tetsunao matsuta
TRANSCRIPT
-
2015 12 4
1 / 56
-
1.
2.
3.
4.
2 / 56
-
1.
3 / 56
-
4 / 56
-
5 / 56
-
John Snow
1854616
John Snow
:
6 / 56
-
7 / 56
-
1.
8 / 56
-
G :V(G) : G
E(G) : G(i, j) E(G) : i j
9 / 56
-
t 0 F (t)
F (t) = 1 et
F
10 / 56
-
t tt
(1 t)t
t
t 0
limt0
(1 t)t
t = et
11 / 56
-
{(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
13 / 56
-
: S(G) V(G) :Cn(, v1) : v1
Cn(, v1) =
GnS(G)
Pn(Gn|v1) Pr{(Gn) = v1}
Pn(Gn|v) v nGn
14 / 56
-
= Gn S(G) V(Gn)
[Shah and Zaman, 2011]
Gn S(G)
v = argmaxvV(Gn)
Pn(Gn|v)
2
Pn(Gn|v)
15 / 56
-
1.
16 / 56
-
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}
}
17 / 56
-
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)}
18 / 56
-
Regular Tree
: Regular Tree
19 / 56
-
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)
23 / 56
-
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)
[Shah and Zaman, 2011]
24 / 56
-
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)
25 / 56
-
100 200 300 400 5000.0
0.2
0.4
0.6
0.8
1.0
n
Correctprob.
" 2
" 3
" 4
" 5
26 / 56
-
[Shah and Zaman, 2012]
= 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,
()
27 / 56
-
50 100 150 2000.300
0.302
0.304
0.306
0.308
lim
Cn
[Shah and Zaman, 2012]
lim
limn
Cn(ML, v1) = 1 ln 2 0.3069
28 / 56
-
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)
29 / 56
-
[Shah and Zaman, 2011]
v = argmaxvV(Gn)
R(v, TBFS(v))p(vnBFS(v))
TBFS(v) v Gn
vn P(v,Gn) p(vn)
30 / 56
-
: Small-World Network
5000 Small-world network 400
) 2%[Shah and Zaman, 2011]
31 / 56
-
: Scale-Free Network
5000 scale-free netowrk 400
5% [Shah and Zaman, 2011]32 / 56
-
33 / 56
-
2.
34 / 56
-
:
2
35 / 56
-
regulartree
36 / 56
-
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)
37 / 56
-
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
38 / 56
-
= 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)
39 / 56
-
= 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
40 / 56
-
= 3
[Matsuta and Uyematsu, 2014]
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.
41 / 56
-
= 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.
42 / 56
-
3
[Matsuta and Uyematsu, 2014]
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.
44 / 56
-
= 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.
46 / 56
-
47 / 56
-
3.
48 / 56
-
[Dong et al., 2013]
Regular tree
Polya
49 / 56
-
SIR[Zhu and Ying, 2013]
Susceptible-Infected-Recovered model: SI + R
(Recovered)
sample pathbased detection
Regular tree
50 / 56
-
[Prakash et al., 2012]
MDL (Minimum description length)
51 / 56
-
[Wang et al., 2014]
Gn L
Regular tree
L 1L 2 1
52 / 56
-
[Luo et al., 2014]
Sample path based detection
Regular tree O(n)
O(n3)
Regular tree
53 / 56
-
4.
54 / 56
-
regular tree
Regular tree
55 / 56
-
[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.
56 / 56