TNETS edition

Susceptible
Infectious
Infectious
Compartmental
models

Infectious
Recovered
Susceptible

Contact
patterns

ID1 ID2 Time
1 2 34
3 4 55
1 5 56
4 2 70
5 4 77
6 1 102
5 6 110
5 7 122
6 7 130
2 5 198
3 4 205
4 2 210
2 7 230

Sociopatterns gallery
P H Y S I C A L P R O X I M I T Y
Prostitution
Sociopatterns conference
Hospital system
N = 16,730, L = 50,632, T = 6.0y
N = 113, L = 20,818, T = 59h
N = 159(8), L = 6,027(350), T = 7.3(1)h
N = 293,878, L = 64,625,283, T = 3,570dReality mining
N = 63, L = 26,260, T = 8.6h

ELECTRONIC COMMUNICATION
N = 57,189, L = 444,162, T = 112.0d
Bornholdt’s e-mail
Eckmann’s e-mail
N = 3,188, L = 115,684, T = 81.6d
Filmtipset forum
N = 7,084, L = 1,412,401, T = 8.61y
Filmtipset messages
Pussokram dating
N = 28,972, L = 529,890, T = 512.0d
QX dating
N = 80,683, L = 4,337,203, T = 63.7d
N = 35,624, L = 472,496, T = 8.27y
Facebook wall posts
N = 293,878, L = 876,993, T = 1591d

TEMPORAL TO
STATIC

tstart
tstop
0 5 10 15 20
1
2
3
4
5
6
t
1
2
3
4
5
6
time-slice networks

tstart
tstop
0 5 10 15 20
1
2
3
4
5
6
t
1
2
3
4
5
6
ongoing networks

0 5 10 15 20
1
2
3
4
5
6
t
1
2
3
4
5
6
1
2
3
4
5
6
Ω
=
2.5
τ = 10
exponential-threshold networks

GOOD REPRESENTATION:
RANKING OF IMPORTANT
VERTICES CONSERVED
FOR ALL PARAMETER VALUES:
MEASURE AVG OUTBREAK SIZE
WHEN SPREADING STARTS AT i
FOR ALL PARAMETER VALUES:
MEASURE DEGREE OF i
FOR ALL PARAMETER VALUES:
MEASURE CORENESS OF i
degree 4
coreness 0
coreness 2
coreness 3
coreness 4
static importance
optimal params.
dynamic
importance
Spearman
rank correlation
coefficent
=
Quality of
representation

E-mail 1
E-mail 2
Dating
Gallery
Conference
Prostitution
Results, Degree
Time-slice Ongoing Exponential-threshold Accumulated

E-mail 1
E-mail 2
Dating
Gallery
Conference
Prostitution
Time-slice Ongoing Exponential-threshold Accumulated
Results, Coreness

Time sliceTime sliceTime slice OngoingOngoingOngoing Expo. thresholdExpo. thresholdExpo. threshold Acc.
ρmax tstart tstop ρmax tstart tstop ρmax τ ΩΩ ρ
E-mail 1 0.73 0 0.42 0.50 0.25 0.25 0.77 0.40 0.30 0.46
E-mail 2 0.91 0 0.25 0.91 0.20 0.20 0.93 1.0 0.26 0.88
Dating 0.82 0 0.65 0.42 0.25 0.25 0.86 0.10 0.16 0.71
Gallery 0.77 0 0.72 0.53 0.39 0.39 0.87 0.70 0.71 0.76
Conference 0.79 0 0.10 0.74 0.10 0.11 0.77 0.04 0.02 0.53
Prostitution 0.71 0 0.77 0.30 0.60 0.60 0.72 0.04 0.20 0.49
Performance & parameter values
Degree

Parameter dependence of performance
0 0.2 0.4 0.6 0.8 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
00
0.2
0.4
0.6
0.8
1
ρ
tstart/ T
tstop/T
Time slice

Parameter dependence of performance
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
ρ
tstart/ T
tstop/T
Concurrency

Parameter dependence of performance
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.5
1
1.5
2
ρ
Ω
τ/T
Exponential threshold

STEP 1 Assign stubs to vertices from a
random number distribution.
1
2
3
4
5
6

STEP 2 Connect random pairs of stubs
to form a simple graph.
1
2
3
4
5
6

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

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.1 1µ
ρ
max
0.50.05
Exponential threshold
Time-slice
Accumulated
Ongoing

break

STATIC TO
TEMPORAL

(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)

(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time
time
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
ONGOINGLINKPICTURE

time
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
(1,2)
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
(5,6)
time
LINKTURNOVERPICTURE

time
Beginning time Interevent times End time
0 T
t1
t2
t3
t4
t5
t6
t7
t8
tB
tE
DEFI
NITI
ONS

Compensate for the size bias on intervals because of finite
T0
t’
t
sampling time (t’ would only be recorded if it starts within [0,T–t’])

