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Preserving PersonalizedPagerank in Subgraphs(Andrea Vattani, Deepayan Chakrabarti, Maxim                  Gurevich)       ...
a       a    bc       c
“Due to spaceconstraints, complete proofs of our claimswill appear in the fullversion of the paper.”
p = αr + (1 − α)A D                t     −1                           ppαrAD
rr[i]
∞                                          [t]pi (j) =                α(1 −          α)pi (j)                t=0          ...
G = (V, E)   S⊂V         p         G     pG[S]         ˜         p   G
˜min d(p , p       G   G[S]                  )
|S| = o(n/ log n)           1/2 − o(1)   Ω(|S|)S ⊂S                                              ∗                        ...
wG (i, j) = 1j∈V
G = (V, wG )        S⊂VH = (S ∪ SIN K, wH )
pi (j) G         = pi (j)            H         ∀i, j ∈ S
aa        sampling a,c    b                      SINK        Remove bc                       c
∞wH (x, y)+ = (1 − α)wG (x, z)wG (z, y)         [(1 − α)wG (z, z)]                                                        ...
wH (x, SIN K)+ = wG (x, z) −                    wH (x, y)                               y=z;wG (z,y)>0
P   H             pi                   H                       =   pi                            G    WH            1     ...
wH (u, v) < τwH (u, SIN K)+ = wH (u, v), wH (u, v) = 0
||P (j) − P (j)||1     H     R                         1−α                    ≤ 2τ      |S| + 1             αP (j) HP (j) R
v=            pj (i),               G                        i∈S     j∈V Sr(j) = r(j ), j, j ∈ V  S               pr (i), ...
SINK       Page       Rank
1    1wH   (SOU RCE, l) =                        pj                                            G                          ...
pr (i) = pr  G           H                  (i), ∀i ∈ S                   r(k),                k∈S r (k) =   ρ|V  S|,   ...
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
Preserving Personalized Pagerank in Subgraphs(ICML 2011)
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Preserving Personalized Pagerank in Subgraphs(ICML 2011)

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http://www.icml-2011.org/papers/434_icmlpaper.pdf

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Preserving Personalized Pagerank in Subgraphs(ICML 2011)

  1. 1. Preserving PersonalizedPagerank in Subgraphs(Andrea Vattani, Deepayan Chakrabarti, Maxim Gurevich) 1
  2. 2. a a bc c
  3. 3. “Due to spaceconstraints, complete proofs of our claimswill appear in the fullversion of the paper.”
  4. 4. p = αr + (1 − α)A D t −1 ppαrAD
  5. 5. rr[i]
  6. 6. ∞ [t]pi (j) = α(1 − α)pi (j) t=0 t [t] 1 pi (j) = d+ (kl ) k1 =i,k2 ,··· ,kt+1 =j l=1
  7. 7. G = (V, E) S⊂V p G pG[S] ˜ p G
  8. 8. ˜min d(p , p G G[S] )
  9. 9. |S| = o(n/ log n) 1/2 − o(1) Ω(|S|)S ⊂S ∗ u S ∗S u ∗ u
  10. 10. wG (i, j) = 1j∈V
  11. 11. G = (V, wG ) S⊂VH = (S ∪ SIN K, wH )
  12. 12. pi (j) G = pi (j) H ∀i, j ∈ S
  13. 13. aa sampling a,c b SINK Remove bc c
  14. 14. ∞wH (x, y)+ = (1 − α)wG (x, z)wG (z, y) [(1 − α)wG (z, z)] t t=0
  15. 15. wH (x, SIN K)+ = wG (x, z) − wH (x, y) y=z;wG (z,y)>0
  16. 16. P H pi H = pi G WH 1 t WH = (I − α(P ) ) H −1 1−α O(E + |S| ) 2
  17. 17. wH (u, v) < τwH (u, SIN K)+ = wH (u, v), wH (u, v) = 0
  18. 18. ||P (j) − P (j)||1 H R 1−α ≤ 2τ |S| + 1 αP (j) HP (j) R
  19. 19. v= pj (i), G i∈S j∈V Sr(j) = r(j ), j, j ∈ V S pr (i), ∀i ∈ S G
  20. 20. SINK Page Rank
  21. 21. 1 1wH (SOU RCE, l) = pj G (k)wG (k.l), l ∈ S |V S| α j,k∈V S wH (SOU RCE, SIN K) = 1 − wH (SOU RCE, i) i∈S
  22. 22. pr (i) = pr G H (i), ∀i ∈ S   r(k), k∈S r (k) = ρ|V S|, k = SOU RCE  0, k = SIN K

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