The document discusses local variation analysis of hashtag spike trains on Twitter. It defines local variation (LV) as the ratio of the difference between the inter-event interval of the forward event and the inter-event interval of the backward event, at discrete time points of the spike train. The document suggests that LV can help characterize whether a time series is Poissonian or not, and can provide insights into the dynamics of hashtag propagation on Twitter.

1. • A typical snapsho
The white spots
of the beads floa
waves.
cedaysan@gmail.com
http://fcxn.wordpress.com
http://xn.unamur.be
r driving want to be mobile. As a
witter users collectively advertise and
orm groups to move together. Both
f-organize and create dynamic
ty.
e, the interpretation of the dynamic
eity of the beads in a critical limit
elp to characterize viral memes
(#hashtags) in twitter.
Refs:
1 C. Sanlı et al. (a
2 L. Berthier (201
• A typical snapshot o
The white spots indi
of the beads floating
waves.
cedaysan@gmail.com
http://fcxn.wordpress.com
http://xn.unamur.be
driving want to be mobile. As a
ter users collectively advertise and
m groups to move together. Both
organize and create dynamic
the interpretation of the dynamic
ty of the beads in a critical limit
p to characterize viral memes
hashtags) in twitter.
Refs:
1 C. Sanlı et al. (arXi
2 L. Berthier (2011).
Local Variation of Collective Attention
in Hashtag Spike Trains
!
(collaboration with Renaud Lambiotte)5mm
• Inset: The displacement
field demonstrates local
heterogeneities in the flow.
• A typical snapshot of an experiment:
The white spots indicate the positions
of the beads floating on surface
waves.
cedaysan@gmail.com
http://fcxn.wordpress.com
http://xn.unamur.be
es social
sages and
As a
ertise and
er. Both
mic
s:
dynamic
al limit
mes
Refs:
1 C. Sanlı et al. (arXiv - 2013).
2 L. Berthier (2011).
International AAAI Conference on Weblogs and Social Media (ICWSM-15) Workshop 3: Modeling and Mining
Temporal Interactions, 26th May 2015, Oxford, the UK.
@CeydaSanli

2. Online
SEPT. 29, 2014
Photo
Credit
Tomi Um
Brendan Nyhan
Information Diffusion in Twitter
Local Variation Spike Trains 1C. Sanli, CompleXity Networks, UNamur
• tweets
• retweets
• mentions
Tomi Um
• following -
followers

3. y Rumors Outrace the Truth
ne
014
Brendan Nyhan
Hashtag Diffusion in Twitter
Local Variation Spike Trains 2C. Sanli, CompleXity Networks, UNamur
Tomi Um
• hashtag
• hashtag spike train
time
count

4. What do we address in this talk?
Local Variation Spike Trains 3C. Sanli, CompleXity Networks, UNamur
• How can we measure local temporal
behaviour of the hashtag diffusion?
• Is there a difference in the dynamics
between popular and less used
hashtags?
• Can we measure (and predict) collective
attention by the hashtag dynamics?

