ネットワーク科学最前線2017 -インフルエンサーと機械学習からの接近-

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1. 2017 - - 29 8 ( ) 2017 1 / 45
2. Collective Influencer (CI) Attacks Belief Propagation (BP) Attacks but + + ( ) 2017 2 / 45
3. 1. 5 1 : 1/3 2 : , 3 4 : 5 : http://blog.btrax.com/jp/2016/06/13/influencer-marketing/ Justin Bieber PPAP ( ) 2017 3 / 45
4. ≈ in : , , . , . market maven : . · , . , · , , 2005 ( ) 2017 4 / 45
5. Klout Twitter Facebook · NPS Net Promoter Score/System ( ) 2017 5 / 45
6. KCORE ANALYTICS http://ww.kcore-analytics.com ( ) 2017 6 / 45
7. CI PageRank ? 2 WWW , AI , / AltaVista, Infoseek, NTT , , . . . , , PageRank Comm. of the ACM, Special Issue on Info. Filtering, 35(12), Dec. 1992 ⇒ ? Infering personal economic status from social network location, nature comm. 5/16 2017 ( ) 2017 7 / 45
8. 2. ( ) 2017 8 / 45
9. 2. (GA) ( ) 2017 8 / 45
10. 2. (GA) , (BP) ( ) 2017 8 / 45
11. 2. (GA) , (BP) ( ) 2017 8 / 45
12. 2. (GA) , (BP) / , ( ) 2017 8 / 45
13. 2. (GA) , (BP) / , Google PageRank ( ) 2017 8 / 45
14. 1 Nonbacktracking Matrix B i → j i ← j . . . k → i k ← i . . .       . . . . . . . . . . . . . . . ni 0 . . . . . . 0 ni . . . . . . . . . . . . . . .       , Zeta ζG (z) = det(I − zB)−1 = exp ∞ m=1 1 m zmTrBm . K.Hashimoto, Advanced Studies in Pure Math. 15, 1989 NB random walks mix faster N.Alon et al., Comm. Contemp. Math. 9(4), 2007 ( ) 2017 9 / 45
15. 1’ TrBm = λm 1 + λm 2 + . . . λm N , log(1 − x) = log(1 + (−x)) = − ∞ m=1 xm m , ( ) = exp ∞ m=1 1 m (λm 1 zm + . . . + λm N zm) = exp (− log(1 − λ1z) − . . . − log(1 − λNz)) = exp N l=1 log 1 1−λl z = exp log ΠN l=1 1 1−λl z = 1 (1−λ1z)...(1−λN z) = det(I − zB)−1. 1, 3 , , 1996 ( ) 2017 10 / 45
16. 2 F(a) = a xt+1 = F(xt) ≈ F(a) + ∂F(x) ∂x (xt − a) + . . . , xt+1 − a xt−1 − a = xt+1 − a xt − a × xt − a xt−1 − a ≈ ∂F(x) ∂x 2 . x0 − a = u1 + u2 + . . . un ui , |λ1 | > |λ2 | > . . . |λn | t → ∞ F(F(F(. . . F(x0) . . .))) ∂F(x) ∂x t (x0 − a) = λt 1 u1 + λt 2 u2 + . . . + λt n un = λt 1 u1 + λt 1 n i=2 λi λ1 t ui → λt 1 u1 . |λ1 | > 1 , |λ1 | < 1 ( ) 2017 11 / 45
17. 2-1 GC i → j νi→j = ni 1 − Πk∈∂i\j (1 − νk→i ) . q , Jacobian M λ(n; q) . ∂νi→j ∂νk→l νi→j =0 = ni Bk→l,i→j . wl (n) def = Ml w0 Power Method: λ(n; q) = lim l→∞ |wl (n)| |w0 | 1/l , F.Morone, and H.A.Makse, Nature 524, 65-68, 2015 ( ) 2017 12 / 45
18. min λ(n; q) 2l- min λ(n; q) 2l- Greedy , CIl (i) i |w2 (n)|2 = i,j,k=i,l=j Aij AjkAkl (ki − 1)(kl − 1)ni nj nknl , |wl (n)|2 ≈ N i=1 (ki − 1) j∈∂Ball(i,2l−1) Πk∈P2l−1(i,j) nk (kj − 1), CIl (i) def = (ki − 1) j∈∂Ball(i,l) (kj − 1). P2l−1 (i, j) 2l − 1 i j , ∂Ball(i, l) i l F.Morone, and H.A.Makse, Nature 524, 65-68, 2015 ( ) 2017 13 / 45
19. CI: Collective Influence F.Morone, and H.A.Makse, Nature 524, 65-68, 2015 ( ) 2017 14 / 45
20. CI F.Morone, and H.A.Makse, Nature 524, 65-68, 2015 ( ) 2017 15 / 45
21. Twitter F.Morone, and H.A.Makse, Nature 524, 65-68, 2015 ( ) 2017 16 / 45
