TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 10, Octobe
r 20
14, pp. 7445
~ 745
1
DOI: 10.115
9
1
/telkomni
ka.
v
12i8.551
9
7445
Re
cei
v
ed
De
cem
ber 2
9
, 2013; Re
vi
sed
Jul
y
30, 201
4
;
Accepte
d
Augu
st 15, 201
4
Bursts of Node
Activation and Asynchronous
Communication in Temporal Networks
Yixin Zhu*, Dong
fen Li, Wenqia
ng G
uo, Fengli Zhang
Schoo
l of Com
puter Scie
nce
and En
gi
neer
in
g, Univers
i
t
y
of
Electronic
Sci
ence a
nd T
e
chnol
og
y of Ch
in
a,
No. 200
6 Xi
yu
an Aven
ue, W
e
st Hi-T
e
ch Z
one, Che
n
g
du, Sichu
an 6
117
3
1
, Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: xjzh
u
y
i
x
i
n
@
1
63.com
A
b
st
r
a
ct
Devel
o
p
m
ent
of sensor tech
nol
ogi
es an
d the pr
ev
al
ence
of electron
ic communic
a
tio
n
service
s
provi
de
us w
i
th a
hu
ge
a
m
ount
of d
a
ta
on
hu
ma
n
c
o
mmu
n
icati
o
n
beh
avior, incl
u
d
in
g
face-to-f
a
ce
convers
a
tions,
e-
mai
l
exc
h
a
nges,
ph
one
c
a
lls,
mess
a
g
e
exch
ang
es
a
nd
other typ
e
s
of i
n
teractio
n
s
in
vario
u
s onl
ine
forums. T
hese
indir
e
ct or dire
ct interact
ion form p
o
tenti
a
l b
r
idg
e
s of the virus sprea
d
. F
o
r a
lon
g
ti
me, the
study of virus
sprea
d
is b
a
se
d on th
e a
ggr
egate static
ne
tw
ork.
How
e
ver, the interacti
o
n
patterns co
ntai
nin
g
div
e
rse te
mp
oral
pro
pert
i
es
may
affect
dyna
mic
proc
esses as
muc
h
as the n
e
tw
ork
topol
ogy
do
es.
So
me
e
m
piric
a
l stu
d
ies
sh
o
w
, the activa
ti
on ti
me
and
d
u
ratio
n
of verti
c
es a
n
d
li
nks
ar
e
hig
h
ly h
e
tero
g
ene
ous, w
h
ich
me
ans
inte
nse
activity may b
e
follow
e
d
by l
o
nger
interv
als i
nactivity. W
e
ta
ke
hetero
g
e
neo
us
distrib
u
tion
of the
no
de i
n
t
e
r-activati
on ti
me
as the r
e
search
backgr
oun
d to b
u
il
d
an
asynchr
ono
us
communic
a
tio
n
mod
e
l. T
h
e t
w
o sides
of th
e co
mmunic
a
ti
on
do
n'
t hav
e
to be
activ
e
at th
e
same ti
me. On
e d
e
rives
the
t
h
resh
old
of v
i
r
u
s spr
e
a
d
in
g
o
n
the
co
mmuni
cation
mo
de
a
nd
an
aly
z
e
s
th
e
reaso
n
the het
erog
ene
ous d
i
stributio
n of
the vertex inter-
activatio
n
time
suppress the
sprea
d
of virus
.
At
last, the ana
lys
i
s and res
u
lts from th
e mode
l are verifi
ed on
the BA netw
o
rk.
Ke
y
w
ords
: com
p
lex networks, epidemic th
resho
l
d, inter-a
ctivation ti
me,
effective trans
miss
ion rat
e
Co
p
y
rig
h
t
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
The net
work
topology whi
c
h is fo
rmed
by t
he interaction b
e
twe
en individu
al
s plays a
fundame
n
tal
role i
n
the
proce
s
s of d
e
termini
ng
the
epidemi
c
sp
read
[1].