5. Key Results: Local Variation
Local Variation Spike Trains 4C. Sanli, CompleXity Networks, UNamur
0
5
10
15
20
25
30
15
20
25
30
<p>=91127
<p>=18553
<p>= 1678
<p>= 318
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
<p>=91127
<p>=18553
<p>= 1678
<p>= 318
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
P()LVP()LV
Real activity(a)
Random activity(b)
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
LV(t1)
LV(t2)
0.4
0.6
0.8
1
(LV(t1),LV(t2))
(a)
(b)
<p
<p
<p
<p
<p
<p
<p
<p
<p
<p
bursty regular
0
5
10
15
20
0 1 2 3 4 5
0
5
10
15
20
25
30
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
<p>=91127
<p>=18553
<p>= 1678
<p>= 318
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
LV
P()LVP()LV
Random activity(b)
FIG. 7. Probability density function (PDF) of the local vari-
ation LV of real hashtag propagation (a) and random hash-
tag time sequence (b). Two distinct shapes are visible: (a)
From high p to low p, the peak position of P(LV ) shifts from
low values of LV to higher values of LV . (b) P(LV ) always
0
5
10
15
20
0 1 2 3 4 5
0
5
10
15
20
25
30
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
<p>=91127
<p>=18553
<p>= 1678
<p>= 318
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
LV
P()LVP()LV
Random activity(b)
FIG. 7. Probability density function (PDF) of the local vari-
ation LV of real hashtag propagation (a) and random hash-
tag time sequence (b). Two distinct shapes are visible: (a)
From high p to low p, the peak position of P(LV ) shifts from
low values of LV to higher values of LV . (b) P(LV ) always
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
LV(t1)
LV(t2
101
102
103
104
105
0
0.2
0.4
0.6
0.8
1
<p>
r(LV(t1),LV(t2))
(b)
<
<
<
<
<
bursty regular
FIG. 8. Linear correlation of LV through real hashta
trains. (a) The linear relation of the first and the
halves of the empirical spike trains, LV (t1) and LV
spectively, are investigated. The legend ranks hpi in d
colors and symbols. (b) The Pearson correlation co
r(LV (t1), LV (t2)) between these quantities show tha
the temporal correlation through moderately popular
is maximum, r reaches the minimum values for both
• hashtag dynamics • artificial dynamics
• collective attention
• C. Sanlı and R. Lambiotte, PLoS ONE 10(7): e0131704 (2015).

6. Hashtag Spike Train
Local Variation Spike Trains 5C. Sanli, CompleXity Networks, UNamur
time
count
t t0 f
f a time delay between successive events, inter-event interva
dent events, the distrubution of inter-event interval is Poissonia
are observed and therefore forward propogation of a signal is a
ry. Thus, quantifying ⌧ is crucial.
is an alternative way to characterize whether a time series is P
For a stationarly process, Lv is a ratio of the di↵erence be
of forward event and the inter-event interval of backward eve
ent intervals. Suppose that a signal propogates in distinct tim
. . . ⌧N . Then, at ⌧i, the inter-event interval of forward event i
r-event interval of backward event is ⌧i = ⌧i ⌧i 1. Conseq
0 50 100 150 200 250 300 350
0
5
1
τh
(hour)
h
< r
h
>= 41
< r
h
>= 35
es versus life time (⌧h) of the corresponsi
domly selected #hashtag activity from r
τ
h
(hour)
FIG. 2. Rank of #hashtag rh versus life ti
2
. . .
τ
h
(hour)
FIG. 2. Rank of #hashtag rh versus life ti
2
hpi =
p

7. Control Parameters
Local Variation Spike Trains 6C. Sanli, CompleXity Networks, UNamur
t
t
0
f
150 200 250 300 350
τh
(hour)
h
< r
h
>= 35
time (⌧h) of the corresponsing
d #hashtag activity from real
τ
h
(hour)
ashtag rh versus life time of #hashtag ⌧h.
2
• .
• .
• .
• .
• .
: number of spikes
: initial time
: final time
: life time
= popularity
τ
h
(hour)
ashtag rh versus life time of #hashtag ⌧h.
2
hpi =
p
clude interaction among agents. Considering online social
ess self-organized optimizing of popularity of information.
of #hashtag propogation, user activity, and user #hashtag
y between successive events, inter-event interval ⌧, is a
he distrubution of inter-event interval is Poissonian. If not,
and therefore forward propogation of a signal is a function
ntifying ⌧ is crucial.
ive way to characterize whether a time series is Poissonian
arly process, Lv is a ratio of the di↵erence between the
: inter-hashtag spike interval

8. Circadian Pattern and Local Signal
Local Variation Spike Trains 7C. Sanli, CompleXity Networks, UNamur

9. Driving Factors in our Twitter Network
Local Variation Spike Trains 8C. Sanli, CompleXity Networks, UNamur
1. circadian human behaviour (internal)
2. political election (external)
+ complex decision-making (both internal
and external)

