Statistical Physics of Complex Dynamics

Hang-Hyun Jo
Asia Paciﬁc Center for Theoretical Physics, Republic of Korea
Dept. of Physics, Pohang University of Science and Technology, Republic of Korea
Statistical Physics of
Complex Dynamics

Junior Research Group
• Title: Statistical Physics of Complex Dynamics (CoDy)
• Period: May 1, 2017—April 30, 2022
• Members: Hang-Hyun Jo (leader),
Byoung-Hwa Lee (PhD student),
Takayuki Hiraoka (postdoc, since July)

Statistical Physics
→ Search for the laws in
many-body systems
(e.g., atom, spin, cell, species, human)

Micro-macro link
→ Macroscopic patterns
emergent from
microscopic elements

Interaction structure of
many-body systems
→ Graph or Network
(e.g., lattice, random graph, complex network)

Real world networks
10
0
10
1
10
2
10
10
6
10
4
10
2
10
0
100 102 104 106 108
10
10 12
10 10
10 8
10 6
10 4
10 2
vi vj
Oij=0 Oij=1/3
Oij=1Oij=2/3
A B
<O>
w
,<O>
b
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
P
cum
(w), P
cum
(b)
C D
Degree k Link weight w (s)
P(k)
P(w)
Fig. 1. Characterizing the large-scale structure and the tie strengths of the
mobile call graph. (A and B) Vertex degree (A) and tie strength distribution (B).
Each distribution was ﬁtted with P(x) ϭ a(x ϩ x0)Ϫx exp(Ϫx/xc), shown as a blue
A
B
1
100
10
Internet by K. C. Claffy
Protein-protein interaction
by H. Jeong
Mobile phone user network
by J.-P. Onnela
Thanks to the large-scale digital datasets or “Big Data”

• Node: elements in a system
• Link: interaction between elements

Local property Global property
Topological
property
Heterogeneous degree
Assortativity
Local clustering
etc.
Small-world effect
Community structure
etc.
Intensity-related
property
Heterogeneous weight
and strength
Neighborhood overlap
etc.
“Strength of weak tie”
hypothesis
etc.
Network properties

3
TABLE I. Stylized facts derived from various datasets with the expected behaviors for the whole social network [20, 21].
The symbol % (&) implies that the overall trend is monotonically increasing (decreasing). The initially increasing and then
decreasing behavior is denoted by %&. For the Granovetterian community structure, see the main text for the details.
Category Property or measure Stylized fact (expectation)
Topological Degree distribution, P(k) & (%&)
Average degree of neighbors as a function of degree, knn(k) %
Local clustering coe cient as a function of degree, c(k) &
Community size distribution, P(g) &
Intensity-related Strength distribution, P(s) & (%&)
Weight distribution, P(w) &
Strength as a function of degree, s(k) %
Neighborhood overlap as a function of weight, o(w) %
Granovetterian community structure, fc > 0
corresponding to ⇠ k2
i in Eq. (3), is typically stronger
than that of ﬁnding new links between neighbors, in re-
lation to ei in Eq. (3). For example, if every new neighbor
of a node i creates a new link to one of node i’s existing
neighbors, then ei ⇠ ki, leading to c(k) ⇠ k 1
. This be-
havior can be measured in terms of the PCC between ci
and ki, which is denoted hereafter as ⇢ck.
The intensities are correlated with topological proper-
ties, which can be called intensity-topology correlation
or weight-topology correlation as used in [9]. A link-
level consequence of intensity-topology correlation can be
measured by the average neighborhood overlap for links
with weight w, denoted by o(w). The neighborhood over-
lap of a link is the fraction of common neighbors of neigh-
Stylized facts in social networks
Jo et al., arXiv:1611.03664 (2016)

Temporal information on
interaction events
→ Temporal network
(e.g., neural network, mobile call network)

Why temporal networks?
• Directionality due to the time asymmetry
• Speed of collective dynamics depends on the
interaction activity.
P. Holme, J. Saramäki / Physics Reports 519 (2012) 97–125 99
the reachability issue and the intransitivity of temporal networks (more specifically a contact sequence). In (a), the times of the
ces A–D are indicated on the edges. Assume that, for example, a disease starts spreading at vertex A and spreads further as soon
e dashed lines and vertices show this spreading process for four different times. The spreading will not continue further than what
1 picture, i.e. D cannot get infected. However, if the spreading started at vertex D, the entire set of vertices would eventually be
he edges into one static graph cannot capture this effect that arises from the time ordering of contacts. Panel (b) visualizes the same
he temporal dimension explicitly. The colors of the lines in (b) match the vertex colors in (a).
e time evolution of the network structure in these windows. Such an approach does not cover all aspects
ructure of contact patterns. For example, the edges between vertices of temporal networks need not be
networks, whether directed or not, if A is directly connected to B and B is directly connected to C, then A
Holme, Saramaki,
Physics Reports (2012)
Information from D reaches A, but not the other way around!

event
time
burst
communitiesindividuals
Threshold, t = 0.20
t = 0.27
0.4
D
0.6 0.8 1
Largest
subcommunity
Remaining
hierarchy
t
0 0.2 0.4 0.6 0.8
Word association
2 0.4 0.6 0.8
drogram threshold, t
Metabolic
Largest
community
Second
largest
Third largest
Ahnetal.,Nature(2010)
Jo (in preparation)

Temporal self-similarity
Ward & Greenwood, Scholarpedia (2007)
~ 1/f noise

Bursts ~ 1/f noise
Nature (2005)
Poissonian
non-Poissonian

earthquakes
neuronal firings
mobile phone calls
Karsai et al., Sci. Rep. (2012)
Bursts in diverse datasets

Topics in bursty dynamics
• Data, measures, and theories [Karsai, Jo, Kaski (Book to be published)]
• De-seasoning method [Jo, Karsai, Kertesz, Kaski, NJP (2012)]
• Novel burstiness measure [Kim, Jo, PRE (2016)]
• Correlated bursts [Jo, Perotti, Kaski, Kertesz, PRE (2015)]
• Contextual bursts [Jo, Pan, Perotti, Kaski, PRE (2013)]
• Effect of bursts on spreading [Jo, Perotti, Kaski, Kertesz, PRX (2014)]

Inter-event times
⌧⌧
time
P(⌧) ⇠ ⌧ ↵
inter-event time distribution
burstiness parameter
Goh & Barabási, EPL (2008)
B =
h⌧i
+ h⌧i
1  B  1

Origin of bursts in human
behavior?

