TNETS edition

TNETS edition
Susceptible
Infectious
Compartmental
models
Infectious
Recovered
Susceptible
Infectious
Contact
patterns
ID1
1
3
1
4
5
6
5
5
6
2
3
4
2
ID2 Time
2
34
4
55
5
56
2
70
4
77
1
102
6
110
7
122
7
130
5
198
4
205
2
210
7
230
PHYSICAL PROXIMITY
y
0
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2
3
50,6
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Ho
N = 293,878, L =
ELECTRONIC COMMUNICATION
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N = 57,18 il
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2
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Film
N = 7,08
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8
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Film
=
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N = 35,624
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5
N = 29
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N=2
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3
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2
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4
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,
3
N = 80,68
TEMPORAL TO
STATIC
time-slice networks
1
4
3
2
3
5
4
5
6
5
10
t
tstop
0
tstart
6
15
20
2
1
ongoing networks
1
4
3
2
3
5
4
5
6
5
10
tstop
0
tstart
6
15
t
20
2
1
exponential-threshold networks
1
4
3
2
τ = 10
3
5
4
5
6
6
0
5
10
15
20
t
Ω
4
5
6
3
2
1
=
5
.
2
2
1
degree 4
dynamic
importance
coreness 2
coreness 4
static importance
optimal params.
GOOD REPRESENTATION:
RANKING OF IMPORTANT
VERTICES CONSERVED
coreness 3
coreness 0
Spearman
rank correlation
coefficent
=
Quality of
representation
FOR ALL PARAMETER VALUES:
MEASURE
AVG OUTBREAK
SIZE
FOR ALL PARAMETER
VALUES:
WHEN
SPREADING
MEASURE
DEGREESTARTS
CORENESS
OFOF
i i AT i
Results, Degree
E-mail 1
E-mail 2
Dating
Gallery
Conference
Prostitution
Time-slice Ongoing Exponential-threshold Accumulated
Results, Coreness
E-mail 1
E-mail 2
Dating
Gallery
Conference
Prostitution
Time-slice Ongoing Exponential-threshold Accumulated
Performance & parameter values
Degree
Time slice
E-mail 1
E-mail 2
Dating
Gallery
Conference
Prostitution
Ongoing
Expo. threshold
Acc.
ρmax
tstart
tstop
ρmax
tstart
tstop
ρmax
τ
ΩΩ
ρ
0.73
0.91
0.82
0.77
0.79
0.71
0
0
0
0
0
0
0.42
0.25
0.65
0.72
0.10
0.77
0.50
0.91
0.42
0.53
0.74
0.30
0.25
0.20
0.25
0.39
0.10
0.60
0.25
0.20
0.25
0.39
0.11
0.60
0.77
0.93
0.86
0.87
0.77
0.72
0.40
1.0
0.10
0.70
0.04
0.04
0.30
0.26
0.16
0.71
0.02
0.20
0.46
0.88
0.71
0.76
0.53
0.49
Parameter dependence of performance
1
0.7
0.6
0.8
0.5
0.4
0.3
ρ
0.4
tstop / T
0.6
0.2
0.2
0.1
Time slice
0
0
0.2
0.4
0.6
tstart/ T
0.8
1
0
Parameter dependence of performance
0.3
1
0.25
0.8
0.2
0.15
ρ
tstop / T
0.6
0.4
0.1
0.2
0.05
Concurrency
0
0
0.2
0.4
0.6
tstart/ T
0.8
1
0
0.7
2
0.6
1.5
0.5
0.4
1
ρ
τ/T
Parameter dependence of performance
0.3
0.2
0.5
0
Exponential threshold
0
0.5
1
1.5
2
Ω
2.5
3
3.5
0.1
0
STEP 1
Assign stubs to vertices from a
random number distribution.
5
1
4
3
2
6
STEP 2
Connect random pairs of stubs
to form a simple graph.
4
5
6
3
2
1
STEP 3
Create active intervals for each
edge.
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time
STEP 4
Create a time series of contacts
from some interevent-time
distribution.
time
STEP 5
Split the time series into
segments proportional to the
intervals and impose the
contacts of the segments to the
intervals.
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time
STEP 6
Forget the active intervals.
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time
0.7
0.6
ρ
max
0.5
0.4
0.3
0.2
0.1
0.05
Time-slice
Ongoing
Exponential threshold
Accumulated
0.1
1
0.5
µ
break
STATIC TO
TEMPORAL
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
ONGOING LINK PICTURE
S
R
KT
SM
M H
RX
KY
V
XI
M
I
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time
LINK TURNOVER PICTURE
X
M
IT
M
X
YH
X
V
RY
V
S
ZI
I
V
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time
DEFI
NITI
ONS
0
tB
t1 t2 t3 t4t5 t6 t7 t8
tE
Beginning time
Interevent times
End time
time
T
Compensate for the size bias on intervals because of finite
sampling time (t’ would only be recorded if it starts within [0,T–t’])
t’
0
t
T
Compensate for the size bias on intervals because of finite
sampling time (t’ would only be recorded if it starts within [0,T–t’])
t’
0
t
T
Compensate for the chance an interevent time t is active
at the start of the sampling is proportional to t
t
0
Compensate for the size bias on intervals because of finite
sampling time (t’ would only be recorded if it starts within [0,T–t’])
t’
0
t
T
Compensate for the chance an interevent time t is active
at the start of the sampling is proportional to t
t
0
Sum up and normalize
ti
i: ti≥t T–ti
∑
/∑
i
ti
T–ti
PROSTITUTION
1
predicted from
interevent times
end times
beginning times
0.8
PB(t)
0.6
0.4
0.2
0
0
500
1000
1500
time t (days)
2000
Predictable 1
edges w.r.t.
