Tetsunao Matsuta

1. 2015 12 4
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2. 1.
2.
3.
4.
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3. 1.
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4. 4 / 56

5. 5 / 56

6. John Snow
1854
616
John Snow
:
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7. 7 / 56

8. 1.
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9. G :
V(G) : G
E(G) : G
(i, j) ∈ E(G) : i j
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10. t ≥ 0 F(t)
F(t) = 1 − e−λt
λ
F
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11. ∆t λ · ∆t
t
(1 − λ · ∆t)
t
∆t
∆t → 0
lim
∆t→0
(1 − λ · ∆t)
t
∆t = e−λt
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12. {τ(i,j)}(i,j)∈E : F
(Susceptible-infected (SI) model)
0 v1
v v v′ τ(v,v′)
†
†
SIS model
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13. S(G) : G
Gn : ( ) n
G
G Gn ∈ S(G)
v1 V(Gn) SI model
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14. ϕ : S(G) → V(G) :
Cn(ϕ, v1) : v1 ϕ
Cn(ϕ, v1) =
Gn∈S(G)
Pn(Gn|v1) Pr{ϕ(Gn) = v1}
Pn(Gn|v) v n
Gn
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15. =⇒ Gn ∈ S(G) V(Gn)
[Shah and Zaman, 2011]
Gn ∈ S(G)
ˆv = argmax
v∈V(Gn)
Pn(Gn|v)
2
Pn(Gn|v)
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16. 1.
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17. N(v) : G v
B(V) : V
B(V)
v∈V
N(v) V
Pn(v1) : v1 n
Pn(v1) vn
∈ Vn
: vi ∈ B({v1, · · · , vi−1})
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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18. 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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19. Regular Tree
: Regular Tree
†
†
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20. Vi : i
Pr{V1 = v1} = 1
v2 ∈ B({v1})
Pr{V2 = v2|V1 = v1} = Pr{τ(v1,v2) = min
v∈B({v1})
{τ(v1,v)}}
=
1
|B({v1})|
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21. †
vn−1 ∈ P(v1) vn ∈ B({v1, · · · , vn−1})
Pr{Vn = vn|V n−1
= vn−1
} =
1
|B({v1, · · · , vn−1})|
†
τ Pr{τ > s + t|τ > s} = Pr{τ > t}
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22. δ(v) : v
|B({v1})| = δ(v1)
|B({v1, v2})| = |B({v1})| − 1 + δ(v2) − 1
= δ(v1) + (δ(v2) − 2)
|B({v1, v2, v3})| = |B({v1, v2})| − 1 + δ(v3) − 1
= δ(v1) + (δ(v2) − 2) + (δ(v3) − 2)
|B({v1, · · · , vn})| = δ(v1) +
n
i=2
(δ(vi) − 2)
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23. Pn(Gn|v1) = Pr{Gn V n |v1}
=
vn∈P(v1,Gn)
Pr{V n
= vn
}
=
vn∈P(v1,Gn)
n
k=2
1
|B({v1, · · · , vk−1})|
=
vn∈P(v1,Gn)
n
k=2
1
δ(v1) + k
i=2(δ(vi) − 2)
vn∈P(v1,Gn)
p(vn
)
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24. Regular Tree
argmax
v∈V(Gn)
Pn(Gn|v) = argmax
v∈V(Gn) vn∈P(v,Gn)
n
k=2
1
δ + k
i=2(δ − 2)
= argmax
v∈V(Gn)
|P(v, Gn)|
argmax
v∈V(Gn)
R(v, Gn)
argmax
v∈V(Gn)
R(v, Gn) O(n)
[Shah and Zaman, 2011]
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25. Regular Tree
[Dong et al., 2013]
ϕML v1 ∈ V(G)
Cn(ϕML, v1) =
⎧
⎪⎪⎨
⎪⎪⎩
1
2n−1
n−1
⌊(n−1)/2⌋ if δ = 2,
1
4 + 3
4
1
2⌊n/2⌋+1 if δ = 3,
1 − δ 1
2 PP´olya(n/2) + x>n/2 PP´olya(x) if δ ≥ 4
PP´olya(x) =
n − 1
x
1(δ−2,x)(δ − 1)(δ−2,n−1−x)
δ(δ−2,n−1)
x(a,b) = x(x + a)(x + 2a) · · · (x + (b − 1)a)
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26. 100 200 300 400 500
0.0
0.2
0.4
0.6
0.8
1.0
n
Correctprob.
∆ 2
∆ 3
∆ 4
∆ 5
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27. [Shah and Zaman, 2012]
δ = 2 v1 ∈ V(G)
Cn(ϕML, v1) = Θ
1
√
n
δ ≥ 3
lim
n→∞
Cn(ϕML, v1) = δ · I1/2
1
δ − 2
,
δ − 1
δ − 2
− (δ − 1)
Ix(a, b)
Ix(a, b)
Γ(a + b)
Γ(a)Γ(b)
x
0
ta−1
(1 − t)b−1
dt,
Γ(·)
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28. 50 100 150 200
0.300
0.302
0.304
0.306
0.308
∆
limCn
[Shah and Zaman, 2012]
lim
δ→∞
lim
n→∞
Cn(ϕML, v1) = 1 − ln 2 ≈ 0.3069
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29. regular tree
[Shah and Zaman, 2011]
ˆv = argmax
v∈V(Gn)
R(v, Gn)p(vn
BFS(v))
vn
BFS(v) v Gn
vn ∈ P(v, Gn) p(vn)
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30. [Shah and Zaman, 2011]
ˆv = argmax
v∈V(Gn)
R(v, TBFS(v))p(vn
BFS(v))
TBFS(v) v Gn
vn ∈ P(v, Gn) p(vn)
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31. : Small-World Network
5000 Small-world network 400
−→ ) 2%
[Shah and Zaman, 2011]
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32. : Scale-Free Network
5000 scale-free netowrk 400
−→ 5% [Shah and Zaman, 2011]
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33. 33 / 56