Why? Priority queuing
Barabási, Nature (2005)
e-mail
time
priority
small
waiting
time
large
waiting
time
waiting time

Cyclic “Poisson” process
Malmgren et al., PNAS (2008)
time-varying rate with weekly cycle for e-mail usage
heavy tail of inter-event time distribution
Are weekly cycles the ONLY reason for bursts?

De-seasoning cycles?
Jo et al., NJP (2012)
mobile call sequence of one user
: weekly cycle (T=7 days)⇢(t)
: no cyclic patterns⇢⇤
(t⇤
) = 1
de-seasoned by weekly cycle
B7 = 0.146
B0 = 0.224
B =
h⌧i
+ h⌧i

Bursts are robust!
Burstiness remains finite after
de-seasoning weekly cycles.
burstiness
de-seasoning period (days)
different activity groupGolden moles have a blue-green sheen to their
coats that is a rare example of iridescence in
mammals, report Matthew Shawkey at the
University of Akron in Ohio and his colleagues.
The group conducted the first detailed study of
iridescent outer hairs and non-iridescent downy
hairs from four species of golden mole. Iridescent
hairs were highly flattened with much smaller
scales than their less eye-catching counterparts.
The scales form multiple layers, which alternate
in colour between light and dark, and probably
produce colour as light passes between layers in a
phenomenon called thin-film interference.
All four mole species are blind, so it is unlikely
that the hairs evolved as sexual ornamentation.
The authors suggest that the iridescence of
these burrowing animals is a by-product of
adaptations for durable, low-friction pelts.
Biol. Lett. http://dx.doi.org/10.1098/rsbl.2011.1168
(2012)
EVOLUTION
Glad rags for a blind mole
cannot currently be cultured,
their genomes may soon be
accessible.
Until now, metagenomic
analyses have been able
to identify only dominant
members of a microbial
community or those sequenced
previously. Virginia Armbrust
and her group at the University
of Washington in Seattle
developed computational
tools to tame the massive
amount of data produced by
next-generation sequencers.
The method successfully
sequenced two of 14 candidate
genomes identified in samples
from Puget Sound, most
notably a microbe of low
abundance but great interest
— a representative of the
mysterious, as yet uncultured
organisms known as marine
group II Euryarchaeota.
Researchers now have a way
to peer into the secret lives of
the uncultured majority.
Science 335, 587–590 (2012)
CANCER DRUGS
Chemo spans
generations
Some commonly used cancer
drugs not only generate
mutations in treated mice,
but scar the genomes of their
NETWORKS
Patchy
communication
People tend to communicate
with each other in bursts,
exchanging clusters of
messages over short time
periods, and following
these up with longer gaps in
communication. But are these
patterns simply the result of a
tendency to talk more during
the day and the working week?
Hang-Hyun Jo of Aalto
University in Finland and
his colleagues found that
these temporal cycles are not
sufficient to explain the bursts.
They analysed 322 million
mobile-phone calls between
more than 5 million users
over 119 days in 2007. After
removing the effects of the
day–night and working-week
cycles, the bursts remained.
The authors suggest that
the patterns reflect something
fundamental in the way that
people communicate.
N. J. Phys. 14, 013055 (2012)
P
C.PFEIFFER&P.HALEY
MATERIALS
Printing tiny
coiled antennas
Typically, the largest circuit
component in wireless
electronic devices such as
mobile phones is the antenna,
which sends and receives
electromagnetic waves. The
tiniest antennas available are
made up of wires twisted into
three-dimensional coils to save
on space while maintaining
high radiation efficiency and
wide bandwidth. But bending
wires is cumbersome and
expensive.
Stephen Forrest and
Anthony Grbic at the
University of Michigan in Ann
Arbor and their colleagues
report a way to rapidly transfer
metallic patterns directly onto
a curved polymer, which can
be pre-moulded to a desired
shape. Stamping the pattern
onto a hemispherical polymer,
for instance, produces
miniature high-performance
antennas curled in spherical
helices (pictured).
Adv. Mater. http://dx.doi.org/10.
1002/adma/201104290 (2012)
Nature (2012)

How to measure
higher-order correlations?

Memory coefﬁcient
⌧⌧
time
Goh & Barabási, EPL (2008)
M =
h(⌧i h⌧i)(⌧i+1 h⌧i)i
2
K.-I. Goh and A. L. Barab´asi
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1
B
M
a
heartbeat
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
B
M
b
human
activities
texts
natural
phenomena
Fig. 4: (Color online) (a) The (M, B) phase diagram. Human activities (red) are captured by activity patte
email (⋆) [5], library loans (◦) [7], and printing ( ) [28] of individuals in Universities, call center record at an
( ) [29], and phone initiation record from a mobile phone company (⋄). Data for natural phenomena (black
records in Japan (•) [26] and daily precipitation records in New Mexico, USA ( ) [27]. Data for written tex

Bursty train size distribution
E = 1E = 5 E = 4E = 2 E = 1 E = 2
time
t
P t(E) ⇠ E
Karsai et al., Sci. Rep. (2012)
P t(E) ⇠ e E/Ec
for uncorrelated inter-event times
for highly correlated inter-event times
: correlated bursts