10. Data Set
Local Variation Spike Trains 9C. Sanli, CompleXity Networks, UNamur
• 9 days of the French election 2012 (May 5th),
• total activity ~ 10 million, hashtag activity~ 3
million, unique hashtags ~ 300.000,
•
!
!
!
• number of total users ~ 475.000,
• number of users tweet or retweet any
hashtags at least ones ~ 230.000.
#ledebat 180946
#hollande 143636
#sarkozy 116906
#votehollande 99908
#france2012 20635
#fh2012 67759

11. Top Most Used Hashtags
Local Variation Spike Trains 10C. Sanli, CompleXity Networks, UNamur
DAILY CYCLE OF #HASHTAGS
00:0012:0000:0012:0000:0012:0000:0012:0000:0012:0000:0012:0000:00
0
10
20
30
40
50
60
70
80
hour
count/min.
#ledebat
#hollande
#sarkozy
#votehollande
#fh2012
#france2012
debate election

12. Local Variation Spike Trains 11C. Sanli, CompleXity Networks, UNamur
Statistics of Hashtags

13. Heterogeneity in Popularity
Local Variation Spike Trains 12C. Sanli, CompleXity Networks, UNamur
3
popu-
quiva-
d the
5,697
f-plot
htags
= 1.
n the
f Fig.
old of
5% of
nally,
ctan-
than
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
0
10
1
10
2
10
3
10
4
10
5
rank hashtag
popularity:p 10
6
(a)
P(p)
83%
0.15%
0.0001% 60%
• C. Sanlı and R. Lambiotte, PLoS ONE 10(7): e0131704 (2015).

14. Heterogeneity in Time
Local Variation Spike Trains 13C. Sanli, CompleXity Networks, UNamur
10−1 100 101 102 10310−3
10−2
10−1
100
101
102
103
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
<p>= 11
<p>= 2
0.8
0.85
0.9
∆τ (hour)
P()∆τCDF
(b)
12 hours
1 day
2 days
3 days
FIG. 3. The cumulative (a), CDF( ⌧), and probability (b),
P( ⌧), distributions of the inter-hashtag spike intervals. We
#hash2#hash3
merged
#hash
ficial
ash
10−3
10−2
10−1
100
101
102
103
<p>=91127
<p>=18553
<p>= 1678
<p>= 318
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
<p>= 11
<p>= 2
0.8
0.85
0.9
0.95
1
P()∆τCDF()∆τ
(a)
(b)
12 hours
1 day
2 days
3 days
#hash1#hash2#hash3
merged
#hash
2
103
<p>=91127
<p>=18553
<p>= 1678
<p>= 318
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
<p>= 11
<p>= 2
0.8
0.85
0.9
0.95
1
CDF()∆τ
popular hashtags
• C. Sanlı and R. Lambiotte, PLoS ONE 10(7): e0131704 (2015).