Th
e origin
al study
o
f
epidemi
o
logy
[2] is based
on homo
gen
eou
s mixing hypothe
sis, a
s
suming that
all people h
a
ve
the
same op
portunity
to contact with other
i
ndividu
al
s in
the
popu
lations. T
he
assumptio
n
a
n
d
the co
rrespo
nding
re
sults were challe
nged
by t
he empiri
cal
study, the inte
ractio
ns i
n
the
popul
ations
can use a me
aningful net
work
stru
ctur
e to better describ
e [3]. A large n
u
mbe
r
o
f
empiri
cal
stu
d
ies sho
w
th
at the n
ode
d
egre
e
di
stri
bu
tion in m
any
of re
ality net
work obey
po
wer-
law di
stributio
n with heavy-tailed, whi
c
h i
s
co
ndu
cive to the sp
read
of virus.
Comm
uni
cati
on b
e
twe
en
individual
s i
s
the b
a
si
s
o
f
the hum
an
so
ciety. No
wad
a
ys
techn
o
logy, such a
s
sen
s
o
r
devices a
n
d
online
comm
unication services p
r
ovid
e
us with a l
a
rge
numbe
r
of re
cords of i
n
teraction
bet
we
en in
divi
dual
s, in
cludi
ng f
a
ce
-to-fa
ce
meeting
s
, E-mail,
and telep
hon
e comm
uni
ca
tion etc.. A traditional
way to
describ
e these d
a
ta is to
represent them
as
an
ag
gre
g
a
te stati
c
net
work, i
n
whi
c
h an
e
dge
is
establi
s
h
ed
if
interactio
n
betwee
n
the
two
end
s of it taked pla
c
e at le
ast on
ce [3].
Another ri
ch
e
r
rep
r
e
s
entati
on of this type of
data is the temporal netwo
rk mo
d
e
l [4-13],
in whi
c
h the conne
ction bet
wee
n
two no
des o
n
ly ex
ist at the time of an event. A large n
u
mb
er
of
these
data
u
s
ually con
s
ist
s
of a
se
quen
ce of inte
ra
ctive event
s. Every event i
s
a
triplet, i.e., the
IDs of two in
dividual
s involved in the event and
th
e time of the event. Some studie
s
of the
temporal n
e
twork fo
cu
se
d on
the i
m
p
a
ct of i
n
terevent time
bursty on the
spread
of info
rm
ation
or viru
s.
Ho
wever, m
any human
intera
ction
s
are not al
ways face to
face or sy
nch
r
on
ou
s
comm
uni
cati
on
mo
de, su
ch as
E-m
a
il excha
nge, sh
ort me
ssag
e, Twitter,
We
Chat
etc.. No
t all
sent i
n
form
ation
can
be
a
c
cepted
by t
he reci
pient, su
ch as
the reci
pient refu
se
s
to ope
n a
su
spi
c
iou
s
mail or refu
se to cli
ck the link re
ceived et
c.. This ki
nd
of asynchro
nous
comm
uni
cati
on mode
ca
n
be rep
r
e
s
ent
ed by a seq
u
ence of two-t
uple
s
, whi
c
h
con
s
i
s
t of the ID
of an individual and the i
ndividual a
c
tivation time
whe
n
the ind
i
vidual sen
d
or acce
pt so
me
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 744
5
– 7451
7446
obje
c
t, such as an E-m
a
il, short me
ssa
ge, to
or from
another. Wh
en a nod
e i send
s a message
to anothe
r n
ode j at the
time t1, nod
e
j in its a
c
tive time t2 to
deci
de
wheth
e
r to a
c
cept
this
messag
e, wh
ere t1
<t2 . It is worthy of at
tenti
on what the hete
r
og
en
eou
s di
stribut
ion of individ
u
a
l
inter-activatio
n
time
com
e
into bei
ng th
e impa
ct o
n
t
he a
s
yn
chron
ous informati
on tra
n
smissi
on
and viru
s p
r
o
pagatio
n and
how the
het
erog
ene
ou
s
behavio
ur p
a
ttern of indivi
dual
s impa
ct
o
n
asy
n
chr
ono
u
s
t
r
an
smi
ssi
o
n
.
