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Temporal networks
of human interaction
Petter Holme
Ntriangles = 3
Among the people we study,
there is a tendency for
triangles to form.
Ntriangles = 3
Rumors would spread slowly
because of the many
triangles.
Time & topology
time
network
Human interaction
What are we interested in?
Something that can:
. . . be measured relatively easy
(who are involved & when).
. . . and give scientific insights.
Examples:
-Two persons being close to each other.
-Two persons doing things together.
-One person sending a message to another.
Human interaction
Two persons being close to each other
-RFID tags.
-Smartphones / Bluetooth.
-Smartphones / GPS.
-Campus Wi-fi.
-Hospital records.
-Co-tagged in images.
-Public transportation.
-Sexual contacts (via Internet mediated prostitution).
Human interaction
Two persons doing things together
(not necessarily close)
-Paper co-authorships.
-Movie actors.
-Criminal co-offenders.
One person sending a message to another
-E-mails.
-Internet forums.
-Instant messaging.
Human interaction
Representations
Numerical representations:
Contact sequences
ID1
2
6
2
10
7
3
5
2
7
10
ID2
4
8
8
11
2
5
3
10
3
2
time
10
10
15
20
22
25
30
30
31
34
Numerical representations:
Interval graphs
ID1
2
6
2
10
7
3
5
2
7
10
ID2
4
8
8
11
2
5
3
10
3
2
time interval
(10,15)
(11,14)
(18,19)
(20,22)
(20,24)
(20,30)
(25,31)
(40,45)
(40,50)
(51,53)
0 5 10 15 20
1
2
3
4
5
6
t
Graphical representations:
Timelines of individuals
(1,2)
(1,3)
(1,4)
(2,3)
Graphical representations:
Timelines of links
Graphical representations:
Annotated graph
E
D
C
B
A
11,20
1,4,8
3,8,10,17
11,15
16
Graphical representations:
Film clip
E
D
C
B
A
E
D
C
B
A
Epidemiology
transmission
probability / rate
after some time /
with some chance
per time unit
susceptible infectious recovered /
susceptible
+
Time matters
Time matters
E
D
C
B
A
11,20
1,4,8
3,8,10,17
11,15
16
Time matters
Rocha, Liljeros, Holme, 2010. PNAS 107: 5706-5711.
Escort/sex-buyer contacts:
16,730 individuals
50,632 contacts
2,232 days
1555
ID1 ID2 time
5 7 1021
20 9 1119
4 30 1539
ID1 ID2 time
5 7 1555
20 9 1021
4 30 1119
4 20 15394 20
Time matters
Rocha, Liljeros, Holme, 2010. PNAS 107: 5706-5711.
Escort/sex-buyer contacts:
16,730 individuals
50,632 contacts
2,232 days
Rocha, Liljeros, Holme, 2011. PLoS Comp. Biol. 7: e1001109.
0
0.2
0.4
0.6
0 200 400 600
Fractionofinfectious
Time (days)
Empirical
800
0
0.2
0.4
0.6
0 200 400 600
Time (days)
Empirical
Randomized
800
Fractionofinfectious
Time matters
Time matters
0
0.2
0.4
0.6
0 200 400 600
Time (days)
Empirical
Randomized
800
Fractionofinfectious
1
0.8
0.6
0.4
0.2
0
0 100 200 300
Time (days)
Fractionofinfectious
Rocha, Liljeros, Holme Karsai, et al.