15. Local Variation Spike Trains 14C. Sanli, CompleXity Networks, UNamur
Local Analysis on Hashtag
Spike Trains

16. Local Variation
Local Variation Spike Trains 15C. Sanli, CompleXity Networks, UNamur
time
count
For a stationarly process, Lv is a
terval of forward event and the int
these inter-event intervals. Suppose
⌧1 . . . , ⌧i 1, ⌧i, ⌧i+1, . . . ⌧N . Then, at
⌧i+1 ⌧i and the inter-event interval o
is
For a stationarly process, Lv is a
terval of forward event and the inter
these inter-event intervals. Suppose
⌧1 . . . , ⌧i 1, ⌧i, ⌧i+1, . . . ⌧N . Then, at ⌧i
⌧i+1 ⌧i and the inter-event interval of
is
✓
For a stationarly process, Lv is a r
terval of forward event and the inter
these inter-event intervals. Suppose t
⌧1 . . . , ⌧i 1, ⌧i, ⌧i+1, . . . ⌧N . Then, at ⌧i,
⌧i+1 ⌧i and the inter-event interval of b
is
N 1X ✓
For a stationarly process, Lv is a
terval of forward event and the int
these inter-event intervals. Suppose
⌧1 . . . , ⌧i 1, ⌧i, ⌧i+1, . . . ⌧N . Then, at ⌧
⌧i+1 ⌧i and the inter-event interval o
is
N 1X ✓
a stationarly process, Lv is a ratio of the di↵erenc
of forward event and the inter-event interval of ba
nter-event intervals. Suppose that a signal propog
⌧i 1, ⌧i, ⌧i+1, . . . ⌧N . Then, at ⌧i, the inter-event inter
⌧i and the inter-event interval of backward event is ⌧
N 1X ✓ ◆2
ocess, Lv is a ratio of the di↵erence between the inter-event in-
and the inter-event interval of backward event to the sum of
als. Suppose that a signal propogates in distinct time such as
⌧N . Then, at ⌧i, the inter-event interval of forward event is ⌧i+1 =
vent interval of backward event is ⌧i = ⌧i ⌧i 1. Consequently, Lv
2
N 1X
i=2
✓
(⌧i+1 ⌧i) (⌧i ⌧i 1)
(⌧i+1 ⌧i) + (⌧i ⌧i 1)
◆2
=
✓
⌧i+1 ⌧i
⌧i+1 + ⌧i
◆2
. (1)
earance of a time series in distinct times. Multiple activity in same
• K. Miura et al. Neural Computation 18, 2359-2386 (2006).
• S. Shinomoto et al. Neural Computation 15, 2823-2842 (2003).
from the nonstationarity of the hashtag propagation. Simila
sed on this distribution, such as its variance or Fano factor,
similar way. For this reason, we consider here the so-called l
y defined to determine intrinsic temporal dynamics of neuro
].
antities such as P( ⌧), LV compares temporal variations w
specifically defined for nonstationary processes [27]
LV =
3
N 2
N 1X
i=2
✓
(⌧i+1 ⌧i) (⌧i ⌧i 1)
(⌧i+1 ⌧i) + (⌧i ⌧i 1)
◆2
he total number of spikes and . . ., ⌧i 1, ⌧i, ⌧i+1, . . . represent
e of a single hashtag spike train. Eq. 1 also takes the form [
LV =
3
N 2
N 1X
i=2
✓
⌧i+1 ⌧i
⌧i+1 + ⌧i
◆2

17. Limits of Local Variation
Local Variation Spike Trains 16C. Sanli, CompleXity Networks, UNamur
v =
3
N 2
N 1X
i=2
✓
(⌧i+1 ⌧i) (⌧i ⌧i
(⌧i+1 ⌧i) + (⌧i ⌧i
total appearance of a time series in d
any #hashtags at least ones = 2
tion 2012
one and tweets of any language
c-
p
1
c-
n
y
s,
,
s
y,
-
LV =
3
N 2
N 1X
i=2
✓
(⌧i+1 ⌧i) (⌧i ⌧i 1)
(⌧i+1 ⌧i) + (⌧i ⌧i 1)
◆2
(1)
Here, N is the total number of spikes and . . ., ⌧i 1, ⌧i,
⌧i+1, . . . represents successive time sequence of a single
hashtag spike train. Eq. 1 also takes the form [27]
LV =
3
N 2
N 1X
i=2
✓
⌧i+1 ⌧i
⌧i+1 + ⌧i
◆2
(2)
where ⌧i+1 = ⌧i+1 ⌧i and ⌧i = ⌧i ⌧i 1. ⌧i+1 quan-
tifies forward delay and ⌧i represents backward waiting
time for an event at ⌧i. Importantly, the denominator
normalizes the quantity such as to account for local vari-
ations of the rate at which events take place. By defini-
tion, LV takes values in the interval [0:3].
The local variation L presents properties making it
time
count
time
count
bursty >1
regular <1
random=1