22. 2-2 LTM ni = 1, ki − 1 mi active i → j νi→j = ni + (1 − ni ) 1 − Π Ph ∈Pmi ∂i\j (1 − Πp∈Ph νp→i ) , νi = ni + (1 − ni ) 1 − Π Ph ∈Pmi ∂i\j (1 − Πp∈Ph νp→i ) . νt+1 = n + Ftνt, Ft = ∂G/∂ν|νt , Ft k→l,i→j = (1 − ni )It k→l,i→j CI-TM1 (i) = ki + j∈∂i (1 − nj ) k∈∂j\i I0 ijjk , . . . Ik→l,i→j = 1 if l = i, k = j, p∈∂i\(k,j) νp→i = mi − 1. S.Pei et al., Scientific Reports 7, 45240, 3/28 2017 ( ) 2017 17 / 45
23. 2-3 l CIp LT M = λLT , MR = λR incoming to i : Lt i→j = 1 |Lt| k∈∂i\j Lt−1 k→i = 1 |Lt| (LT M)i→j , outgoting from j : Rt i→j = 1 |Rt| k∈∂j\i Rt−1 j→k = 1 |Rt| (MR)i→j . δλ = L(δMR) LT R = 1 LT R i→j,k→l Li→j δMi→j,k→l Rk→l , k → i → j, k ← i ← j, i → k → j, i ← k ← j i CIp (i) = j,l (Li→j Rj→l + Lj→i Ri→l ) + k,j Lk→j Rj→i . F.Morone et al., Scientific Reports 6, 2016 ( ) 2017 18 / 45
24. 2-4 Bond-Percolation Threshold pc = k k2 − k , R.Cohen, D.S.Callaway PRL 2000 pc = maxv vT Av vT v −1 , B.Bollob´ as Ann.Probab. 2010 pc = maxw wT Mw wT w −1 = 1 λM , F.Radicchi PRE 2015 M def = A 1 − D 1 0 , M w1 w2 = λM w1 w2 . B ⇔ M λM , u = (1, . . . , 1), d = (k1 , . . . , kn ) λM = dT w1 uT w1 − 1. NB xj def = i Aij Ri→j T.Martin, X.Zhang, M.E.J.Newman, Phys.Rev. E 90, 2014 ( ) 2017 19 / 45
25. Percolation on Sparse Nets πi (s) = sj :j∈Ni [Πj∈Ni πj←i (sj )] δ(s − 1, j∈Ni sj ), Gi (z) def = s πi (s)zs = zΠj∈Ni Hi←j (z), s πi (s) = Gi (1). S = 1 N N i=1 [1 − Gi (1)] = 1 − 1 N N i=1 Πj∈Ni Hi←j (1) Hi←j (1) = 1 − p + p k∈Nj \i Hj←k (1). s = s sπi (s) s πi (s) = G′ i (1) Gi (1) = 1 + j∈Ni H′ i←j (1) Hi←j (1) , H′ i←j (1) = p   1 + k∈Nj \i H′ j←k (1) Hj←k (1)   Πk∈Nj \i Hj←k (1). B.Karrer, M.E.J.Newman, L.Zdeborov´ a, Phys.Rev.Lett. 113, 2014 ( ) 2017 20 / 45
26. Beyond the locally treelike approx. si = p [1 − Πj∈Ni (1 − ti→j )] , ti→j = p 1 − Πk∈Qi→j (1 − tj→k ) . ⇒ t = pGt , G A-based Qi→j = Nj , Gi→j,l→k = δj,l M-based Qi→j = Nj \i, G = M, Mi→j,l→k = δj,l (1 − δi,k ) nonbacktrack W-based Qi→j = Nj \[i ∪ (Nj ∩ Ni )], G = W , Wi→j,l→k = δj,l (1 − δi,k )(1 − Aik ) F.Radicchi, C.Castellano, Phys.Rev. E 93, 2016 G.Tim´ ar, R.A. da Costa, S.N.Dorogovtev, and J.F.F.Mendes, Non-backtracking expansion of finite graphs, Phys.Rev. E 95, 2017 ( ) 2017 21 / 45
27. 2-5 NP Decycling G decycling θdec (G) Dismantling( , ) G dismantling θdis (G) GC C {pk } θdec (pk ) = lim N→∞ E[θdec (G)], θdis (pk ) = lim N→∞ lim C→∞ E[θdis (G, C)]. θdis (pk ) ≤ θdec (pk ) k2 < ∞ , θdis (pk ) = θdec (pk ) A.Braunstein et al., PNAS 113(44), 12368-12373, 2016 ( ) 2017 22 / 45
28. 2-6 Feedback Vertex Set NP INPUT: Digraph H, positive integer k PROPERTY: There is a set R ⊆ V such that every (directed) cycle of H contains a node in R. R.M.Karp, Reducibility among combinatorial problems, In Complexity of Computer Communications, E.Miller et al.(eds), pp.85-103, NY Plenum Press, 1972 ⇒ , kv − comp(G − v) v V.V.Vazirani, Approximation Algorithm, Chapter 6, pp.54-60, Springer, 2002 ( ) 2017 23 / 45
29. Cavity( ) Bethe-Peirls joint prob. i j k l m j k l m P\i (Aj : j ∈ ∂i) ≈ Πj∈∂i qAj j→i . 1 Ai = 0: i ⇒ 2 Ai = i: i . i , j ∈ ∂i Aj = j , i j Aj = i 3 Ai = l: i , l ∈ ∂i , k ∈ ∂i or , l i H.-J.Zhou, Euro.Phys. J. B 86, 2013 ( ) 2017 24 / 45