Autocorrelation function
x(t) = 0, 1
time
A(td) =
hx(t)x(t + td)i hxi2
2
x
⇠ td

Figure 2.1: Schematic diagramme of an event sequence, where each vertical line
indicates the timing of the event. (a) The inter-event time ⌧ is the time interval
between two consecutive events. The residual time ⌧r is the time interval from
a random moment (e.g., the timing annotated by the vertical arrow) to the next
event. (b) For a given time window t, a bursty train is determined by a set
of events separated by ⌧  t, while events in di↵erent trains are separated by
⌧ > t. The number of events in each train, i.e., burst size, is denoted by E. In
most empirical datasets, both distributions of ⌧ and E are heavy-tailed.
for i = 1, · · · , n 1. From this we construct the sequence of inter-event times, i.e.,
{⌧1, · · · , ⌧n 1}. By ignoring the order of inter-event times, we obtain the inter-
event time distribution P(⌧). For the completely regular time series, all inter-event
times are the same as the mean inter-event time denoted by h⌧i. The inter-event
time distribution then reads P(⌧) = (⌧ h⌧i), where (·) denotes the Dirac delta
function. Here the standard deviation of inter-event times, denoted by , is zero.
P(⌧) ⇠ ⌧ ↵
A(td) ⇠ tdP t(E) ⇠ E
⇠ td for 0 < < 1, then one ﬁnds the scaling
f) ⇠ f ⌘
with
⌘ = 1 . (4)
n the interevent times are i.i.d. random variables
P(⌧) ⇠ ⌧ ↵
, implying no interdependency between
event times, the power-law exponent ⌘ is obtained
unction of ↵ as follows [11, 12]:
⌘ =
8
<
:
↵ 1 for 1 < ↵  2,
3 ↵ for 2 < ↵  3,
0 for ↵ > 3.
(5)
bining Eqs. (4) and (5), we have
↵ + = 2 for 1 < ↵  2, (6)
↵ = 2 for 2 < ↵  3. (7)
e power-law exponents can also be related via Hurst
nent H, i.e., = 2 2H [13] or ⌘ = 2H 1 [12, 14].
for uncorrelated inter-event times
What if inter-event times are correlated?

Other result: Exponential P(E)
Two-state Markov-chain Poisson nature of individual cellphone call statistics
Jiang et al., J. Stat. Mech. (2016)

PHYSICAL REVIEW E 94, 032311 (2016)
Measuring burstiness for finite event sequences
Eun-Kyeong Kim1
and Hang-Hyun Jo2,3,*
1
GeoVISTA Center, Department of Geography, Pennsylvania State University, PA 16802, USA
2
BK21plus Physics Division and Department of Physics, Pohang University of Science and Technology, Pohang 37673, Republic of Korea
3
Department of Computer Science, Aalto University School of Science, P. O. Box 15500, Espoo, Finland
(Received 4 April 2016; published 15 September 2016)
Characterizing inhomogeneous temporal patterns in natural and social phenomena is important to understand
underlying mechanisms behind such complex systems and, hence, even to predict and control them. Temporal
inhomogeneities in event sequences have been described in terms of bursts that are rapidly occurring events in
short time periods alternating with long inactive periods. The bursts can be quantified by a simple measure, called
the burstiness parameter, which was introduced by Goh and Barabási [Europhys. Lett. 81, 48002 (2008)]. The
burstiness parameter has been widely used due to its simplicity, which, however, turns out to be strongly affected
by the finite number of events in the time series. As the finite-size effects on burstiness parameter have been
largely ignored, we analytically investigate the finite-size effects of the burstiness parameter. Then we suggest
an alternative definition of burstiness that is free from finite-size effects and yet simple. Using our alternative
burstiness measure, one can distinguish the finite-size effects from the intrinsic bursty properties in the time
series. We also demonstrate the advantages of our burstiness measure by analyzing empirical data sets.
DOI: 10.1103/PhysRevE.94.032311
I. INTRODUCTION spatiotemporal organization of aftershocks in seismology [23].
In addition, higher-order correlations between interevent times
Novel burstiness measure

Burstiness parameter
• regular time series:
• Poisson or random time series:
• bursty time series:
r = 0, B = 1
r = 1, B = 0
r ! 1, B ! 1
B =
µ
+ µ
=
r 1
r + 1
r =
µ
: coefficient of variation (CV)
➜ only when # of events is infinite!
(µ = h⌧i)

Motivation
• # of events = n
• All empirical datasets have ﬁnite n.
• Elements of small n have been arbitrarily ignored.
• How to isolate ﬁnite-size effects from intrinsic
bursty dynamics?

Single burst model
t1 tn
n events
: total period
: lower bound of inter-event time
inter-event times under
periodic boundary condition:
we conclude
n events, each
time interval
ings, and the
i = 1, · · · , n.
condition in
oundary con-
ent times are
(2)
ime distribu-
= 1
FIG. 1. Schematic diagram of the localized model: n events
are localized in the period beginning at t0 in [0, T), and
they are separated from each other at least by ⌧0.
B. Localized model
We now consider the general case that all events are
localized in the interval [t0, t0 + ) with t0 0 and
t0+ < T, indicating that events do not take place in the
intervals [0, t0) and [t0 + , T), as depicted in Fig. 1. A
similar model has been studied in a di↵erent context [27].
The localization parameter is introduced to simulate
the bursty limit for ⌧ T. Since we use periodic bound-
ary condition, t0 can be ignored. In addition, the lower
bound of interevent time, ⌧0, is introduced, implying that
events must be separated from each other at least by ⌧0.
Accordingly, it is assumed that
(n 1)⌧0   T ⌧0, (8)
leading to ⌧0  T
n . If ⌧0 = T
n , one gets the regular time
series.
⌧i =
(
T tn + t1 if i = 1
ti ti 1 if i = 2, · · · , n

