.edu/˜pholme/ – p.35/47

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SLIDE 48

Prediction algorithm

? ? ? ? ?

Assign a functional similarity between classified vertices. (If vertices can have more than one function, we can use the Jaccard index |Fi ∩ Fj|/|Fi ∪ Fj|.) Set Sij = δij if i or j is unclassificed. Update Sij (i or j is unclassificed) iteratively. For an unclassified vertex i let the functions of the most similar classified vertex be your guess.

http://www-personal.umich.edu/˜pholme/ – p.35/47

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SLIDE 49

Testing prediction: Model

supply A-distributor delivery B-distributor A-edge B-edge C-edge assembler * * * * * * *

http://www-personal.umich.edu/˜pholme/ – p.36/47

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SLIDE 50

Testing prediction: Model

supply A-distributor delivery B-distributor A-edge B-edge C-edge assembler (

  • r

) and add supply vertex * * * * * * *

http://www-personal.umich.edu/˜pholme/ – p.37/47

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SLIDE 51

Testing prediction: Model

supply A-distributor delivery B-distributor A-edge B-edge C-edge assembler add assembler vertex ( ) and (

  • r
  • r

) * * * * * * *

http://www-personal.umich.edu/˜pholme/ – p.38/47

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SLIDE 52

Testing prediction: Model

supply A-distributor delivery B-distributor A-edge B-edge C-edge assembler add delivery vertex )

  • r

and ( * * * * * * *

http://www-personal.umich.edu/˜pholme/ – p.39/47

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SLIDE 53

Testing prediction: Model

supply A-distributor delivery B-distributor A-edge B-edge C-edge assembler add A-distributor vertex ( ) and (

  • r
  • r

) * * * * * * *

http://www-personal.umich.edu/˜pholme/ – p.40/47

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SLIDE 54

Testing prediction: Model

supply A-distributor delivery B-distributor A-edge B-edge C-edge assembler add B-distributor vertex ( ) and (

  • r
  • r

) * * * * * * *

http://www-personal.umich.edu/˜pholme/ – p.41/47

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SLIDE 55

Testing prediction: Model

supply A-distributor delivery B-distributor A-edge B-edge C-edge assembler rewire the edges with probability r * * * * * * *

http://www-personal.umich.edu/˜pholme/ – p.42/47

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SLIDE 56

Testing prediction: Model

supply A-distributor delivery B-distributor A-edge B-edge C-edge assembler * * * * * * *

http://www-personal.umich.edu/˜pholme/ – p.43/47

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SLIDE 57

Testing prediction: Model

(a) (b)

0.25 0.5 0.75 1 0.25 0.75 0.5 1 s s 0.25 0.5 0.75 1 0.25 0.5 0.75 1 r r N = 500 N = 5000 N = 50

http://www-personal.umich.edu/˜pholme/ – p.44/47

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SLIDE 58

Testing prediction: Yeast

YNL041c YBR068c YOR133w YDR385w YJL191w YCR031c

protein synthesis requirement (structural or catalytic) transport routes cellular transport, transport facilitation and protein with binding function or cofactor

http://www-personal.umich.edu/˜pholme/ – p.45/47

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SLIDE 59

Testing prediction: Yeast

s+ = nc f∗

  • (precision) and s− =

nc f

  • (recall)

where nc is the number of correctly predicted functions, f is the real number of functions and f∗ is the number of predicted functions.

▽http://www-personal.umich.edu/˜pholme/ – p.46/47

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SLIDE 60

Testing prediction: Yeast

s+ = nc f∗

  • (precision) and s− =

nc f

  • (recall)

where nc is the number of correctly predicted functions, f is the real number of functions and f∗ is the number of predicted functions. level 1 level 2 NCM

  • ur I
  • ur II

NCM

  • ur I
  • ur II

s+ 0.269 0.392 0.337 0.199 0.238 0.220 s− 0.354 0.291 0.346 0.252 0.199 0.231

http://www-personal.umich.edu/˜pholme/ – p.46/47

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SLIDE 61

Conclusions

Vertex similarity measures serve to show which vertex-pairs that have the same function in the network.

▽http://www-personal.umich.edu/˜pholme/ – p.47/47

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SLIDE 62

Conclusions

Vertex similarity measures serve to show which vertex-pairs that have the same function in the network. If it is formulated as a linear algebra problem it turns into a measure based on path counts.

▽http://www-personal.umich.edu/˜pholme/ – p.47/47

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SLIDE 63

Conclusions

Vertex similarity measures serve to show which vertex-pairs that have the same function in the network. If it is formulated as a linear algebra problem it turns into a measure based on path counts. Sij = 2mλ1 kikj

  • I − α

λ1 A −1

ij

▽http://www-personal.umich.edu/˜pholme/ – p.47/47

slide-64
SLIDE 64

Conclusions

Vertex similarity measures serve to show which vertex-pairs that have the same function in the network. If it is formulated as a linear algebra problem it turns into a measure based on path counts. Sij = 2mλ1 kikj

  • I − α

λ1 A −1

ij

Similarity measures can be turned into classification and prediction algorithms.

▽http://www-personal.umich.edu/˜pholme/ – p.47/47

slide-65
SLIDE 65

Conclusions

Vertex similarity measures serve to show which vertex-pairs that have the same function in the network. If it is formulated as a linear algebra problem it turns into a measure based on path counts. Sij = 2mλ1 kikj

  • I − α

λ1 A −1

ij

Similarity measures can be turned into classification and prediction algorithms. Good in general, but for specific systems there can be smarter prediction / classification algorithms.

http://www-personal.umich.edu/˜pholme/ – p.47/47