Tetsunao Matsuta

2015 12 4

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1.

2.

3.

4.

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1.

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John Snow

1854616

John Snow

:

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1.

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G :V(G) : G

E(G) : G(i, j) E(G) : i j

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t 0 F (t)

F (t) = 1 et

F

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t tt

(1 t)t

t

t 0

limt0

(1 t)t

t = et

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{(i,j)}(i,j)E : F

(Susceptible-infected (SI) model)

0 v1

v v v (v,v)

SIS model12 / 56

S(G) : GGn : ( ) n

G

G Gn S(G)v1 V(Gn) SI model

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: S(G) V(G) :Cn(, v1) : v1

Cn(, v1) =

GnS(G)

Pn(Gn|v1) Pr{(Gn) = v1}

Pn(Gn|v) v nGn

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= Gn S(G) V(Gn)

[Shah and Zaman, 2011]

Gn S(G)

v = argmaxvV(Gn)

Pn(Gn|v)

2

Pn(Gn|v)

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1.

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N (v) : G vB(V) : V

B(V) !{

vVN (v)

}\V

Pn(v1) : v1 n

Pn(v1) !{vn Vn : vi B({v1, , vi1})

}

vn = (v1, v2, , vn) Vn = V V V n

Pn(v1, Gn) : V(Gn) Pn(v1)

Pn(v1, Gn) !{vn Pn(v1) : V(Gn) = {v1, v2, , vn}

}

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N (1) = {2, 3, 4}B({1, 2}) = {3, 4, 5, 6}

P2(2) = {(2, 1), (2, 5), (2, 6)}P3(2) = {(2, 1, 4), (2, 1, 3), (2, 1, 5), (2, 1, 6),

(2, 5, 1), (2, 5, 6), (2, 6, 1), (2, 6, 5)}P3(2, Gn) = {(2, 5, 6), (2, 6, 5)}

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Regular Tree

: Regular Tree

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Vi : i

Pr{V1 = v1} = 1

v2 B({v1})

Pr{V2 = v2|V1 = v1} = Pr{(v1,v2) = minvB({v1})

{(v1,v)}}

=1

|B({v1})|20 / 56

vn1 P(v1) vn B({v1, , vn1})

Pr{Vn = vn|V n1 = vn1} =1

|B({v1, , vn1})|

Pr{ > s+ t| > s} = Pr{ > t}21 / 56

(v) : v

|B({v1})| = (v1)|B({v1, v2})| = |B({v1})| 1 + (v2) 1

= (v1) + ((v2) 2)|B({v1, v2, v3})| = |B({v1, v2})| 1 + (v3) 1

= (v1) + ((v2) 2) + ((v3) 2)

|B({v1, , vn})| = (v1) +n

i=2

((vi) 2)22 / 56

Pn(Gn|v1) = Pr{Gn V n |v1}

=

vnP(v1,Gn)

Pr{V n = vn}

=

vnP(v1,Gn)

n

k=2

1

|B({v1, , vk1})|

=

vnP(v1,Gn)

n

k=2

1

(v1) +k

i=2((vi) 2)

!

vnP(v1,Gn)

p(vn)

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Regular Tree

argmaxvV(Gn)

Pn(Gn|v) = argmaxvV(Gn)

vnP(v,Gn)

n

k=2

1

+k

i=2( 2)

= argmaxvV(Gn)

|P(v,Gn)|

! argmaxvV(Gn)

R(v,Gn)

argmaxvV(Gn)

R(v,Gn) O(n)


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Regular Tree

[Dong et al., 2013]

ML v1 V(G)

Cn(ML, v1) =

12n1

( n1(n1)/2

)if = 2,

14 +

34

12n/2+1 if = 3,

1 (12PPolya(n/2) +

x>n/2PPolya(x)

)if 4

PPolya(x) =

(n 1x

)1(2,x)( 1)(2,n1x)

(2,n1)

x(a,b) = x(x+ a)(x+ 2a) (x+ (b 1)a)

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100 200 300 400 5000.0

0.2

0.4

0.6

0.8

1.0

n

Correctprob.

" 2

" 3

" 4

" 5

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= 2 v1 V(G)

Cn(ML, v1) =

(1n

)

3

limn

Cn(ML, v1) = I1/2(

1

2 , 1 2

) ( 1)

Ix(a, b)

Ix(a, b) !(a+ b)

(a)(b)

x

0ta1(1 t)b1dt,

()

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50 100 150 2000.300

0.302

0.304

0.306

0.308

lim

Cn


lim

limn

Cn(ML, v1) = 1 ln 2 0.3069

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regular tree[Shah and Zaman, 2011]

v = argmaxvV(Gn)

R(v,Gn)p(vnBFS(v))

vnBFS(v) v Gn

vn P(v,Gn) p(vn)

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v = argmaxvV(Gn)

R(v, TBFS(v))p(vnBFS(v))

TBFS(v) v Gn

vn P(v,Gn) p(vn)

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: Small-World Network

5000 Small-world network 400

) 2%[Shah and Zaman, 2011]

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: Scale-Free Network

5000 scale-free netowrk 400

5% [Shah and Zaman, 2011]32 / 56

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2.