Order statistics analysis
placed by ⌘ (n 1)⌧0 to define interevent times
as
⌧i ⌘
⇢
⌧1, + T if i = 1
⌧i, + ⌧0 if i 6= 1.
(9)
Using Eq. (3), we get
h⌧ii =
⇢
T n 1
n+1 ( + 2⌧0) if i = 1
+2⌧0
n+1 if i 6= 1,
(10)
and
h⌧2
i i =
(
6 2
(n+1)(n+2) + 4(T )
n+1 + (T )2
if i = 1
2 2
(n+1)(n+2) + 2⌧0
n+1 + ⌧2
0 if i 6= 1.
(11)
Then we calculate the mean µn and the variance 2
n of
interevent times to get the coe cient of variation rn =
n
µn
:
µn = 1
n [h⌧1i + (n 1)h⌧i6=1i], (12)
2
n = 1
n [h⌧2
1 i + (n 1)h⌧2
i6=1i] µ2
n, (13)
rn(x, y) =
q
(n 1)[1+n(1 x)2
+n(n+1)y2
2n(2 x)y]
n+1 .(14)
Here we have defined
(4)
(5)
(6)
each
tisfy
erage
(7)
tion.
as
⌧i ⌘
⇢
⌧1, + T if i = 1
⌧i, + ⌧0 if i 6= 1.
(9)
h⌧ii =
⇢
T n 1
n+1 ( + 2⌧0) if i = 1
+2⌧0
n+1 if i 6= 1,
(10)
and
h⌧2
i i =
(
6 2
(n+1)(n+2) + 4(T )
n+1 + (T )2
if i = 1
2 2
(n+1)(n+2) + 2⌧0
n+1 + ⌧2
0 if i 6= 1.
(11)
n of
n
µn
:
µn = 1
n [h⌧1i + (n 1)h⌧i6=1i], (12)
2
n = 1
n [h⌧2
1 i + (n 1)h⌧2
i6=1i] µ2
n, (13)
rn(x, y) =
q
(n 1)[1+n(1 x)2
+n(n+1)y2
2n(2 x)y]
n+1 .(14)
x ⌘ T , y ⌘ ⌧0
T , (15)
discussed in Appendix A. Interevent times are
s
⌧i,d ⌘
⇢
d tn + t1 if i = 1
ti ti 1 if i 6= 1.
(2)
rder statistics [25, 26], interevent time distribu-
written as follows:
P(⌧i,d) =
8
<
:
(⌧1,d/d)(1 ⌧1,d/d)n 2
B(2,n 1)d if i = 1
(1 ⌧i,d/d)n 1
B(1,n)d if i 6= 1,
(3)
(n, m) denotes the beta function,
m) =
Z 1
0
zn 1
(1 z)m 1
dz = (n 1)!(m 1)!
(n+m 1)! . (4)
ion values of ⌧i,d and ⌧2
i,d are obtained as
h⌧i,di =
⇢ 2d
n+1 if i = 1
d
n+1 if i 6= 1,
(5)
(
the bursty limit for ⌧ T. Since we us
ary condition, t0 can be ignored. In ad
bound of interevent time, ⌧0, is introduc
events must be separated from each oth
Accordingly, it is assumed that
(n 1)⌧0   T ⌧0
leading to ⌧0  T
n . If ⌧0 = T
n , one gets
series.
Then, we use definitions in Eq. (2)
placed by ⌘ (n 1)⌧0 to define
as
⌧i ⌘
⇢
⌧1, + T if i =
⌧i, + ⌧0 if i 6=
h⌧ii =
⇢
T n 1
n+1 ( + 2⌧0) if
+2⌧0
n+1 if
and
h⌧2
i i =
(
6 2
(n+1)(n+2) + 4(T )
n+1 + (T
2 2
+ 2⌧0
+ ⌧2
0
h⌧ii =
⇢
T n 1
n+1 ( + 2⌧0) if i = 1
+2⌧0
n+1 if i 6= 1,
(10)
and
h⌧2
i i =
(
6 2
(n+1)(n+2) + 4(T )
n+1 + (T )2
if i = 1
2 2
(n+1)(n+2) + 2⌧0
n+1 + ⌧2
0 if i 6= 1.
(11)
n of
n
µn
:
µn = 1
n [h⌧1i + (n 1)h⌧i6=1i], (12)
2
n = 1
n [h⌧2
1 i + (n 1)h⌧2
i6=1i] µ2
n, (13)
rn(x, y) =
q
(n 1)[1+n(1 x)2
+n(n+1)y2
2n(2 x)y]
n+1 .(14)
x ⌘ T , y ⌘ ⌧0
T , (15)
!
. (4)
s
(5)
(6)
of each
satisfy
average
(7)
ection.
h⌧ii =
⇢
T n 1
n+1 ( + 2⌧0) if i = 1
+2⌧0
n+1 if i 6= 1,
(10)
and
h⌧2
i i =
(
6 2
(n+1)(n+2) + 4(T )
n+1 + (T )2
if i = 1
2 2
(n+1)(n+2) + 2⌧0
n+1 + ⌧2
0 if i 6= 1.
(11)
n of
n
µn
:
µn = 1
n [h⌧1i + (n 1)h⌧i6=1i], (12)
2
n = 1
n [h⌧2
1 i + (n 1)h⌧2
i6=1i] µ2
n, (13)
rn(x, y) =
q
(n 1)[1+n(1 x)2
+n(n+1)y2
2n(2 x)y]
n+1 .(14)
x ⌘ T , y ⌘ ⌧0
T , (15)
CV:
h⌧i,di =
⇢ 2d
n+1 if i = 1
d
n+1 if i 6= 1,
(5)
di =
(
6d2
(n+1)(n+2) if i = 1
2d2
(n+1)(n+2) if i 6= 1.
(6)
ed that ⌧i,ds are independent of each
they are not independent but to satisfy
n
i=1 ⌧i,d = d. Instead we find on average
i = h⌧1,di + (n 1)h⌧i6=1,di = d. (7)
e discussed later in the next Subsection.
i +2⌧0
n+1 if i 6= 1
and
h⌧2
i i =
(
6 2
(n+1)(n+2) + 4(T )
n+1 + (T )2
2 2
(n+1)(n+2) + 2⌧0
n+1 + ⌧2
0
Then we calculate the mean µn and the var
interevent times to get the coe cient of var
n
µn
:
µn = 1
n [h⌧1i + (n 1)h⌧i6=1i],
2
n = 1
n [h⌧2
1 i + (n 1)h⌧2
i6=1i] µ2
n,
rn(x, y) =
q
(n 1)[1+n(1 x)2
+n(n+1)y2
2n
n+1
x ⌘ T , y ⌘ ⌧0
T ,
satisfying the conditions that
(n 1)y  x  1 y, y  1
n . (16)
It is straightforward to show that rn(x, y) is a non-
increasing function of x and y, respectively.
In order to study the strong bias due to the finite size
of event sequences, we define the burstiness parameter
using x and y as follows:
Bn(x, y) ⌘ rn(x,y) 1
rn(x,y)+1 . (17)
satisfying the conditions that
(n 1)y  x  1 y, y  1
n .
It is straightforward to show that rn(x, y) i
In order to study the strong bias due to the
of event sequences, we define the burstiness p

