From temporal to static networks, and back

TNETS edition

Susceptible

Infectious

Infectious

Comp!rtment!l models

Infectious

RecoveredSusceptible

Cont!ct p!tterns

ID1 ID2 Time1 2 343 4 551 5 564 2 705 4 776 1 1025 6 1105 7 1226 7 1302 5 1983 4 2054 2 2102 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 miningN = 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 datingN = 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 TOSTATIC

t st!

rt

t sto

p0 5 10 15 20

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6

t

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2

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6

time-slice networks

t st!

rt

t sto

p0 5 10 15 20

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2

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t

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34

5

6

ongoing networks

0 5 10 15 20

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2

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t

1

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34

5

6

1

2

34

5

6

! = 2.5

! = 10

exponential-threshold networks

GOOD REPRESENTATION:RANKING OF IMPORTANTVERTICES CONSERVED

FOR ALL PARAMETER VALUES:MEASURE AVG OUTBREAK SIZE WHEN SPREADING STARTS AT i

FOR ALL PARAMETER VALUES:MEASURE DEGREE OF iFOR ALL PARAMETER VALUES:MEASURE CORENESS OF i

degree 4

coreness 0

coreness 2

coreness 3

coreness 4

static importanceoptimal params.

dyna

mic

impo

rtan

ce

Spearmanrank correlationcoefficent =Quality ofrepresentation

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 ρE-m!il 1 0.73 0 0.42 0.50 0.25 0.25 0.77 0.40 0.30 0.46E-m!il 2 0.91 0 0.25 0.91 0.20 0.20 0.93 1.0 0.26 0.88D!tin" 0.82 0 0.65 0.42 0.25 0.25 0.86 0.10 0.16 0.71G!ller# 0.77 0 0.72 0.53 0.39 0.39 0.87 0.70 0.71 0.76Conference 0.79 0 0.10 0.74 0.10 0.11 0.77 0.04 0.02 0.53Prostitution 0.71 0 0.77 0.30 0.60 0.60 0.72 0.04 0.20 0.49

Perform!nce & p!r!meter v!lues De"ree

P!r!meter dependence of perform!nce

0 0.2 0.4 0.6 0.8 1

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00

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!

tstart/ T

t sto

p/

T

Time slice


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0 0.2 0.4 0.6 0.8 10

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0.25

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!

tstart/ T

t sto

p/

T

Concurrency


0 0.5 1 1.5 2 2.5 3 3.50

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1

1.5

2

!

!

" /

T

Exponential threshold

STEP 1 Assign stubs to vertices from a random number distribution.

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6

STEP 2 Connect random pairs of stubs to form a simple graph.

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34

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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

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0.1 1µ

! max

0.50.05

Exponential threshold

Time-slice

Accumulated

Ongoing

STATIC TOTEMPORAL

(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)

ON

GO

ING

LIN

K P

ICT

UR

E

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

LIN

K T

UR

NO

VER

PIC

TU

RE

timeBeginning time Interevent times End time

0 Tt1 t2 t3 t4t5 t6t7 t8tB 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’])


T0

t’

t


Compensate for the chance an interevent time t is active

0

tat the start of the sampling is proportional to t


T0

t’

t


Compensate for the chance an interevent time t is active

0

tat the start of the sampling is proportional to t

tiT–tii: ti!t

! / tiT–tii

!Sum up and normalize

0

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1

0 500 1000 1500 2000time t (days)

predicted frominterevent timesend timesbeginning times

PROSTITUTION

P (t

)B

Dating 2

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0.5

0.75

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ࢥ

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Forum

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Prostitution

Hospital

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End Times

Beginning Times

Predictableedges 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:identic!l interevent times

(1,2)

(1,3)

(1,4)

(2,3)

(1,2)

(1,3)

(1,4)

(2,3)

reference network:identic!l be"innin" times

(1,2)

(1,3)

(1,4)

(2,3)

(1,2)

(1,3)

(1,4)

(2,3)

reference network:identic!l end times

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E-mail 1

Film

Dating 1

Forum

E-mail 2

Facebook Prostitution

Hospital

Gallery

Conference

Dating 2

end

times

begin

nin

g times

intereven

t times

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

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E-mail 1

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Film

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Facebook

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Dating 1

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ForumDating 2

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Conference

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Prostitution

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E-mail 1

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E-mail 2

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Dating 1

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Facebook

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0ForumDating 2

–0.005

0

0.005

end

times

–0.5

0

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Conference

Hospital

–0.001

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Prostitution

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Gallery0

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Film

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From temporal to static networks, and back

Spiritual

Transcript of From temporal to static networks, and back