34. 2.
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35. :
2
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36. regular
tree
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37. Dn(d) : v1 ˆv d
Dn(d) Pr ˆV ∈ v
(d)
1 , v
(d)
2 · · · , v
(d)
δ·(δ−1)d−1
ˆV :
v
(d)
1 , · · · , v
(d)
δ·(δ−1)d−1 : v1 d(≥ 1)
( δ · (δ − 1)d−1 )
Dn(0) = Cn(ϕML, v1)
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38. 1
1
k
l
(k − 1)
k − 1
l
+
k − 1
l − 1
xk = x(x + 1)(x + 2) · · · (x + k − 1)
xk
=
n
l=0
k
l
xl
1
s(k, l) (−1)k−l k
l
xk = x(x − 1)(x − 2) · · · (x − k + 1)
xk
=
n
l=0
s(k, l)xl
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39. δ = 3
[Matsuta and Uyematsu, 2014]
d ≥ 1 n ≥ 3
Dn(d) = 3 · 2d−1
(n+1)/2
k=d+1
2
k + 1
(n+3)/2
k+1
n+1
k+1
(−1)d+k
(k − 1)!
d
l=1
s(k, l)
n ≥ 2
Dn(d) = 3 · 2d−1
n/2+1
k=d+1
2
k + 1
n/2+1
k+1 + n
2(n+2)
n/2+1
k
n+1
k+1
(−1)d+k
(k − 1)!
d
l=1
s(k, l)
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40. δ = 3
50 100 150 200
0.0
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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41. δ = 3
[Matsuta and Uyematsu, 2014]
d ≥ 2
lim
n→∞
Dn(d)
= 3 · 2d−1
(−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 6
0.0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5 6
d
DistanceProb.
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42. δ = 3
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6
d
CumulativeProb.
3
d=0
lim
n→∞
Dn(d) ≈ 0.9676.
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43. δ ≥ 3
[Matsuta and Uyematsu, 2014]
d ≥ 1 δ ≥ 3 m ∈ N
0 ≤ lim
n→∞
Dn(d) − f(δ, d, m) ≤ e2
(3 + m)24−m
f(δ, d, m) δ(δ − 1)d−1
m
k=d+1
p(δ, d, k) I1/2 k − 1 +
1
δ − 2
,
δ − 1
δ − 2
− (δ − 1)I1/2 k − 1 +
δ − 1
δ − 2
,
1
δ − 2
p(δ, d, k)
2
(δ − 2)d
1
δ−2
k−1
2
δ−2
k
ζd−1
k−2
1
δ − 2
ζd
k (x)
1≤j1<j2<···<jd≤k
d
i=1
1
ji + x
( )
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44. δ = 6
m = 35 f(δ, d, 35)
lim
n→∞
Dn(d) − f(δ, d, m) ≤ e2
(3 + m)24−m
≈ 1.3075 · 10−7
0 1 2 3 4 5 6
0.0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4 5 6
d
DistanceProb.
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45. δ = 6
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6
d
CumulativeProb.
3
d=0
lim
n→∞
Dn(d) ≈ lim
n→∞
Cn(ϕML, v1) +
3
d=1
f(6, d, 35)
≈ 0.9854.
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46. 2.
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47. 47 / 56