Reference cases
• regular time series:
• random time series:
• extremely bursty time series:
hat ⌧i,ds are independent of each
are not independent but to satisfy
i,d = d. Instead we find on average
⌧1,di + (n 1)h⌧i6=1,di = d. (7)
cussed later in the next Subsection.
n
µn
:
µn = 1
n [h⌧1i + (n 1)h⌧i6=1i],
2
n = 1
n [h⌧2
1 i + (n 1)h⌧2
i6=1i] µ2
n,
rn(x, y) =
q
(n 1)[1+n(1 x)2
+n(n+1)y2
2n(2 x)
n+1
x ⌘ T , y ⌘ ⌧0
T ,
(n 1)y  x  1 y, y  1
n . (16)
It is straightforward to show that rn(x, y) is a non-
rn(x,y)+1 . (17)
We discuss three reference cases. Firstly, the regular time
series means that all interevent times are the same as
µn, implying that x = 1 1
n and y = 1
n . Since rn = 0
independent of n, we get
Bn(1 1
n , 1
n ) = 1. (18)
Secondly, the Poissonian or random time series corre-
sponds to the case with = T and ⌧0 = 0, i.e., x = 1
and y = 0, leading to rn =
q
n 1
n+1 . We get
Bn(1, 0) =
p
n 1
p
n+1p
n 1+
p
n+1
. (19)
Note that B1(1, 0) = 1 and that Bn(1, 0) is always nega-
tive but approaches 0 as n increases, i.e., Bn(1, 0) ⇡ 1
2n
for large n, as shown in Fig. 2(a). Since this result
FIG. 2. (Color onlin
three reference cases:
for random time serie
merical results for the
to the analytic result
ness parameter An(r
parameter B(r) in Eq
plot of B(r) for indivi
but using Twitter dat
C. Novel defi
rn(x,y)+1 . (17)
We discuss three reference cases. Firstly, the regular time
series means that all interevent times are the same as
µn, implying that x = 1 1
n and y = 1
n . Since rn = 0
independent of n, we get
Bn(1 1
n , 1
n ) = 1. (18)
q
n 1
n+1 . We get
Bn(1, 0) =
p
n 1
p
n+1p
n 1+
p
n+1
. (19)
2n
is based on the assumption of independence of ⌧is, we
test our result by comparing it to numerical values of
burstiness parameter. For this, we generate 105
event se-
quences for each n to obtain the burstiness parameter as
FIG. 2. (Color on
three reference case
for random time ser
merical results for th
to the analytic resu
ness parameter An
parameter B(r) in E
plot of B(r) for indi
but using Twitter d
C. Novel de
In order to fix
rameter due to th
a novel definition
by An(r), where i
Bn(1 1
n , 1
n ) = 1. (18)
q
n 1
n+1 . We get
Bn(1, 0) =
p
n 1
p
n+1p
n 1+
p
n+1
. (19)
2n
is based on the assumption of independence of ⌧is, we
test our result by comparing it to numerical values of
burstiness parameter. For this, we generate 105
event se-
quences for each n to obtain the burstiness parameter as
depicted in Fig. 2(a). We find that the deviation of our
analytic results from the simulations is negligible.
Finally, the extremely bursty time series corresponds
to the case that all events occur asymptotically at the
same time, i.e., x = y = 0, leading to rn =
p
n 1. Thus
one gets
Bn(0, 0) =
p
n 1 1p
n 1+1
. (20)
Note that B1(0, 0) = 1 and B2(0, 0) = 0. Bn(0, 0) be-
comes positive for n 3, and then approaches 1 as n
FIG. 2. (Color onlin
three reference cases:
for random time serie
merical results for the
to the analytic result
ness parameter An(r
parameter B(r) in Eq
plot of B(r) for indivi
but using Twitter dat
C. Novel defi
In order to fix th
rameter due to the
a novel definition o
by An(r), where it
r = n
µn
. Then An(
tions:
A
A
which correspond t
extremely bursty ti
was originally defin
An(r) = anr bn
r+cn
wit
ying the conditions that
(n 1)y  x  1 y, y  1
n . (16)
straightforward to show that rn(x, y) is a non-
sing function of x and y, respectively.
order to study the strong bias due to the finite size
nt sequences, we define the burstiness parameter
x and y as follows:
rn(x,y) 1