30. Marginal Probability 1 i q0 i = 1 zi , 2 j or i qi i = ex Πj∈∂i (q0 j→i + qj j→i ) zi , 3 l k or i l ql i = ex (1 − q0 l→i )Πk∈∂i (q0 k→i + qk k→i ) zi , q0 i + qi i + l∈∂i ql i = 1 zi def = 1 + ex Πj∈∂i (q0 j→i + qj j→i ) + l∈∂i (1 − q0 l→i )Πk∈∂i\l (q0 k→i + qk k→i ) . ( ) 2017 25 / 45
31. BP t , (BP) q0 i q0 i def = 1 1 + ex k∈∂i(t) 1−q0 k→i q0 k→i +qk k→i Πj∈∂i(t) q0 j→i + qj j→i q0 i→j = 1 zi→j (t) , qi i→j = ex Πk∈∂i(t)\j q0 k→i + qk k→i zi→j (t) , q0 i→j + qi i→j + l∈∂i ql i→j = 1, ∂i(t) t i , ∂i(t)\j ∂i(t) j , x > 0 . S.Mugisha, H.-J.Zhou, Phy.Rev. E 94, 2016 ( ) 2017 26 / 45
32. 3 = C.M.Shneider et al., PNAS 810, 2011, T.Tanizawa, S.Havlin, and H.E.Stanley, PRE 85, 046109, 2012 ⇒ Z.-X.Wu, and P.Holme, PRE 84, 026116, 2011 Y.Hayashi, IEEE Xplore Digital Library SASO 2014 ! ( ) 2017 27 / 45
33. 3-1 deletion randomly chosen + mutual link new j i , , , Y.Hayashi, Physica A 457, 255-269, 2016 ( ) 2017 28 / 45
34. Rewired ← Onion-like( , ) and Tree-like(△, ▽, ♦) Y.Hayashi, IEEE Xplore Digital Library SASO 2014 pp.50-59 ( ) 2017 29 / 45
35. deletion randomly chosen + mutual link new j i X ( : Y.Hayashi, Physica A 457 pp.255-269, 2016 ( ) 2017 30 / 45
36. Diffusion Limited Aggregation: , Invasion Percolation: , Eden Model: , ⇒ , : Y.Hayashi, Physica A 457 pp.255-269, 2016 ( ) 2017 31 / 45
37. DLA,IP,Eden Y.Hayashi, Physica A 457 pp.255-269, 2016 ( ) 2017 32 / 45
38. S(q): q GC , s(q) :GC : R def = 1 N 1 q=1/N S(q) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 S(q)/N q δ = 0.1 δ = 0.3, psc =0.015 δ = 0.5, psc =0.026 δ = 0.7, psc =0.035 δ = 0.9, psc =0.042 DLA 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.2 0.4 0.6 0.8 1 <s(q)> q 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 <s(q)> q ( ) 2017 33 / 45
39. , ⇒ 0 0.1 0.2 0.3 0.4 0 500 1000 1500 2000 r Step: N(t) δ = 0.3, psc = 0.015 δ = 0.5, psc = 0.026 δ = 0.7, psc = 0.035 δ = 0.9, psc = 0.042 Deg-deg correlation with moderate r ≈ 0.2 0.05 0.1 0.15 0.2 0.25 0.3 0 500 1000 1500 2000 R Step: N(t) Strong Robustness with increasing R Y.Hayashi, Physica A 457 pp.255-269, 2016 ( ) 2017 34 / 45
40. , , ⇒ , , ... ( ) 2017 35 / 45
41. 3-2 RLD-A: m m/2 MED: µ µ + 1 ⇒ Y. Hayashi, To appear in Network Science, Open Access , arXiv:1706.03910 , 2017 ( ) 2017 36 / 45
42. HDA, CI, BP m = 4 m = 2 ⇒ m ≥ 4 ( ) 2017 37 / 45
43. :R r ( ) 2017 38 / 45
44. ( ) 2017 39 / 45
45. Assortativity r vs N ( ) 2017 40 / 45
46. Small-World O(log N) or ⇒ ( ) 2017 41 / 45
47. p(k) N = 5000 ( ) 2017 42 / 45
48. p(k) ki (t) , ti t = h(ki ) h(k) , p(ki (t) < k) = p ti > h(ki ) h(k) t , p(k) = ∂p(ki (t) < k) ∂k = ∂ ∂k 1 − const. h(k) t N0 + t ∼ h′(k) h2(k) . 1 2 3 4 5 6 7 8 9 10 100 101 102 103 ki (t) t 10-5 10-4 10-3 10-2 10-1 100 5 10 15 20 25 30 35 40 p(k) k Y.Hayashi, Network Science 4(3), 385-399, Open Access , 2016 ( ) 2017 43 / 45
49. 4. Copying , , , , ... ( ) 2017 44 / 45
50. 5. → or Y , ( ) 2017 45 / 45