Physics Reports 519 (2012) 97–125
Contents lists available at SciVerse ScienceDirect
Physics Reports
journal homepage: www.elsevier.com/locate/physrep
Temporal networks
Petter Holmea,b,c,⇤
, Jari Saramäkid
a
IceLab, Department of Physics, Umeå University, 901 87 Umeå, Sweden
b
Department of Energy Science, Sungkyunkwan University, Suwon 440–746, Republic of Korea
c
Department of Sociology, Stockholm University, 106 91 Stockholm, Sweden
d
Department of Biomedical Engineering and Computational Science, School of Science, Aalto University, 00076 Aalto, Espoo, Finland
a r t i c l e i n f o
Article history:
Accepted 1 March 2012
Available online 6 March 2012
editor: D.K. Campbell
a b s t r a c t
A great variety of systems in nature, society and technology – from the web of sexual
contacts to the Internet, from the nervous system to power grids – can be modeled as
graphs of vertices coupled by edges. The network structure, describing how the graph is
wired, helps us understand, predict and optimize the behavior of dynamical systems. In
many cases, however, the edges are not continuously active. As an example, in networks
of communication via e-mail, text messages, or phone calls, edges represent sequences
of instantaneous or practically instantaneous contacts. In some cases, edges are active for
non-negligible periods of time: e.g., the proximity patterns of inpatients at hospitals can
be represented by a graph where an edge between two individuals is on throughout the
time they are at the same ward. Like network topology, the temporal structure of edge
activations can affect dynamics of systems interacting through the network, from disease
contagion on the network of patients to information diffusion over an e-mail network. In
1
ISBN 978-3-642-36460-0
Understanding Complex Systems
Petter Holme
Jari Saramäki Editors
Temporal
Networks
TemporalNetworksHolme·SaramäkiEds.
Understanding Complex Systems
Petter Holme · Jari Saramäki Editors
Temporal Networks
The concept of temporal networks is an extension of complex networks as a modeling
framework to include information on when interactions between nodes happen. Many
studies of the last decade examine how the static network structure affect dynamic
systems on the network. In this traditional approach the temporal aspects are pre-
encoded in the dynamic system model. Temporal-network methods, on the other hand,
lift the temporal information from the level of system dynamics to the mathematical
representation of the contact network itself. This framework becomes particularly
useful for cases where there is a lot of structure and heterogeneity both in the timings
of interaction events and the network topology. The advantage compared to common
static network approaches is the ability to design more accurate models in order to
explain and predict large-scale dynamic phenomena (such as, e.g., epidemic outbreaks
and other spreading phenomena). On the other hand, temporal network methods are
mathematically and conceptually more challenging. This book is intended as a first
introduction and state-of-the art overview of this rapidly emerging field.
Physics
9 7 8 3 6 4 2 3 6 4 6 0 0
Randomization
Randomization
Times shuffled
Original
1
0.8
0.6
0.4
0.2
0
0 100 200 300
Time (days)
Fractionofinfectious
Karsai, et al.,
PRE, 2011.
Randomization
Times shuffled
Original
1
0.8
0.6
0.4
0.2
0
0 100 200 300
Time (days)
Fractionofinfectious
Karsai, et al.,
PRE, 2011.
Random times
Times shuffled
Original
Contact sequences of
links shuffled among
links similar weight
Contact sequences
of links shuffled
1
0.8
0.6
0.4
0.2
0
0 100 200 300
Time (days)
Fractionofinfectious
Temporal structure
Temporal structure
Fat-tailed interevent time distributions
Slowing down of spreading.
10-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
Poisson
Power-law
Time
Incidence/N
Min, Goh, Vazquez, 2011. PRE 83, 036102.
But both the cell phone and
the prostitution data are
bursty. So why are they
different w.r.t. spreading?
Europhys. Lett., 64 (3), pp. 427–433 (2003)
EUROPHYSICS LETTERS 1 November 2003
Network dynamics of ongoing social relationships
P. Holme(∗
)
Department of Physics, Ume˚a University - 901 87 Ume˚a, Sweden
(received 21 July 2003; accepted in final form 22 August 2003)
PACS. 89.65.-s – Social and economic systems.
PACS. 89.75.Hc – Networks and genealogical trees.
PACS. 89.75.-k – Complex systems.
Abstract. – Many recent large-scale studies of interaction networks have focused on networks
of accumulated contacts. In this letter we explore social networks of ongoing relationships with
an emphasis on dynamical aspects. We find a distribution of response times (times between
consecutive contacts of different direction between two actors) that has a power law shape over a
large range. We also argue that the distribution of relationship duration (the time between the
first and last contacts between actors) is exponentially decaying. Methods to reanalyze the data
to compensate for the finite sampling time are proposed. We find that the degree distribution
for networks of ongoing contacts fits better to a power law than the degree distribution of
the network of accumulated contacts do. We see that the clustering and assortative mixing
coefficients are of the same order for networks of ongoing and accumulated contacts, and that
the structural fluctuations of the former are rather large.