18. Classification of Spike Trains
Local Variation Spike Trains 17C. Sanli, CompleXity Networks, UNamur
• S. Shinomoto et al. PLoS Comput. Biol. 15, 2823-2842 (2003).
e collected from awake,
our of the 15 areas were
sets were generated in
used to record neuronal
nter-trial intervals. All
perimentation were in
onal Institutes of Health
ent committee at the
nts were performed.
d spike train for each
task trial periods and
ring rate differs greatly.
000 ISIs, or those with
ignored; 1,307 neurons
computed for the entire
for each neuron. They
for analyzing fractional
n individual neuron was
mputed for 20 fractional
the spike data.
In comparison with Cv, local metrics, such as Lv, SI, Cv2, and IR,
detect firing irregularities fairly invariantly with firing rate
fluctuations. However, these metrics are still somewhat dependent
on firing rate fluctuations. Assuming that rate dependence is
caused by the refractory period that follows a spike, we can
Figure 1. Spike sequences that have identical sets of inter-
spike intervals. Intervals are aligned (A) in a regular order, (B)
randomly, and (C) alternating between short and long.
doi:10.1371/journal.pcbi.1000433.g001
ploscompbiol.org 2 July 2009 | Volume 5 | Issue 7 | e1000433
ts were performed.
d spike train for each
task trial periods and
ing rate differs greatly.
000 ISIs, or those with
ignored; 1,307 neurons
computed for the entire
for each neuron. They
or analyzing fractional
individual neuron was
puted for 20 fractional
he spike data.
Figure 1. Spike sequences that have identical sets of inter-
spike intervals. Intervals are aligned (A) in a regular order, (B)
randomly, and (C) alternating between short and long.
doi:10.1371/journal.pcbi.1000433.g001
oscompbiol.org 2 July 2009 | Volume 5 | Issue 7 | e1000433
dual neurons can be
nalyzed differences in
rtical areas and found a
t closely corresponded
area; neuronal firing is
er-order motor areas,
prefrontal area. Thus,
al areas that may be
utations.
collected from awake,
ur of the 15 areas were
sets were generated in
sed to record neuronal
nter-trial intervals. All
erimentation were in
nal Institutes of Health
nt committee at the
ts were performed.
d spike train for each
task trial periods and
ng rate differs greatly.
000 ISIs, or those with
ignored; 1,307 neurons
computed for the entire
for each neuron. They
instantaneous ISI variability, SI, the geometric average of the
rescaled cross-correlation of ISIs [37,38], Cv2, the coefficient of
variation for a sequence of two ISIs [39], and IR, the difference of
the log ISIs [34] were also used.
Figure 1 displays three types of spike sequences comprising
identical sets of exponentially distributed ISIs. In terms of the ISI
distributions, all of these are regarded as Poisson processes,
accordingly Cv values are all identical at 1. However, these
sequences clearly differ in how their ISIs are arranged; Lv may be
able to detect these differences.