Compensate for the size bias on intervals because of finite
T0
t’
t
sampling time (t’ would only be recorded if it starts within [0,T–t’])
Compensate for the chance an interevent time t is active
0
t
at the start of the sampling is proportional to t

Compensate for the size bias on intervals because of finite
T0
t’
t
sampling time (t’ would only be recorded if it starts within [0,T–t’])
Compensate for the chance an interevent time t is active
0
t
at the start of the sampling is proportional to t
ti
T–tii: ti≥t
∑ /
ti
T–tii
∑
Sum up and normalize

0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000
time t (days)
predicted from
interevent times
end times
beginning times
PROSTITUTION
P(t)B

Dating 2
1
0
0.5
0.75
0.25
Dating 1
Forum
1
0
0.5
0.75
0.25
1
0
0.5
0.75
0.25
Prostitution
Hospital
1
0
0.5
0.75
0.25
1
0
0.5
0.75
0.25
E-mail 2
Facebook
1
0
0.5
0.75
0.25
1
0
0.5
0.75
0.25
E-mail 1
Film
1
0
0.5
0.75
0.25
1
0
0.5
0.75
0.25
Conference
Gallery
1
0
0.5
0.75
0.25
1
0
0.5
0.75
0.25
End Times
Beginning Times
Predictable
edges w.r.t.
beginning /
end times

(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

0
0.1
0.2
0.3
0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
fractionofinfectives
O r i g i n a l d a t a S I R

0
0.1
0.2
0.3
0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
fractionofinfectives
I n t e r e v e n t t i m e s S I R

0
0.1
0.2
0.3
0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
fractionofinfectives
B e g i n n i n g t i m e s S I R

0
0.1
0.2
0.3
0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
fractionofinfectives
E n d t i m e s S I R

0
0.02
0.04
0.06
E-mail 1
0.1
0
0.05
Film
0
0.05
0.1
Dating 1
0.05
0.1
0.15
0.2
0
Forum
0
0.02
0.04
0.06
E-mail 2
0
0.02
0.06
0.08
0.04
Facebook
0
0.01
0.02
0.03
0.04
Prostitution
0
0.1
0.2
0.3
Hospital
0
0.04
0.06
0.08
0.02
Gallery
0
0.02
0.04
0.06
Conference
0.05
0.1
0
Dating 2
endtimes
beginningtimes
intereventtimes

0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
O r i g i n a l d a t a
0
0.2
0.3
0.4
averagenumberofinfections
S I S
0.1

0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
I n t e r e v e n t t i m e s
0
0.2
0.3
0.4
averagenumberofinfections
S I S
0.1

0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
B e g i n n i n g t i m e s
0
0.2
0.3
0.4
averagenumberofinfections
S I S
0.1

0
0.2
0.3
0.4
0.1 0.2 0.90.8 10.70.60.50.40.3
0.1
1
0.01
0.001
per-contact transmission probability
durationofinfectivestage
averagenumberofinfections
E n d t i m e s S I S
0.1

0
0.01
0.02
0
0.0005
0.001
0.0015
0
0.05
0.1
0
0.01
0.02
0.03
0.04
0
0.0005
0.001
0.0005
0.001
0 0
0.05
0.1
0
0.001
0.0015
0.0005
0.05
0.1
0.15
0.2
0 0
0.02
0.04
0.06
0.08
0
0.01
0.03
0.04
0.02
E-mail 1
Film
Dating 1
Forum
E-mail 2
Facebook Prostitution
Hospital
Gallery
Conference
Dating 2
endtimes
beginningtimes
intereventtimes

Science by: Illustrations by:
Petter Holme Fredrik Liljeros Mi Jin Lee
P Holme, 2013, PLoS Comp. Biol. 9:e1003142.
P Holme, F Liljeros, 2013, arxiv:1307.6436.

0
0.2
0.4
0.6
0.8
E-mail 1
0
0.2
0.6
0.4
Film
0
0.1
0.2
0.3
0.4
E-mail 2
0
0.2
0.4
Facebook
0
0.4
0.6
0.2
Dating 1
0.8
0.2
0.4
0.6
0
ForumDating 2
0.2
0.8
0
0.4
0.6
–0.4
–0.2
0.2
0.8
0
0.4
0.6
Conference
Hospital
0
0.2
0.4
Prostitution
0
0.1
0.2
Gallery
0
0.2
0.8
0.4
0.6endtimes
beginningtimes
intereventtimes

0
0.1
0.2
0.3
0.4
E-mail 1
0
0.2
0.4
0.6
E-mail 2
0
0.4
0.6
0.2
Dating 1
0
0.005
0.01
Facebook
0.8
0.2
0.4
0
ForumDating 2
–0.005
0
0.005
endtimes
–0.5
0
0.5
Conference
Hospital
–0.001
0
0.001
Prostitution
0
0.1
0.2
beginningtimes
intereventtimes
Gallery
0
0.2
0.8
0.4
0.6
0
0.2
0.5
0.4
Film
0.3
0.1