Introduction. – The recent development in database technology has allowed researchers
to extract very large data sets of human interaction sequences. These large data sets are
suitable to the methods and modeling techniques of statistical physics, and thus, the last years
have witnessed the appearance of an interdisciplinary field between physics and sociology [1–3].
More specifically, these studies have focused on network structure —in what ways the networks
Limited communication capacity unveils
strategies for human interaction
Giovanna Miritello1,2
, Rube´n Lara2
, Manuel Cebrian3,4
& Esteban Moro1,5
1
Departamento de Matema´ticas & GISC, Universidad Carlos III de Madrid, 28911 Legane´s, Spain, 2
Telefo´nica Research, 28050
Madrid, Spain, 3
NICTA, Melbourne, Victoria 3010, Australia, 4
Department of Computer Science & Engineering, University of
California at San Diego, La Jolla, CA 92093, USA, 5
Instituto de Ingenierı´a del Conocimiento, Universidad Auto´noma de Madrid,
28049 Madrid, Spain.
Connectivity is the key process that characterizes the structural and functional properties of social networks.
However, the bursty activity of dyadic interactions may hinder the discrimination of inactive ties from large
interevent times in active ones. We develop a principled method to detect tie de-activation and apply it to a
large longitudinal, cross-sectional communication dataset (<19 months, <20 million people). Contrary to
the perception of ever-growing connectivity, we observe that individuals exhibit a finite communication
capacity, which limits the number of ties they can maintain active in time. On average men display higher
capacity than women, and this capacity decreases for both genders over their lifespan. Separating
communication capacity from activity reveals a diverse range of tie activation strategies, from stable to
exploratory. This allows us to draw novel relationships between individual strategies for human interaction
and the evolution of social networks at global scale.
any different forces govern the evolution of social relationships making them far from random. In recent
years, the understanding of what mechanisms control the dynamics of activating or deactivating social
SUBJECT AREAS:
SCIENTIFIC DATA
COMPLEX NETWORKS
APPLIED MATHEMATICS
STATISTICAL PHYSICS
Received
15 January 2013
Accepted
2 May 2013
Published
6 June 2013
Correspondence and
requests for materials
should be addressed to
E.M. (emoro@math.
time
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)
Ongoing link picture
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
Link turnover picture
T0
T0
Beginning interval neutralized
T0
T0
Interevent times neutralized
End interval neutralized
T0
T0
Reference models
SIR on prostitution data
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
durationofinfection
Ω
SIR on prostitution data
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
durationofinfection
Ω
Interevent times neutralized
SIR on prostitution data
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
durationofinfection
Ω
Beginning times neutralized
SIR on prostitution data
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
durationofinfection
Ω
End times neutralized
SIR, average deviations
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
Holme, Liljeros, 2014. Scientific Reports 4: 4999.
Temporal to static
Time-slice networks
tstart
tstop
0 5 10 15 20
1
2
3
4
5
6
t
1
2
3
4
5
6
Ongoing (concurrent) networks
tstart
tstop
0 5 10 15 20
1
2
3
4
5
6
t
1
2
3
4
5
6
Exponential threshold 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
Static Temporal
Evaluate by comparing ranking of vertices
Run SIR and measure
the expected outbreak
size Ωi if the i is the
source.
Measure predictors of
i’s importance. (Degree
ki and coreness ci.)
Calculate the rank correlation between Ωi and ki.
Higher correlation → better static representation.
E-mail 1
E-mail 2
Dating
Gallery
Conference
Prostitution
Results, Degree
Time-slice Ongoing Exponential-threshold Accumulated
Holme, 2013. PLoS Comput. Biol. 9: e1003142.
R₀
R₀ — basic reproductive number,
reproduction ratio, reproductive
ratio, ...
e expected number of secondary
infections of an infectious individual in
a population of susceptible individuals.