48. 3.
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49. [Dong et al., 2013]
Regular tree
P´olya
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50. SIR
[Zhu and Ying, 2013]
Susceptible-Infected-Recovered model: SI + R
(Recovered)
sample path
based detection
Regular tree
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51. [Prakash et al., 2012]
MDL (Minimum description length)
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52. [Wang et al., 2014]
Gn L
Regular tree
L → ∞ 1
L ≥ 2 δ → ∞ 1
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53. [Luo et al., 2014]
Sample path based detection
Regular tree O(n)
O(n3)
Regular tree
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54. 4.
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55. regular tree
Regular tree
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from suspects,” ISIT 2013, pp.2671–2675, 7-12 July 2013.
[Kuba and Prodinger, 2010] M. Kuba and H. Prodinger, “A note on Stirling series,”
Integers, vol. 10, no. 4, pp. 393–406, 2010.
[Luo et al., 2014] W. Luo, W. P. Tay, and M. Leng, “How to identify an infection source
with limited observations,” IEEE Journal of Selected Topics in Signal Processing, vol. 8,
no. 4, pp. 586–597, Aug. 2014
[Matsuta and Uyematsu, 2014] T. Matsuta and T. Uyematsu, “Probability distributions of
the 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 in
epidemics: How many and which ones?,” ICDM 2012, pp. 11–20, 10-13 Dec. 2012.
[Shah and Zaman, 2011] D. Shah and T. Zaman, “Rumors in a network: Who’s the
culprit?,” IEEE Trans. Inform. Theory, vol. 57,no. 8, pp. 5163–5181, Aug. 2011.
[Shah and Zaman, 2012] D. Shah and T. Zaman, “Rumor centrality: A universal source
detector,” SIGMETRICS Perform. Eval. Rev., vol. 40, no. 1, pp. 199–210, Jun. 2012.
[Steyn, 1951] H. S. Steyn, “On discrete multivariate probability functions,”
Proc. Koninklijke Nderlandse Akademie van Wetenschappen, Ser. A, vol. 54, pp. 23–30.
[Wang et al., 2014] Z. Wang, W. Dong, and W. Zhang and C.W. Tan, “Rumor source
detection with multiple observations: Fundamental limits and algorithms,” ACM
SIGMETRICS 2014, pp. 1–13, 16-20 June 2014.
[Zhu and Ying, 2013] K. Zhu and L. Ying, “Information source detection in the SIR
model: A sample path based approach,” ITA 2013, pp. 1–9, 10-15 Feb. 2013.
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