2. Model
In som
e
te
mporal net
work literatu
r
e
,
any two
a
c
tive nod
es
are li
kely to
build
a
temporal
edg
e. But the reality is that the
node
s conta
c
ted
with a node, which a
r
e
call
ed
neigh
bor
nod
es of the
nod
e, are at a
ce
rtain sco
pe.
The vari
ou
s factors d
e
ci
de
the ran
ge of
an
individul co
ntact, su
ch a
s
geog
rap
h
ical area
s of indi
v
i
dual a
c
tivity,
the so
cial ci
rcle of individ
ual
life and lea
r
ning, ki
nshi
p
and ho
bby
and
so on.
Between
one
node a
nd a
ll of its possible
intera
ction n
ode
s are e
s
tabli
s
he
d links, whi
c
h
con
s
titutes a
static ag
gregation n
e
twork
descri
p
ting n
ode a
c
tivity range
and i
s
denote
d
by
G. Email excha
nge
syste
m
, for exam
ple,
node
s i
s
fo
rm
ed by
email
a
c
count
add
re
ss in
sy
st
em
and
edg
es a
r
e e
s
tabli
s
he
d
between
ea
ch
email u
s
e
r
an
d users of hi
s or h
e
r e
m
ail
address li
s
t, whi
c
h
con
s
titute a stati
c
n
e
twork. So in
the
netwo
rk, th
e
vast majo
rity of activities
ar
e
ca
rri
ed o
u
t betwe
en t
he adj
acent
node
s. Doe
s
not
rule out, a very small amo
unt of interact
ions d
on’t take place between the adja
c
ent node
s, it
will
lead to som
e
small ch
ang
es in ori
g
inall
y
static
network
stru
cture. When a no
d
e
is activated
,
it
can
interact
with its nei
gh
bours
rath
er
than a
n
y oth
e
r
node
in t
h
e net
work. T
he
static
net
work
topology a
n
d
node
activa
tion se
que
nce prope
rtie
s
affect the
sp
read
beh
avio
r on
networks
together.
The math
em
atical e
p
idem
iologi
cal mo
d
e
l that is p
r
obably the
most
widely
use
d
for
theori
z
ing
ab
out an
d em
ulating
epide
mics is
the
so-call
ed th
e
SIR (Su
s
ce
ptible-Infe
c
te
d-
Re
covered
)
model. In the SIR model, with whi
c
h we are concerned in the prese
n
t repo
rt, each
individual bel
ong
s to either a S (susce
ptible), I
(infected), o
r
R (recove
r
ed
) st
ate at any given
time. Wh
en
a susce
p
tible
individu
al
contact
with
a
n
infe
cted
in
dividual, the
former may
be
infected at an
infection rate
.
In our m
odel,
an a
c
tion of a
n
individual,
such
as
se
ndi
ng a
sho
r
t me
ssage
or
re
ce
iving a
email, is
calle
d as
an a
c
tivation event of
the nod
e. Th
ere i
s
a
difference in m
ean
ing bet
wee
n
the
inter-activatio
n
time of a node an
d the i
n
ter-event
time of an edg
e. The forme
r
is ba
sed o
n
the
behavio
r of an individual a
nd the latter is ba
sed
o
n
the intera
ction
betwe
en two i
ndividual
s.
In the mod
e
l
of the bu
rst
s
of node i
n
ter-activat
ion tim
e
from a
re
cent literatu
r
e
[11], a
t
each time p
o
i
nt, an a
c
tivated no
de
ch
o
o
se
ra
ndomly
anothe
r a
c
ti
vated nod
e t
o
build
an
ed
ge
betwe
en th
e
m
. If one of
the two n
ode
s is I
state
no
de a
nd th
e ot
her i
s
S
state
nod
e, the I
state
node
will infe
ct the S stat
e node
with
some
pro
bab
ility. Clearly, the model a
n
d
the previou
s
model
s have
one thing in
comm
on, that is, the sy
n
c
hrono
us i
n
tera
ction, such as p
hon
e
call,
video me
etin
g, real
-time
files, etc.