One of few concepts that
went from mathematical
to medical epidemiology
Disease R₀
Measles 12–18
Pertussis 12–17
Diphtheria 6–7
Smallpox 5–7
Polio 5–7
Rubella 5–7
Mumps 4–7
SARS 2–5
Influenza 2–4
Ebola 1–2
SIR model
ds
dt
= –βsi—
di
dt
= βsi – νi—
= νi
dr
dt
—
S I I I
I R
Ω = r(∞) = 1 – exp[–R₀ Ω]
where R₀ = β/ν
Ω > 0 if and only if R₀ > 1
e epidemic threshold
Problems with R₀
Hard to
estimate
Can be hard for
models
& even harder for outbreak data
and many datasets lack
the important early period
e threshold isn’t R₀ = 1 in practice
e meaning of a threshold in a finite population.
In temporal networks, the outbreak size
needn’t be a monotonous function of R₀
Problems with R₀
Hard to
estimate
Can be hard for
models
& even harder for outbreak data
and many datasets lack
the important early period
e threshold isn’t R₀ = 1 in practice
e meaning of a threshold in a finite population.
e topic of this project
In temporal networks, the outbreak size
needn’t be a monotonous function of R₀
Plan
Use empirical
contact data
Simulate the entire
parameter space of
the SIR model
Plot Ω vs R₀
Figure out what temporal network
structure that creates the deviations
1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Averageoutbreaksize,Ω
Basic reproductive number, R0
Conference
1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Averageoutbreaksize,Ω
Basic reproductive number, R0
Conference
0.001 0.01 0.1 1
1
0.1
0.01
0.001
transmission probability
diseaseduration
1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Averageoutbreaksize,Ω
Basic reproductive number, R0
Conference
Shape index (example)—
discordant pair separation in Ω
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Basic reproductive number, R0
Averageoutbreaksize,Ω
μΩ=0.304
ρΩ = 2.663
avg. fraction of nodes present when 50% of contact happened
avg. fraction of links present when 50% of contact happened
avg. fraction of nodes present at 50% of the sampling time
avg. fraction of links present at 50% of the sampling time
frac. of nodes present 1st and last 10% of the contacts
frac. of links present 1st and last 10% of the contacts
frac. of nodes present 1st and last 10% of the sampling time
frac. of links present 1st and last 10% of the sampling time
Time evolution
degree distribution, mean
degree distribution, s.d.
degree distribution, coefficient of variation
degree distribution, skew
Degree distribution
link duration, mean
link duration, s.d.
link duration, coefficient of variation
link duration, skew
link interevent time, mean
link interevent time, s.d.
link interevent time, coefficient of variation
link interevent time, skew
Link activity
Node activity
node duration, mean
node duration, s.d.
node duration, coefficient of variation
node duration, skew
node interevent time, mean
node interevent time, s.d.
node interevent time, coefficient of variation
node interevent time, skew
Other network structure
number of nodes
clustering coefficient
assortativity
Temporal network structure
Correlation between point-cloud shape & temporal
network structure
*
*
** ** ** **
**
*
**
**
**
*
∆R0
0
0.2
0.4
0.6
0.8
1
R²
Time evolution
Node activity Link activity
Degree
distribution
Network
structure
fLT
fNT
fLC
fNC
FLT
FNT
FLC
FNC
γNt
σNt
cNt
µNt
γNτ
σNτ
cNτ
µNτ
γLt
σLt
cLt
µLt
γLτ
σLτ
cLτ
µLτ
γk
σk
ck
µk
N C r
***
**
∆Ω
0
0.2
0.4
0.6
0.8
1
R²
Time evolution
Node activity
Link activity
Network
structure
fLT
fNT
fLC
fNC
FLT
FNT
FLC
FNC
γNt
σNt
cNt
µNt
γNτ
σNτ
cNτ
µNτ
γLt
σLt
cLt
µLt
γLτ
σLτ
cLτ
µLτ
γk
σk
ck
µk
N C r
Degreedistribution
Correlation between point-cloud shape & temporal
network structure
Holme & Masuda, 2015,
PLoS ONE 10:e0120567.
Beyond
epidemiology
Information / opinion spreading
“Viral videos doesn’t spread like viruses.”