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:

2

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regulartree

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Dn(d) : v1 v d

Dn(d) ! Pr{V

{v(d)1 , v

(d)2 , v

(d)(1)d1

}}

V :{v(d)1 , , v

(d)(1)d1

}: v1 d( 1)

( ( 1)d1 )

Dn(0) = Cn(ML, v1)

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1

1[k

l

]! (k 1)

[k 1l

]+

[k 1l 1

]

xk = x(x+ 1)(x+ 2) (x+ k 1)

xk =n

l=0

[k

l

]xl

1

s(k, l) ! (1)kl[k

l

]

xk = x(x 1)(x 2) (x k + 1)

xk =n

l=0

s(k, l)xl

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= 3

[Matsuta and Uyematsu, 2014]

d 1 n 3

Dn(d) = 3 2d1(n+1)/2

k=d+1

2

k + 1

((n+3)/2k+1

)(n+1k+1

) (1)d+k

(k 1)!

d

l=1

s(k, l)

n 2

Dn(d) = 3 2d1n/2+1

k=d+1

2

k + 1

(n/2+1k+1

)+ n2(n+2)

(n/2+1k

)

(n+1k+1

) (1)d+k

(k 1)!

d

l=1

s(k, l)

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= 3

50 100 150 2000.0

0.1

0.2

0.3

0.4

0.5

0.6

n

DistanceProb.

d ! 0

d ! 1

d ! 2

d ! 3

d ! 4

d ! 5

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= 3


d 2

limn

Dn(d)

= 3 2d1(1)d(

d

l=1

(1)l(lnl 2

l! 2 +

l

m=0

(ln 2)m

m!

)+

1

4

)

!

!

!

!

!! !

0 1 2 3 4 5 60.0

0.1

0.2

0.3

0.4

0.50 1 2 3 4 5 6

d

DistanceProb.

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= 3

!

!

!

!! ! !

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.00 1 2 3 4 5 6

d

CumulativeProb.

3

d=0

limn

Dn(d) 0.9676.

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3


d 1 3 m N0 lim

nDn(d) f(, d,m) e2(3 +m)24m

f(, d,m) !( 1)d1m

k=d+1

p(, d, k)

(I1/2

(k 1 + 1

2 , 1 2

)

( 1)I1/2(k 1 + 1

2 ,1

2

))

p(, d, k) ! 2( 2)d

(1

2

)k1

(2

2

)k d1k2

( 1 2

)

dk(x) !

1j1

= 6

m = 35 f(, d, 35)

limn

Dn(d) f(, d,m) e2(3 +m)24m 1.3075 107

!

!

!

!

! ! !

0 1 2 3 4 5 60.0

0.1

0.2

0.3

0.4

0.50 1 2 3 4 5 6

d

DistanceProb.

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= 6

!

!

!

! ! ! !

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.00 1 2 3 4 5 6

d

CumulativeProb.

3

d=0

limn

Dn(d) limn

Cn(ML, v1) +3

d=1

f(6, d, 35)

0.9854.45 / 56

2.

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3.

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[Dong et al., 2013]

Regular tree

Polya

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SIR[Zhu and Ying, 2013]

Susceptible-Infected-Recovered model: SI + R

(Recovered)

sample pathbased detection

Regular tree

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[Prakash et al., 2012]

MDL (Minimum description length)

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[Wang et al., 2014]

Gn L

Regular tree

L 1L 2 1

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[Luo et al., 2014]

Sample path based detection

Regular tree O(n)

O(n3)

Regular tree

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4.

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regular tree

Regular tree

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[Dong et al., 2013] W. Dong, W. Zhang, and C. W. Tan, Rooting out the rumor culpritfrom suspects, ISIT 2013, pp.26712675, 7-12 July 2013.[Kuba and Prodinger, 2010] M. Kuba and H. Prodinger, A note on Stirling series,Integers, vol. 10, no. 4, pp. 393406, 2010.[Luo et al., 2014] W. Luo, W. P. Tay, and M. Leng, How to identify an infection sourcewith limited observations, IEEE Journal of Selected Topics in Signal Processing, vol. 8,no. 4, pp. 586597, Aug. 2014[Matsuta and Uyematsu, 2014] T. Matsuta and T. Uyematsu, Probability distributions ofthe distance between the rumor source and its estimation on regular trees, SITA 2014,pp. 605-610, Dec. 2014.[Prakash et al., 2012] B. A. Prakash, J. Vreeken, and C. Faloutsos, Spotting culprits inepidemics: How many and which ones?, ICDM 2012, pp. 1120, 10-13 Dec. 2012.[Shah and Zaman, 2011] D. Shah and T. Zaman, Rumors in a network: Whos theculprit?, IEEE Trans. Inform. Theory, vol. 57,no. 8, pp. 51635181, Aug. 2011.[Shah and Zaman, 2012] D. Shah and T. Zaman, Rumor centrality: A universal sourcedetector, SIGMETRICS Perform. Eval. Rev., vol. 40, no. 1, pp. 199210, Jun. 2012.[Steyn, 1951] H. S. Steyn, On discrete multivariate probability functions,Proc. Koninklijke Nderlandse Akademie van Wetenschappen, Ser. A, vol. 54, pp. 2330.[Wang et al., 2014] Z. Wang, W. Dong, and W. Zhang and C.W. Tan, Rumor sourcedetection with multiple observations: Fundamental limits and algorithms, ACMSIGMETRICS 2014, pp. 113, 16-20 June 2014.[Zhu and Ying, 2013] K. Zhu and L. Ying, Information source detection in the SIRmodel: A sample path based approach, ITA 2013, pp. 19, 10-15 Feb. 2013.

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Science

Tetsunao Matsuta