In comparison with Cv, local metrics, such as Lv, SI, Cv2, and IR,
detect firing irregularities fairly invariantly with firing rate
fluctuations. However, these metrics are still somewhat dependent
on firing rate fluctuations. Assuming that rate dependence is
caused by the refractory period that follows a spike, we can
LV =
N 2 i=2
(⌧i+1 ⌧i) + (⌧i ⌧i 1)
e, N is the total number of spikes and . . ., ⌧i 1, ⌧i, ⌧i+1, . . . represen
e sequence of a single hashtag spike train. Eq. 1 also takes the form [
LV =
3
N 2
N 1X
i=2
✓
⌧i+1 ⌧i
⌧i+1 + ⌧i
◆2
ere ⌧i+1 = ⌧i+1 ⌧i and ⌧i = ⌧i ⌧i 1. ⌧i+1 quantifies forward
resents backward waiting time for an event at ⌧i. Importantly, the de
malizes the quantity such as to account for local variations of the rat
nts take place. By definition, LV takes values in the interval [0:3].
The local variation LV presents properties making it an interesting can
lysis of hashtag spike trains [23–27]. In particular, LV is on average eq
random process is either a stationary or a non-stationary Poisson pro
only condition that the time scale over which the firing rate ⇠(t) fluct
n the typical time between spikes. Deviations from 1 originate from l
relations in the underlying signal, either under the form of pairwise c
=1.4
Unlike quantities such as P( ⌧), LV compares temporal variations w
es and is specifically defined for nonstationary processes [27]
LV =
3
N 2
N 1X
i=2
✓
(⌧i+1 ⌧i) (⌧i ⌧i 1)
(⌧i+1 ⌧i) + (⌧i ⌧i 1)
◆2
e, N is the total number of spikes and . . ., ⌧i 1, ⌧i, ⌧i+1, . . . represen
e sequence of a single hashtag spike train. Eq. 1 also takes the form [
LV =
3
N 2
N 1X
i=2
✓
⌧i+1 ⌧i
⌧i+1 + ⌧i
◆2
ere ⌧i+1 = ⌧i+1 ⌧i and ⌧i = ⌧i ⌧i 1. ⌧i+1 quantifies forward
resents backward waiting time for an event at ⌧i. Importantly, the de
malizes the quantity such as to account for local variations of the rat
nts take place. By definition, LV takes values in the interval [0:3].
The local variation LV presents properties making it an interesting can
lysis of hashtag spike trains [23–27]. In particular, LV is on average eq
random process is either a stationary or a non-stationary Poisson pro
=1.0
cted in a similar way. For this reason, we consider here the so-called
, originally defined to determine intrinsic temporal dynamics of neuro
ns [23–27].
Unlike quantities such as P( ⌧), LV compares temporal variations w
es and is specifically defined for nonstationary processes [27]
LV =
3
N 2
N 1X
i=2
✓
(⌧i+1 ⌧i) (⌧i ⌧i 1)
(⌧i+1 ⌧i) + (⌧i ⌧i 1)
◆2
e, N is the total number of spikes and . . ., ⌧i 1, ⌧i, ⌧i+1, . . . represen
e sequence of a single hashtag spike train. Eq. 1 also takes the form [
LV =
3
N 2
N 1X
i=2
✓
⌧i+1 ⌧i
⌧i+1 + ⌧i
◆2
ere ⌧i+1 = ⌧i+1 ⌧i and ⌧i = ⌧i ⌧i 1. ⌧i+1 quantifies forward
resents backward waiting time for an event at ⌧i. Importantly, the de
malizes the quantity such as to account for local variations of the rat
=0.1
bursty
irregular!
(random)
regular