Actors does not necessarily get infected by only one other.
Karimi, Holme, 2013. Physica A 392: 3476–3483.
reshold models:
Takaguchi, Masuda, Holme, 2013. PLoS ONE 8: e68629.
Time can be incorporated in many ways.
Major conclusion: burstiness can speed up spreading.
ank you!
ank you!
Collaborators:
Naoki Masuda
Jari Saramäki
Fredrik Liljeros
Luis Rocha
Sungmin Lee
Fariba Karimi
Juan Perotti
Taro Takaguchi
Hang-Hyun Jo
Illustrations by:
Mi Jin Lee

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Temporal Networks of Human Interaction

  • 1. Temporal networks of human interaction Petter Holme
  • 2.
  • 3. Ntriangles = 3 Among the people we study, there is a tendency for triangles to form.
  • 4. Ntriangles = 3 Rumors would spread slowly because of the many triangles.
  • 6.
  • 10. What are we interested in? Something that can: . . . be measured relatively easy (who are involved & when). . . . and give scientific insights. Examples: -Two persons being close to each other. -Two persons doing things together. -One person sending a message to another. Human interaction
  • 11. Two persons being close to each other -RFID tags. -Smartphones / Bluetooth. -Smartphones / GPS. -Campus Wi-fi. -Hospital records. -Co-tagged in images. -Public transportation. -Sexual contacts (via Internet mediated prostitution). Human interaction
  • 12. Two persons doing things together (not necessarily close) -Paper co-authorships. -Movie actors. -Criminal co-offenders. One person sending a message to another -E-mails. -Internet forums. -Instant messaging. Human interaction
  • 15. Numerical representations: Interval graphs ID1 2 6 2 10 7 3 5 2 7 10 ID2 4 8 8 11 2 5 3 10 3 2 time interval (10,15) (11,14) (18,19) (20,22) (20,24) (20,30) (25,31) (40,45) (40,50) (51,53)
  • 16. 0 5 10 15 20 1 2 3 4 5 6 t Graphical representations: Timelines of individuals
  • 20.
  • 21.
  • 23.
  • 24.
  • 25.
  • 26.
  • 27. transmission probability / rate after some time / with some chance per time unit susceptible infectious recovered / susceptible
  • 28. +
  • 31. Time matters Rocha, Liljeros, Holme, 2010. PNAS 107: 5706-5711. Escort/sex-buyer contacts: 16,730 individuals 50,632 contacts 2,232 days
  • 32. 1555 ID1 ID2 time 5 7 1021 20 9 1119 4 30 1539 ID1 ID2 time 5 7 1555 20 9 1021 4 30 1119 4 20 15394 20 Time matters Rocha, Liljeros, Holme, 2010. PNAS 107: 5706-5711. Escort/sex-buyer contacts: 16,730 individuals 50,632 contacts 2,232 days Rocha, Liljeros, Holme, 2011. PLoS Comp. Biol. 7: e1001109. 0 0.2 0.4 0.6 0 200 400 600 Fractionofinfectious Time (days) Empirical 800 0 0.2 0.4 0.6 0 200 400 600 Time (days) Empirical Randomized 800 Fractionofinfectious
  • 34. Time matters 0 0.2 0.4 0.6 0 200 400 600 Time (days) Empirical Randomized 800 Fractionofinfectious 1 0.8 0.6 0.4 0.2 0 0 100 200 300 Time (days) Fractionofinfectious Rocha, Liljeros, Holme Karsai, et al.