Novel burstiness measure
which correspond to the cases of regular, random, and
extremely bursty time series, respectively. Since B(r)
was originally defined as r 1
r+1 , we similarly assume that
An(r) = anr bn
r+cn
with coe cients an, bn, and cn. Using a
general formula of Eq. (B2) in Appendix B, we get
An(r) =
p
n+1r
p
n 1
(
p
n+1 2)r+
p
n 1
(22)
for 0  r 
p
n 1. Our novel burstiness parameter
An has no longer a upper bound due to the finite n, as
depicted in Fig. 2(b), where the curves for di↵erent ns
are described by
An(r) =
p
n+1
p
n 1+(
p
n+1+
p
n 1)B(r)
p
n+1+
p
n 1 2+(
p
n+1
p
n 1 2)B(r)
. (23)
Then let us consider two event sequences with the same
value of r but with di↵erent ns. The original burstiness
nce cases: Eq. (18) for regular time series, Eq. (19)
time series, and Eq. (20) for the bursty limit. Nu-
ults for the random case are plotted for comparison
ytic results. (b) Comparison of the novel bursti-
eter An(r) in Eq. (22) to the original burstiness
B(r) in Eq. (1) for several values of n. (c) Scatter
) for individual Twitter users. (d) The same as (b)
Twitter dataset.
Novel definition of burstiness parameter
to fix the bias in the original burstiness pa-
ue to the finite number of events, we suggest
finition of the burstiness parameter, denoted
where it is assumed to be a function of only
Then An(r) must satisfy the following condi-
An(0) = 1,
An
⇣q
n 1
n+1
⌘
= 0, (21)
An
p
n 1 = 1,
respond to the cases of regular, random, and
bursty time series, respectively. Since B(r)
ally defined as r 1
r+1 , we similarly assume that
nr bn
r+cn
with coe cients an, bn, and cn. Using a
mula of Eq. (B2) in Appendix B, we get
p p
: regular time series
: random time series
: bursty time series
3
(16)
non-
e size
meter

Correlated bursts
Correlated bursts and the role of memory range
Hang-Hyun Jo,1,2
Juan I. Perotti,2,3
Kimmo Kaski,2
and János Kertész2,4
1
BK21plus Physics Division and Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea
2
Department of Computer Science, School of Science, Aalto University, P.O. Box 15500, Espoo, Finland
3
IMT Institute for Advanced Studies Lucca, Piazza San Francesco 19, I-55100 Lucca, Italy
4
Center for Network Science, Central European University, Nádor utca 9, H-1051 Budapest, Hungary
(Received 8 May 2015; published 20 August 2015)
Inhomogeneous temporal processes in natural and social phenomena have been described by bursts that are
rapidly occurring events within short time periods alternating with long periods of low activity. In addition to the
analysis of heavy-tailed interevent time distributions, higher-order correlations between interevent times, called
correlated bursts, have been studied only recently. As the underlying mechanism behind such correlated bursts is
far from being fully understood, we devise a simple model for correlated bursts using a self-exciting point process
with a variable range of memory. Whether a new event occurs is stochastically determined by a memory function
that is the sum of decaying memories of past events. In order to incorporate the noise and/or limited memory
capacity of systems, we apply two memory loss mechanisms: a fixed number or a variable number of memories.
By analysis and numerical simulations, we find that too much memory effect may lead to a Poissonian process,
implying that there exists an intermediate range of memory effect to generate correlated bursts comparable to
empirical findings. Our conclusions provide a deeper understanding of how long-range memory affects correlated
bursts.
DOI: 10.1103/PhysRevE.92.022814 PACS number(s): 89.75.Da, 05.40.−a, 89.20.−a
I. INTRODUCTION
Many natural phenomena and human activities are ex-
M and the burstiness parameter B, defined as
B =
σ − ⟨τ⟩
. (3)

Self-exciting point process
• Memory function = sum of memory kernel of the past
events
• cf.) Epidemic Type Aftershock Sequence (ETAS)
models for earthquakes
t1 tn
m(t) =
nX
i=1
1
t ti
for t > tn p[m(t)] = 1 e µm(t) ✏
: prob. of a new event in time t

• Memory loss due to the limited capacity, noise, etc.
• Sequential memory loss: ﬁxed L
• Preferential memory loss: variable L
Memory loss mechanism
µ = 0.1, ✏ = 10 6
, ✏L = 10 6
: memory initialization probability whenever
an event occurs
m(t) =
nX
i=n L+1
1
t ti
m(t) =
nX
i=n L+1
1
t ti

~ event rate
→ approaching Poisson process?
Sequential case: m(t)
m(t) =
nX
i=n L+1
1
t ti

L events new event
Sequential case: P(τ)
⌧L 1 · · · ⌧2 ⌧1 t
P(⌧|{⌧i}) =
"⌧ 1Y
t=1
e µm(t|{⌧i}) ✏
#
h
1 e µm(⌧|{⌧i}) ✏
i
p[m(t)] = 1 e µm(t) ✏

