This document discusses methods for identifying the source node of information spread in networks based on the observed spread over time. It begins by introducing epidemic models like SIS and SI for modeling information spread over networks. It then discusses maximum likelihood methods for identifying the source node on regular tree networks based on the observed subgraph. The accuracy of these methods increases with network size and degree. Extensions to other network structures and SIR models are also proposed. Overall, the document reviews mathematical models and algorithms for source identification in networks from limited observations of information spread.

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.
48 / 56

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
50 / 56

51. [Prakash et al., 2012]
MDL (Minimum description length)
51 / 56

52. [Wang et al., 2014]
Gn L
Regular tree
L → ∞ 1
L ≥ 2 δ → ∞ 1
52 / 56

53. [Luo et al., 2014]
Sample path based detection
Regular tree O(n)
O(n3)
Regular tree
53 / 56

54. 4.
54 / 56

55. regular tree
Regular tree
55 / 56

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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.
56 / 56