  • 35. Physics Reports 519 (2012) 97–125 Contents lists available at SciVerse ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Temporal networks Petter Holmea,b,c,⇤ , Jari Saramäkid a IceLab, Department of Physics, Umeå University, 901 87 Umeå, Sweden b Department of Energy Science, Sungkyunkwan University, Suwon 440–746, Republic of Korea c Department of Sociology, Stockholm University, 106 91 Stockholm, Sweden d Department of Biomedical Engineering and Computational Science, School of Science, Aalto University, 00076 Aalto, Espoo, Finland a r t i c l e i n f o Article history: Accepted 1 March 2012 Available online 6 March 2012 editor: D.K. Campbell a b s t r a c t A great variety of systems in nature, society and technology – from the web of sexual contacts to the Internet, from the nervous system to power grids – can be modeled as graphs of vertices coupled by edges. The network structure, describing how the graph is wired, helps us understand, predict and optimize the behavior of dynamical systems. In many cases, however, the edges are not continuously active. As an example, in networks of communication via e-mail, text messages, or phone calls, edges represent sequences of instantaneous or practically instantaneous contacts. In some cases, edges are active for non-negligible periods of time: e.g., the proximity patterns of inpatients at hospitals can be represented by a graph where an edge between two individuals is on throughout the time they are at the same ward. Like network topology, the temporal structure of edge activations can affect dynamics of systems interacting through the network, from disease contagion on the network of patients to information diffusion over an e-mail network. In
  • 36. 1 ISBN 978-3-642-36460-0 Understanding Complex Systems Petter Holme Jari Saramäki Editors Temporal Networks TemporalNetworksHolme·SaramäkiEds. Understanding Complex Systems Petter Holme · Jari Saramäki Editors Temporal Networks The concept of temporal networks is an extension of complex networks as a modeling framework to include information on when interactions between nodes happen. Many studies of the last decade examine how the static network structure affect dynamic systems on the network. In this traditional approach the temporal aspects are pre- encoded in the dynamic system model. Temporal-network methods, on the other hand, lift the temporal information from the level of system dynamics to the mathematical representation of the contact network itself. This framework becomes particularly useful for cases where there is a lot of structure and heterogeneity both in the timings of interaction events and the network topology. The advantage compared to common static network approaches is the ability to design more accurate models in order to explain and predict large-scale dynamic phenomena (such as, e.g., epidemic outbreaks and other spreading phenomena). On the other hand, temporal network methods are mathematically and conceptually more challenging. This book is intended as a first introduction and state-of-the art overview of this rapidly emerging field. Physics 9 7 8 3 6 4 2 3 6 4 6 0 0
  • 38. Randomization Times shuffled Original 1 0.8 0.6 0.4 0.2 0 0 100 200 300 Time (days) Fractionofinfectious Karsai, et al., PRE, 2011.
  • 39. Randomization Times shuffled Original 1 0.8 0.6 0.4 0.2 0 0 100 200 300 Time (days) Fractionofinfectious Karsai, et al., PRE, 2011. Random times Times shuffled Original Contact sequences of links shuffled among links similar weight Contact sequences of links shuffled 1 0.8 0.6 0.4 0.2 0 0 100 200 300 Time (days) Fractionofinfectious
  • 41. Temporal structure Fat-tailed interevent time distributions Slowing down of spreading. 10-12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Poisson Power-law Time Incidence/N Min, Goh, Vazquez, 2011. PRE 83, 036102. But both the cell phone and the prostitution data are bursty. So why are they different w.r.t. spreading?
  • 42.
  • 43.