Sequential case: L=1, 10
↵ + = 2
O, PEROTTI, KASKI, AND KERT´ESZ PHYSICAL REVIEW E 92, 022814 (2015)
10-6
10
-4
10
-2
10
0
0 4 8 12 16
P∆t(E)
E
(b)
∆t=1
4
16
64
10-6
10-4
10
-2
10
0
0 4 8 12 16 20 24 28
P∆t(E)
E
(e)
∆t=1
16
64
256
10
-2
100
)
(h)
β=1.55(5)
10
-6
10
-4
10
-2
10
0
100
101
102
103
104
105
A(td)
td
(c)
numeric
γ=0.90(1)
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
100
101
102
103
104
105
A(td)
td
(f)
numeric
γ=0.81(1)
10
-2
10
-1
100
)
(i)
numeric
γ=0.58(1)
10
-12
10
-9
10
-6
10
-3
10
0
100
102
104
106
108
P(τ)
τ
(a)
L=1
α=1.1
numeric
analysis
10-12
10
-9
10-6
10-3
10
0
100
102
104
106
108
P(τ)
τ
(d)
L=2
α=1.1 α=1.2
numeric
analysis
10
-3
100(g)
L=10
numeric
10-6
10
-4
10
-2
10
0
0 4 8 12 16
P∆t(E)
E
(b)
∆t=1
4
16
64
10-6
10-4
10
-2
10
0
0 4 8 12 16 20 24 28
P∆t(E)
E
(e)
∆t=1
16
64
256
10-6
10
-4
10
-2
100
100
101
102
P∆t(E)
E
(h)
β=1.55(5)
∆t=16
64
256
1024
-2
10
0(k)
∆t=16
64
256
10
-6
10
-4
10
-2
10
0
100
101
102
103
104
105
A(td)
td
(c)
numeric
γ=0.90(1)
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
100
101
102
103
104
105
A(td)
td
(f)
numeric
γ=0.81(1)
10-5
10-4
10-3
10
-2
10
-1
100
100
101
102
103
104
105
A(td)
td
(i)
numeric
γ=0.58(1)
10
-1
10
0
(l)
numeric
γ=0.53(1)
10
-12
10
-9
10
-6
10
-3
10
0
100
102
104
106
108
P(τ)
τ
(a)
L=1
α=1.1
numeric
analysis
10-12
10
-9
10-6
10-3
10
0
100
102
104
106
108
P(τ)
τ
(d)
L=2
α=1.1 α=1.2
numeric
analysis
10-12
10
-9
10
-6
10
-3
100
100
102
104
106
108
P(τ)
τ
(g)
L=10
numeric
-3
10
0(j)
numeric
1
2
100
103
106
αlocal
τ
Ec ⇠ L
↵ + 6= 2

inter-event time distribution
for sequential case with L
From sequential case to
preferential case

for large ν, L=1 dominant
for small ν, ???
Preferential case
↵ + = 2↵ + = 2?
P t(E) ⇠ EP(⌧) ⇠ ⌧ ↵
A(td) ⇠ td

Preferential case: ν=0.1
B ⇡ 1
M ⇡ 0
Poisson process?

Contextual bursts
Contextual analysis framework for bursty dynamics
Hang-Hyun Jo,*
Raj Kumar Pan, Juan I. Perotti, and Kimmo Kaski
Department of Biomedical Engineering and Computational Science, Aalto University School of Science, P. O. Box 12200, Espoo, Finland
(Received 20 March 2013; published 20 June 2013)
To understand the origin of bursty dynamics in natural and social processes we provide a general analysis
framework in which the temporal process is decomposed into subprocesses and then the bursts in subprocesses,
called contextual bursts, are combined to collective bursts in the original process. For the combination of
subprocesses, it is required to consider the distribution of different contexts over the original process. Based on
minimal assumptions for interevent time statistics, we present a theoretical analysis for the relationship between
contextual and collective interevent time distributions. Our analysis framework helps to exploit contextual
information available in decomposable bursty dynamics.
DOI: 10.1103/PhysRevE.87.062131 PACS number(s): 05.40.−a, 89.75.Da, 89.20.−a
I. INTRODUCTION
In a wide range of natural and social phenomena, inho-
mogeneous or non-Poissonian temporal processes have been
observed. They are described in terms of 1/f noise [1,2], or in
terms of bursts that are rapidly occurring events within short
time periods alternating with long periods of low activity [3–5].
In studies of inhomogeneous temporal processes one ﬁnds a
crucial for the process than their real timings or when the real
timings are not available, such as the sequence of words in the
text [21]. In addition, the origin of bursts can be explored more
explicitly as the effect of any intrinsic temporal patterns, such
as circadian and weekly cycles of humans [22], is excluded.
Moreover, the human bursty dynamics has often been modeled
in terms of the ordinal time frame by ignoring the real time

decompose!
friend A
friend B
friend C
burst
contexts
burst
Decomposition of time series
contextual burst
P(l) ⇠ l ↵
P(⌧) ⇠ ⌧ ↵0
⌧
l

collective real inter-event time
P(l) ⇠ l ↵
contextual real inter-event time
P(⌧) ⇠ ⌧ ↵0
contextual ordinal inter-event time
P(n) ⇠ n
irrelevant context
irrelevant time-frame
⌧ =
nX
i=1
li ↵0
= min{(↵ 1)( 1) + 1, ↵, }

Overview of analysis
→ probability of making one τ
from n collective inter-event times
⌧ =
nX
i=1
li P(⌧) =
1X
n=1
P(n)Fn(⌧)
Fn(⌧) =
nY
i=1
Z 1
l0
dliP(li) ⌧
nX
i=1
li
!
P(l) ⇠ l ↵
P(n) ⇠ n
↵0
= min{(↵ 1)( 1) + 1, ↵, }