19. Analysis of Hashtag Spike Trains
Local Variation Spike Trains 18C. Sanli, CompleXity Networks, UNamur
time
#hash1
time
time
time
#hash2#hash3
merged
#hash

20. Generating Random Spike Trains
Local Variation Spike Trains 19C. Sanli, CompleXity Networks, UNamur
time
merged
#hash
htags. Each activity time gives us a spike in a merged
we set the appearance to 1 even though we observe mu
in a second.
mization is satisfied by a random permutation. We us
f MatLab such as randperm(T, p). Here, T represents th
pike train which we keep all information of real #hash
rank, the total exact appearance, of individual #hashtag
uniformly distributed unique numbers out of T. For ou
5
time
artificially
generated
#hash
B. Local variable of #hashtag s
For a stationarly process, Lv is a
terval of forward event and the int
these inter-event intervals. Suppose
⌧1 . . . , ⌧i 1, ⌧i, ⌧i+1, . . . ⌧N . Then, at
⌧i+1 ⌧i and the inter-event interval o
is
B. Local variable of #hashtag sp
For a stationarly process, Lv is a
terval of forward event and the inter
these inter-event intervals. Suppose
⌧1 . . . , ⌧i 1, ⌧i, ⌧i+1, . . . ⌧N . Then, at ⌧i
⌧i+1 ⌧i and the inter-event interval of
is
B. Local variable of #hashtag sp
For a stationarly process, Lv is a r
terval of forward event and the inter
these inter-event intervals. Suppose t
⌧1 . . . , ⌧i 1, ⌧i, ⌧i+1, . . . ⌧N . Then, at ⌧i,
⌧i+1 ⌧i and the inter-event interval of b
is
B. Local variable of #hashtag s
For a stationarly process, Lv is a
terval of forward event and the int
these inter-event intervals. Suppose
⌧1 . . . , ⌧i 1, ⌧i, ⌧i+1, . . . ⌧N . Then, at
⌧i+1 ⌧i and the inter-event interval o
is
Local variable of #hashtag spike trains
a stationarly process, Lv is a ratio of the di↵erenc
of forward event and the inter-event interval of b
nter-event intervals. Suppose that a signal propog
⌧i 1, ⌧i, ⌧i+1, . . . ⌧N . Then, at ⌧i, the inter-event inter
⌧i and the inter-event interval of backward event is ⌧
[ ]
=

21. Distribution of Local Variation
Local Variation Spike Trains 20C. Sanli, CompleXity Networks, UNamur
0
5
10
15
20
25
30
15
20
25
30
<p>=91127
<p>=18553
<p>= 1678
<p>= 318
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
<p>=91127
<p>=18553
<p>= 1678
<p>= 318
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
P()LVP()LV
Real activity(a)
Random activity(b)
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
LV(t1)
LV(t2)
0.4
0.6
0.8
1
(LV(t1),LV(t2))
(a)
(b)
<p
<p
<p
<p
<p
<p
<p
<p
<p
<p
bursty regular
0
5
10
15
20
0 1 2 3 4 5
0
5
10
15
20
25
30
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
<p>=91127
<p>=18553
<p>= 1678
<p>= 318
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
LV
P()LVP()LV
Random activity(b)
FIG. 7. Probability density function (PDF) of the local vari-
ation LV of real hashtag propagation (a) and random hash-
tag time sequence (b). Two distinct shapes are visible: (a)
From high p to low p, the peak position of P(LV ) shifts from
low values of LV to higher values of LV . (b) P(LV ) always
0
5
10
15
20
0 1 2 3 4 5
0
5
10
15
20
25
30
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
<p>=91127
<p>=18553
<p>= 1678
<p>= 318
<p>= 174
<p>= 117
<p>= 86
<p>= 68
<p>= 56
<p>= 47
<p>= 41
<p>= 35
LV
P()LVP()LV
Random activity(b)
FIG. 7. Probability density function (PDF) of the local vari-
ation LV of real hashtag propagation (a) and random hash-
tag time sequence (b). Two distinct shapes are visible: (a)
From high p to low p, the peak position of P(LV ) shifts from
low values of LV to higher values of LV . (b) P(LV ) always
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
LV(t1)
LV(t2
101
102
103
104
105
0
0.2
0.4
0.6
0.8
1
<p>
r(LV(t1),LV(t2))
(b)
<
<
<
<
<
bursty regular
FIG. 8. Linear correlation of LV through real hashta
trains. (a) The linear relation of the first and the
halves of the empirical spike trains, LV (t1) and LV
spectively, are investigated. The legend ranks hpi in d
colors and symbols. (b) The Pearson correlation co
r(LV (t1), LV (t2)) between these quantities show tha
the temporal correlation through moderately popular
is maximum, r reaches the minimum values for both
• hashtag dynamics • artificial dynamics
• C. Sanlı and R. Lambiotte, PLoS ONE 10(7): e0131704 (2015).