  • 44. Europhys. Lett., 64 (3), pp. 427–433 (2003) EUROPHYSICS LETTERS 1 November 2003 Network dynamics of ongoing social relationships P. Holme(∗ ) Department of Physics, Ume˚a University - 901 87 Ume˚a, Sweden (received 21 July 2003; accepted in final form 22 August 2003) PACS. 89.65.-s – Social and economic systems. PACS. 89.75.Hc – Networks and genealogical trees. PACS. 89.75.-k – Complex systems. Abstract. – Many recent large-scale studies of interaction networks have focused on networks of accumulated contacts. In this letter we explore social networks of ongoing relationships with an emphasis on dynamical aspects. We find a distribution of response times (times between consecutive contacts of different direction between two actors) that has a power law shape over a large range. We also argue that the distribution of relationship duration (the time between the first and last contacts between actors) is exponentially decaying. Methods to reanalyze the data to compensate for the finite sampling time are proposed. We find that the degree distribution for networks of ongoing contacts fits better to a power law than the degree distribution of the network of accumulated contacts do. We see that the clustering and assortative mixing coefficients are of the same order for networks of ongoing and accumulated contacts, and that the structural fluctuations of the former are rather large. Introduction. – The recent development in database technology has allowed researchers to extract very large data sets of human interaction sequences. These large data sets are suitable to the methods and modeling techniques of statistical physics, and thus, the last years have witnessed the appearance of an interdisciplinary field between physics and sociology [1–3]. More specifically, these studies have focused on network structure —in what ways the networks
  • 45. Limited communication capacity unveils strategies for human interaction Giovanna Miritello1,2 , Rube´n Lara2 , Manuel Cebrian3,4 & Esteban Moro1,5 1 Departamento de Matema´ticas & GISC, Universidad Carlos III de Madrid, 28911 Legane´s, Spain, 2 Telefo´nica Research, 28050 Madrid, Spain, 3 NICTA, Melbourne, Victoria 3010, Australia, 4 Department of Computer Science & Engineering, University of California at San Diego, La Jolla, CA 92093, USA, 5 Instituto de Ingenierı´a del Conocimiento, Universidad Auto´noma de Madrid, 28049 Madrid, Spain. Connectivity is the key process that characterizes the structural and functional properties of social networks. However, the bursty activity of dyadic interactions may hinder the discrimination of inactive ties from large interevent times in active ones. We develop a principled method to detect tie de-activation and apply it to a large longitudinal, cross-sectional communication dataset (<19 months, <20 million people). Contrary to the perception of ever-growing connectivity, we observe that individuals exhibit a finite communication capacity, which limits the number of ties they can maintain active in time. On average men display higher capacity than women, and this capacity decreases for both genders over their lifespan. Separating communication capacity from activity reveals a diverse range of tie activation strategies, from stable to exploratory. This allows us to draw novel relationships between individual strategies for human interaction and the evolution of social networks at global scale. any different forces govern the evolution of social relationships making them far from random. In recent years, the understanding of what mechanisms control the dynamics of activating or deactivating social SUBJECT AREAS: SCIENTIFIC DATA COMPLEX NETWORKS APPLIED MATHEMATICS STATISTICAL PHYSICS Received 15 January 2013 Accepted 2 May 2013 Published 6 June 2013 Correspondence and requests for materials should be addressed to E.M. (emoro@math.
  • 48. T0 T0 Beginning interval neutralized T0 T0 Interevent times neutralized End interval neutralized T0 T0 Reference models
  • 49. SIR on prostitution data 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 durationofinfection Ω
  • 50. SIR on prostitution data 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 durationofinfection Ω Interevent times neutralized
  • 51. SIR on prostitution data 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 durationofinfection Ω Beginning times neutralized
  • 52. SIR on prostitution data 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 durationofinfection Ω End times neutralized
  • 53. SIR, average deviations 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 Holme, Liljeros, 2014. Scientific Reports 4: 4999.
  • 55. Time-slice networks tstart tstop 0 5 10 15 20 1 2 3 4 5 6 t 1 2 3 4 5 6
  • 56. Ongoing (concurrent) networks tstart tstop 0 5 10 15 20 1 2 3 4 5 6 t 1 2 3 4 5 6
  • 57. Exponential threshold 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
  • 58. Static Temporal Evaluate by comparing ranking of vertices Run SIR and measure the expected outbreak size Ωi if the i is the source. Measure predictors of i’s importance. (Degree ki and coreness ci.) Calculate the rank correlation between Ωi and ki. Higher correlation → better static representation.
  • 59. E-mail 1 E-mail 2 Dating Gallery Conference Prostitution Results, Degree Time-slice Ongoing Exponential-threshold Accumulated Holme, 2013. PLoS Comput. Biol. 9: e1003142.
  • 60. R₀
  • 61. R₀ — basic reproductive number, reproduction ratio, reproductive ratio, ... e expected number of secondary infections of an infectious individual in a population of susceptible individuals.
