ネットワーク科学 中心性とGoogleのPageRank

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1. Google PageRank 2 2019 ( ) Google PageRank 1 / 23
2. 1. , , ( ) Google PageRank 2 / 23
3. , , , 6 , , 2007 ( ) Google PageRank 3 / 23
4. 2. (Degree Centrality): i ki ki /(N − 1) . N − 1 , i N − 1 . , . (Information Centrality): i j ( ) . ( ) Google PageRank 4 / 23
5. (Closeness Centrality): i j : +1 Hij , j Hij N − 1 −1 = N − 1 j Hij . . N − 1 . , . (Flow Centrality): i . ( ) Google PageRank 5 / 23
6. (betweenness centrality): v s,t δst (v), σst (s, t) , σst (v) v δst (v) def = σst (v) σst L.C. Freeman, Sociometry 40, 1977 http://www.geocities.jp/woodone3831/kanntou/c-4-11-sekisyo-HAKONE.html ( ) Google PageRank 6 / 23
7. BC http://astamuse.com/ja/published/JP/No/2010141442 ( ) Google PageRank 7 / 23
8. Brandes BC s u σsu , δs,•(w) w ∼ w′ t , w ∼ w′ t δs,•(v) = {w|v∈Ps (w)} σsv σsw (1 + δs,•(w)), Ps (w) def = {v ∈ V |(v, w) ∈ E, d(s, w) = d(s, v) + 1}, w ∈ ∂v s v w w’ Ps(w) t t’ : : : : σ sw σ sv : U.Brandes, Journal of Math. Sociology 25, 2001 ( ) Google PageRank 8 / 23
9. , δst (v) = u∈Preds , t (v) δst (u) × R(s, u, v, t), Preds,t (v) s-t v 1 {u}, R(s, u, v, t) s t (u, v) , . s-t T(s, t) , v ∈ V . δ•,•(v) = s,t∈V δst (v) × T(s, t). S.Dolev et al., Journal of the ACM 57, 2010 ( ) Google PageRank 9 / 23
10. R(s , u , v , t) T(s , t) s u-v t u s t v Pred T(s, t) , ( ) Google PageRank 10 / 23
11. (Bonacichi Centrality): . x = αA1 + βAx, x , 1 1−z = 1 + z + z2 + z3 + . . . x = (I − βA)−1(αA1) = α ∞ k=0 βkAk+11, = α(A + βA2 + β2A3 + . . .)1. ⇒ i j , 1 , 2 , 3 , . . . β . α = 0, β = 1/λ , [aij ] . ( ) Google PageRank 11 / 23
12. , , i aij 1/ki , , PageRank . , Katz Hubbel . ⇒ , . ( ) Google PageRank 12 / 23
13. 3. PageRank WWW , ( ) Google PageRank 13 / 23
14. r ← Pr, r P rv ← d v′∈Nv rv′ kv′ + 1 − d N , d ≈ 0.85, 1 − d ⇒ ( ) Google PageRank 14 / 23
15. Google i ri ⇔ πi , i πi = 1 , r ← d × Pr + (1 − d)/N              π1 . . . πi . . . πj . . . πn              T                1−d N . . . . . . . . . . . . . . . 1−d N . . . . . . . . . . . . . . . . . . . . . . . . . . . 1−d N . . . d ki + 1−d N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1−d N . . . 1−d N . . . . . . . . . . . . . . . . . . . . . . . . . . . 1−d N . . . . . . . . . . . . . . . 1−d N                =              π1 . . . πi . . . πj . . . πn              T A.N.Langville and C.D.Meyer, Google’s Page Rank and Beyond, Princeton Univ. Press, 2006 ( ) Google PageRank 15 / 23
16. WWW ( ) Google PageRank 16 / 23
17. πi = d j Aij πj kout j + (1 − d)β j πj . π = dAD−1π + (1 − d)β1 → π = (1 − d)β(I − dAD−1)−11. Katz xi = α j Aij xi + β′. x = αAx + β′1 → x = β′(I − αA)−11 = β′ ∞ k=0 (αA)k1. A : x = 1 λ1 Ax = PageRank ! xi (t) = j Aij kj xj (t − 1), xi = ki j kj . x = AD−1x → (I − AD−1)x = (D − A)D−1x = 0, D−1x = 1. M.E.J.Newman, Networks -An Introduction-, OXFORD Univ. Press, 2010. ( ) Google PageRank 17 / 23
18. , , , , Gram-Schmidt Householder R Q QR Rayleigh-Ritz Krylov x, Ax, A2x, A3x, . . . Arnoldi , Lanczos , Jacobi , , , , [ 2], 1994, ( ) Google PageRank 18 / 23
19. Google L.E.Page . , . WWW , . , WWW , PC . ( ) Google PageRank 19 / 23
20. Indexing Hadoop https://enterprisezine.jp/dbonline/detail/4254 ( ) Google PageRank 20 / 23
21. MapReduce Map Key Value , Reduce Key http://www.dineshonjava.com/2014/11/mapreduce-flow-chart-sample-example.html ⇒ ( ) Google PageRank 21 / 23
22. 4. HITS J.M.Kleinberg HITS (Hyperlink-induced topic search) Authority: x ← AT y ← AT Ax, x AT A Hub: y ← Ax ← AAT y. y AAT . A , AT . J.M.Kleinberg, Journal of the ACM 46, 1999 ( ) Google PageRank 22 / 23
23. 5. i Li , . i Li N . ¯ L def = i Li /N . i ¯ L , ∆Li = Li − ¯ L . ∆Li < 0 . . , ( ) Google PageRank 23 / 23