22. Statistics of Local Variation
Local Variation Spike Trains 21C. Sanli, CompleXity Networks, UNamur
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
real
random
µ()LV
real hashtags:
decay in µ
with increasing p
30
40
50
es
µ (LV) = 1
real
random
0
(a)
1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
log (<p>)10
real hashtags:
decay in µ
with increasing p
−20
−10
0
10
20
30
40
50z−values
µ (LV) = 1
real
random
0
µ0µ =
(a)
(b)
bursty
regular
• C. Sanlı and R. Lambiotte, PLoS ONE 10(7): e0131704 (2015).

23. Local Variation Spike Trains 22C. Sanli, CompleXity Networks, UNamur
Local Variation of Collective Attention

24. Trains in Collective Attention
Local Variation Spike Trains 23C. Sanli, CompleXity Networks, UNamur
8 am 12 pm 4 pm 8 pm 0 am4 am0 am
hour
debate day
regular day
election day
hashtagspiketrains:#ledebat
• #ledebat: debate at
7-11 pm

25. Local Variation of Collective Attention
Local Variation Spike Trains 24C. Sanli, CompleXity Networks, UNamur
0
0.5
1
1.5
2
2.5
3
#ledebat
#hollande
#sarkozy
#votehollande
#avecsarkozy
#ledebat
(a) debate day
0 am 3 am 6 am 9 am 12 pm3 pm 6 pm 9 pm 0 am
0
500
1000
1500
2000
2500
3000
3500
4000
(b)
0 am 3 am 6 am 9 am 12 pm3 pm 6 pm 9 pm 0 am
#hollande
#sarkozy
0 am 3 am 6 am 9 am 12 pm3 pm 6 pm 9 pm 0 am
regular day election day
#ledebat
#hollande
#sarkozy
#votehollande
#avecsarkozy
#ledebat
hour hour hour
L(t)Vtweetcount/hour
announcement
of the result
at 7 pm
debate
at 7-11 pm

26. What we understand
Local Variation Spike Trains 25C. Sanli, CompleXity Networks, UNamur
• Local variation is very simple, but powerful to
quantify local temporal characteristics of complex
hashtag dynamics.
• There is a direct relation between the popularity and
the inter-hashtag spike interval: While popular
hashtags present regular activation, less used
hashtags indicate bursty spiking.
• Collective attention can be determined by the local
variation: From random irregular spiking to regular
signal. Further detail analysis will help to predict
collective attention.

27. • Online analysis tool for the optimisation of social
media campaigns (EU project),
Acknowledgement
Local Variation Spike Trains 26C. Sanli, CompleXity Networks, UNamur
ceday
http://fcx!
Month 6 General Meeting
htt
• Restricted amount of sources forces social
and physical systems to present
emergence of order.
• Twitter users want to spread their messages and
beads under driving want to be mobile. As a
result, the twitter users collectively advertise and
the beads form groups to move together. Both
systems self-organize and create dynamic
heterogeneity.
Therefore, the interpretation of the dynamic
heterogeneity of the beads in a critical limit
would help to characterize viral memes
(#hashtags) in twitter.
Refs:
1 H. Simon (1971).
2 L. Weng et al. (2012).
3 J. P. Gleeson et al. (2014).
0 12 24 36 48 60 72 84
0
10
20
30
40
time (hours)
numberoftweets/unitt
!
Month 6 General Meeting
beads under driving want to be
result, the twitter users collecti
the beads form groups to mov
systems self-organize and cre
heterogeneity.
Therefore, the interpretatio
heterogeneity of the beads in
would help to characterize
(#hashtags) in twit
Refs:
1 H. Simon (1971).
2 L. Weng et al. (2012).
3 J. P. Gleeson et al. (2014).
0 12 24 36 48 60 72 84
0
10
time (hours)
number
• FNRS (le Fonds de la Recherche Scientifique),
Wallonie, Belgium.
• Takaaki Aoki (Kagawa University, Japan) and
Taro Takaguchi (Nat. Inst. of Informatics, Japan),