  • 62. One of few concepts that went from mathematical to medical epidemiology
  • 63. Disease R₀ Measles 12–18 Pertussis 12–17 Diphtheria 6–7 Smallpox 5–7 Polio 5–7 Rubella 5–7 Mumps 4–7 SARS 2–5 Influenza 2–4 Ebola 1–2
  • 64. SIR model ds dt = –βsi— di dt = βsi – νi— = νi dr dt — S I I I I R Ω = r(∞) = 1 – exp[–R₀ Ω] where R₀ = β/ν Ω > 0 if and only if R₀ > 1 e epidemic threshold
  • 65. Problems with R₀ Hard to estimate Can be hard for models & even harder for outbreak data and many datasets lack the important early period e threshold isn’t R₀ = 1 in practice e meaning of a threshold in a finite population. In temporal networks, the outbreak size needn’t be a monotonous function of R₀
  • 66. Problems with R₀ Hard to estimate Can be hard for models & even harder for outbreak data and many datasets lack the important early period e threshold isn’t R₀ = 1 in practice e meaning of a threshold in a finite population. e topic of this project In temporal networks, the outbreak size needn’t be a monotonous function of R₀
  • 67. Plan Use empirical contact data Simulate the entire parameter space of the SIR model Plot Ω vs R₀ Figure out what temporal network structure that creates the deviations
  • 68. 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Averageoutbreaksize,Ω Basic reproductive number, R0 Conference
  • 69. 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Averageoutbreaksize,Ω Basic reproductive number, R0 Conference 0.001 0.01 0.1 1 1 0.1 0.01 0.001 transmission probability diseaseduration
  • 70. 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Averageoutbreaksize,Ω Basic reproductive number, R0 Conference
  • 71. Shape index (example)— discordant pair separation in Ω 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Basic reproductive number, R0 Averageoutbreaksize,Ω μΩ=0.304 ρΩ = 2.663
  • 72. avg. fraction of nodes present when 50% of contact happened avg. fraction of links present when 50% of contact happened avg. fraction of nodes present at 50% of the sampling time avg. fraction of links present at 50% of the sampling time frac. of nodes present 1st and last 10% of the contacts frac. of links present 1st and last 10% of the contacts frac. of nodes present 1st and last 10% of the sampling time frac. of links present 1st and last 10% of the sampling time Time evolution degree distribution, mean degree distribution, s.d. degree distribution, coefficient of variation degree distribution, skew Degree distribution link duration, mean link duration, s.d. link duration, coefficient of variation link duration, skew link interevent time, mean link interevent time, s.d. link interevent time, coefficient of variation link interevent time, skew Link activity Node activity node duration, mean node duration, s.d. node duration, coefficient of variation node duration, skew node interevent time, mean node interevent time, s.d. node interevent time, coefficient of variation node interevent time, skew Other network structure number of nodes clustering coefficient assortativity Temporal network structure
  • 73. Correlation between point-cloud shape & temporal network structure * * ** ** ** ** ** * ** ** ** * ∆R0 0 0.2 0.4 0.6 0.8 1 R² Time evolution Node activity Link activity Degree distribution Network structure fLT fNT fLC fNC FLT FNT FLC FNC γNt σNt cNt µNt γNτ σNτ cNτ µNτ γLt σLt cLt µLt γLτ σLτ cLτ µLτ γk σk ck µk N C r
  • 74. *** ** ∆Ω 0 0.2 0.4 0.6 0.8 1 R² Time evolution Node activity Link activity Network structure fLT fNT fLC fNC FLT FNT FLC FNC γNt σNt cNt µNt γNτ σNτ cNτ µNτ γLt σLt cLt µLt γLτ σLτ cLτ µLτ γk σk ck µk N C r Degreedistribution Correlation between point-cloud shape & temporal network structure Holme & Masuda, 2015, PLoS ONE 10:e0120567.
  • 76. Information / opinion spreading “Viral videos doesn’t spread like viruses.” Actors does not necessarily get infected by only one other. Karimi, Holme, 2013. Physica A 392: 3476–3483. reshold models: Takaguchi, Masuda, Holme, 2013. PLoS ONE 8: e68629. Time can be incorporated in many ways. Major conclusion: burstiness can speed up spreading.
  • 78. ank you! Collaborators: Naoki Masuda Jari Saramäki Fredrik Liljeros Luis Rocha Sungmin Lee Fariba Karimi Juan Perotti Taro Takaguchi Hang-Hyun Jo Illustrations by: Mi Jin Lee