0.75
beginning /
end times 0.5
1
1
1
1
0.75
0.75
0.75
0.75
0.5
0.5
0.5
0.5
0.25
0.25
0.25
0.25
0.25
0
0
0
0
0
Beginning Times
End Times
E-mail 1
E-mail 2
Dating 1
Conference
1
1
1
1
1
1
0.75
0.75
0.75
0.75
0.75
0.75
0.5
0.5
0.5
0.5
0.5
0.5
0.25
0.25
0.25
0.25
0.25
0.25
0
0
0
0
0
0
Dating 2
Film
Facebook
Forum
Gallery
Hospital
Prostitution
(1,2)
(1,3)
(1,4)
(2,3)
reference networks
(1,2)
(1,3)
(1,4)
(2,3)
(1,2)
(1,3)
(1,4)
(2,3)
reference network:
identical interevent times
(1,2)
(1,3)
(1,4)
(2,3)
(1,2)
(1,3)
(1,4)
(2,3)
reference network:
identical beginning times
(1,2)
(1,3)
(1,4)
(2,3)
(1,2)
(1,3)
(1,4)
(2,3)
reference network:
identical end times
Original data
SIR
1
0.1
0.2
0.01
0.001
0.1
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8 0.9 1
per-contact transmission probability
0
fraction of infectives
duration of infective stage
0.3
Interevent times SIR
1
0.1
0.2
0.01
0.001
0.1
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8 0.9 1
per-contact transmission probability
0
fraction of infectives
duration of infective stage
0.3
Beginning times SIR
1
0.1
0.2
0.01
0.001
0.1
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8 0.9 1
per-contact transmission probability
0
fraction of infectives
duration of infective stage
0.3
End times
SIR
1
0.1
0.2
0.01
0.001
0.1
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8 0.9 1
per-contact transmission probability
0
fraction of infectives
duration of infective stage
0.3
end times
beginning times
interevent times
0.06
0.06
0.3
0.06
0.1
0.04
0.04
0.02
0.02
0
0
0.05
E-mail 2
E-mail 1
0.2
0.04
0
Dating 1
0.02
0.1
0
0
Conference
0.04
0.08
0.1
0.08
0.2
0.1
0.03
0.06
0.06
0.15
0.02
0.05
0.05
0.04
0
0
0
Dating 2
Film
Hospital
Facebook
0.02
0.1
0.04
0.05
0.02
0.01
0
0
0
Forum
Gallery
Prostitution
Original data
SIS
1
0.3
0.1
0.2
0.01
0.001
0.1
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8 0.9 1
per-contact transmission probability
0
average number of infections
duration of infective stage
0.4
Interevent times SIS
1
0.3
0.1
0.2
0.01
0.001
0.1
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8 0.9 1
per-contact transmission probability
0
average number of infections
duration of infective stage
0.4
Beginning times SIS
1
0.3
0.1
0.2
0.01
0.001
0.1
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8 0.9 1
per-contact transmission probability
0
average number of infections
duration of infective stage
0.4
End times
SIS
1
0.3
0.1
0.2
0.01
0.001
0.1
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8 0.9 1
per-contact transmission probability
0
average number of infections
duration of infective stage
0.4
end times
beginning times
interevent times
0.04
0.02
0.0015
0.001
0.1
0.03
0.001
0.02
0.01
0.0005
0.05
0.0005
0
E-mail 1
0
0.01
E-mail 2
0
Dating 1
Conference
Hospital
0.2
0.1
0.03
0.06
0.15
0.001
0.0005
0
0.04
0.08
0.0015
0.001
0
0.04
0.02
0.05
0.02
0.01
0
0
0
0.1
0.05
0.0005
0
0
0
Dating 2
Film
Facebook
Forum
Gallery
Prostitution
Science by:
Petter Holme Fredrik Liljeros
Illustrations by:
Mi Jin Lee
P Holme, 2013, PLoS Comp. Biol. 9:e1003142.
P Holme, F Liljeros, 2013, arxiv:1307.6436.
0.8
0.4
0.8
0.8
0.6
end times
beginning times
interevent times
0.6
0.6
0.3
0.4
0.4
0.4
0.2
0.6
0.2
0.4
0.2
0.2
0.1
0
0
0.2
–0.2
0
E-mail 1
0
E-mail 2
Dating 1
–0.4
Conference
0
Gallery
0.6
0.8
0.8
0.2
0.4
0.6
0.4
0.4
0.6
0.1
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0
0
0
0
0
Dating 2
Film
0
Facebook
Forum
Hospital
Prostitution
end times
beginning times
interevent times
0.4
0.8
0.6
0.6
0.3
0.5
0.4
0.6
0.4
0.2
0.4
0.2
0
0.2
0.1
0.2
0
0
0
E-mail 2
E-mail 1
–0.5
Dating 1
Conference
0
Gallery
0.8
0.5
0.005
0.01
0.4
0.2
0
0.1
–0.001
0
0.4
0.3
0
0.001
0.005
0.2
0.2
0.1
–0.005
0
0
Dating 2
Film
0
Facebook
Forum
Hospital
Prostitution