Numerical confirmation
with αc ≡ (α − 1)(β − 1) + 1 and crossovers n× and x× =
(τ − n×l0)n−ν
× . For derivation, (τ − nl0)n−ν
has been replaced
by x and then approximated as x ≈ τn−ν
. While the first term
in the parentheses is independent of τ, the second term is
obtained as ταc−α
− xαc−α
× , leading to
P(τ) ∝ c1τ−αc
+ c2τ−α
, (7)
with coefficients c1 and c2. Thus, we obtain
α′
= min{αc,α} if 1 < α < 2. (8)
The condition for αc = α is β = 2, when the second term in
Eq. (6) gives the logarithmic correction as ln τ. That is, if the
tail of P(n) is sufficiently small, α′
= α is obtained, implying
that contextual bursts in a real time frame are determined only
by collective bursts. In any case, we get α′
< β, implying that
contextual bursts in a real time frame are stronger than those
in an ordinal time frame due to large fluctuations of collective
interevent times.
Figures 3(a) and 3(b) show that our analysis is confirmed
by the numerical simulations (to be described later) for α = 3
2
and l0 = 1. We find that the numerically obtained Fn(τ) for
different ns collapse into one curve corresponding to g−
1 (x)
for x < x× and g+
1 (x) for x > x×. Then, based on the simple
scaling form P(τ) ∼ τ−α′
, we estimate the value of α′
, which
10-8
10
-6
10
-4
10-2
10
0
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
Fn(τ)n
2
(τ-n)n
-2
(a)
n=10
20
40
80
160
320
10
-6
10
-4
10
-2
10
0
100
101
102
103
104
P(τ)
τ
(b)
β=1.1
2
3
1
1.5
1 2 3 4
β
α’
10
-6
10
-4
10
-2
10
0
0 20 40 60 80 100
Fn(τ)n2/3
(τ-τc)n-2/3
-x0
(c)
n=10
20
40
80
160
320
10-6
10-4
10-2
10
0
10
0
10
1
10
2
10
3
P(τ)
τ
(d)
β=1.1
2.5
4
1
2
2.5
1 2 3 4
β
α’
100(e) 100(f)
↵ = 3/2
↵ = 5/2
Fn(⌧) P(⌧)

Relation to transport models
→ partition function of mass transport models
Majumdar et al., PRL (2005)
al Review R199
Particle:
Site:
u(1)
u(2)
u(3)
u(1)u(3) u(2)
1 2 3 4
1 2 3 4
5
5
(a)
(b)
Figure 1. Mapping between the zero-range process and the asymmetric exclusion process.
ZRP with a corresponding conﬁguration of particles in an exclusion model. (The mapping
Evans, Hanney, JPA (2005)
Fn(⌧) =
nY
i=1
Z 1
l0
dliP(li) ⌧
nX
i=1
li
!

Effect of bursts on spreading
Analytically Solvable Model of Spreading Dynamics with Non-Poissonian Processes
Hang-Hyun Jo,1,*
Juan I. Perotti,1
Kimmo Kaski,1
and János Kertész2,1
1
BECS, Aalto University School of Science, FI-00076 Espoo, Finland
2
Center for Network Science, Central European University, H-1051 Budapest, Hungary
(Received 22 November 2013; revised manuscript received 5 February 2014; published 17 March 2014)
Non-Poissonian bursty processes are ubiquitous in natural and social phenomena, yet little is known
about their effects on the large-scale spreading dynamics. In order to characterize these effects, we devise
an analytically solvable model of susceptible-infected spreading dynamics in infinite systems for arbitrary
inter-event time distributions and for the whole time range. Our model is stationary from the beginning, and
the role of the lower bound of inter-event times is explicitly considered. The exact solution shows that for
early and intermediate times, the burstiness accelerates the spreading as compared to a Poisson-like process
with the same mean and same lower bound of inter-event times. Such behavior is opposite for late-time
dynamics in finite systems, where the power-law distribution of inter-event times results in a slower and
algebraic convergence to a fully infected state in contrast to the exponential decay of the Poisson-like
process. We also provide an intuitive argument for the exponent characterizing algebraic convergence.
DOI: 10.1103/PhysRevX.4.011041 Subject Areas: Complex Systems
I. INTRODUCTION
Events of the dynamical processes of various complex
systems are often not distributed homogeneously in time
but have intermittent or bursty character. This is ubiqui-
by model calculations [8,14,20–22]. In those studies, the
bursty character of an event sequence was found to slow
down the late-time dynamics of spreading, evidenced also
by a heavy tail in the inter-event time distribution.
PHYSICAL REVIEW X 4, 011041 (2014)

Ebola outbreak
Gomes et al., PLOS Currents Outbreaks (2014)

Do individual bursts speed up
or down spreading in a
population?

Conﬂicting results
Spreading on the mobile
phone network
Karsai et al., PRE (2011)
Spreading on the sexual
network
Rocha et al., PLOS CB (2011)
slower
faster
vs.

Susceptible-Infected (SI)
spreading dynamics
P(w) =
1
µ
Z 1
w
P(l)dl, µ = hli
l0
w
w0
w00
l00
l
l
S
I
S
S
S
: # infected nodes in time t
branching process
l = inter-event time
w = residual time
P(l) ! n(t)

Exact solution for an inﬁnite
system
˜n(s) =
1
s
+
˜P(s)
(s µ 1)[1 ˜P(s)]
P(l) =
l↵ 1
c
(1 ↵, l0
lc
)
l ↵
e l/lc
✓(l l0)
for arbitrary inter-event time distribution
and for the entire range of time
Laplace transform
x ⌘
l0
µ
, y ⌘
l0
lc
,
x
y
=
(1 ↵, y)
(2 ↵, y)
C(x, y, ↵) ⌘ µ
dn(t)
dt t=l+
0
=
y1 ↵
e y
x (1 ↵, y)
C(x, y, 0)
Bursts speed up the initial spreading!

Bursts speed up initial spreading
but slow down at ﬁnal stage!
a
t
i
a
w
e
α
t
d
100
101
10
2
103
0 1 2 3 4
n0(t)
t
(a)
power law
power law + cutoff
shifted Poisson
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
n0(t)/N
t
(b)
10
-2
10
-1
(c)
α=2.001
2.2
2.4 1.6
1.8(d)
estimated
β=α-1
ANALYTICALLY SOLVABLE MODEL OF SPREADING …P(l) =
l↵ 1
c
(1 ↵, l0
lc
)
l ↵
e l/lc
✓(l l0)
All cases with the same mean and lower bound of inter-event times

Future works
• How to characterize correlated contextual bursts
(CCB)?
• Effects of CCB on collective dynamics (avalanche,
diffusion, etc.) on temporal networks?
• Interplay between bursts and network topology?
• e.g., Jo et al., PLOS ONE (2011)
• Origin of bursts and temporal scale-invariance?

Statistical Physics of Complex Dynamics

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