Ho
wever,
many
ca
se
s a
r
e
clo
s
er to the
asyn
ch
ron
o
u
s
comm
uni
cati
on, such as E-mail exchange,
SMS, Twitter, BBS and other net
work
comm
uni
cati
on
way, which the t
w
o
sid
e
s
of the
co
m
m
unication
can b
e
a
c
tive
at different
times.
At each time
t, each a
c
tive node in t
he mod
e
l ca
n accept fro
m
neigh
bori
n
g node
s
so
me
informatio
n or send some i
n
formatio
n to a neighb
or
. In reality, user may send o
r
receive a grou
p
of information
to or from more u
s
e
r
s at the sam
e
time. For simpli
city, as long as
the time scal
e
is
small
eno
ugh
, it ca
n b
e
co
nsid
ere
d
that
informatio
n i
s
only
sent
to
one
of its adj
ace
n
t no
de f
r
om
an activated
node at a time.
In ou
r mo
del,
all n
ode
s
are S
state at i
n
itia
l mom
ent
exce
pt from
a n
ode
i
wh
ich i
s
I
state. Wh
en t
he initial infe
cted nod
e i is
activat
ed, it choo
se rand
o
m
ly
one of its neigh
bor
no
des
j and
send
n
ode j
a
me
ssage
co
ntainin
g
infe
ction
co
ntent no
matt
er
wh
ether n
ode j
is
curre
n
tly
activated . Th
en the no
de i
become
s
ina
c
tive stat
e at
next time. At
each time t, every activat
ed
node i
will
accept o
ne
or mo
re m
e
ssage
s
cont
ai
ning virus
in acco
rda
n
ce with a
ce
rtain
prob
ability for ea
ch me
ssage of them
and then c
hang
e from
S state to I
state at the next
moment if it h
a
s received
messag
es
co
ntaining
virus sent fro
m
its
neigh
bor
nod
es a
nd the n
o
de
i is S state before time t; If the activated node i is
I state, it will choo
se a neig
hbor fro
m
so
me
address bo
ok, such
as E-mail add
re
ss
boo
k, t
he telepho
ne com
m
unication b
ook, MSN fri
end
s
list, to send a
message
co
ntaining viru
s. At each tim
e
t, an infected node recover to R state
with
some p
r
o
babi
lity.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Burst
s
of Nod
e
Activation a
nd Asynchron
ous
Co
m
m
unication in Te
m
poral Net
w
orks (Yi
x
in Zh
u)
7447
To fa
cilitate t
he n
a
rrative
of nod
e
state
tran
si
tion
in
the m
odel,
we
will
di
stingui
sh th
e S
state nod
es
not re
ceived
the messa
g
e
s
co
ntainin
g
virus fro
m
the S states n
o
des
re
ceived
the
messag
es
co
ntaining
viru
s. Whe
n
a
S
state n
ode
receive
d
me
ssag
es contai
ning viruse from
other
nod
es, t
he no
de i
s
at
the ri
sk of inf
e
cted. Its
stat
e is
den
oted
by D.
When
a
D
state n
ode
is
activated, it h
a
s th
e pote
n
tial to a
c
cept t
h
is
su
spi
c
iou
s
me
ssag
e a
nd the
n
its
st
ate chan
ge from
D into I.
At each time t, for each a
c
ti
vated node i,
it is subje
c
t to the following
rule:
1) if the
no
de
i is I state, it
se
nd
a me
ss
age
co
ntainin
g
viru
se to
a
n
its
neig
hbo
r nod
e j
rand
omly cho
o
se
d. If the n
ode j i
s
S
sta
t
e at
pr
e
s
ent,
it become
D state at n
e
xt moment t
+
1;
If
the node j is
D state, I state or R
state,
it will maintain
the current st
ate.
2) if the n
o
d
e
i is
D
state
,
that is it re
ceived
on
e
o
r
more me
ssa
ges co
ntainin
g
viru
se
from nei
ghb
o
r
s
at one
po
int
t
(
t
t
), it turn into I state
if it acce
pte the me
ssa
ge with
probability
β
, whi
c
h th
e transmi
ssion
time del
ay is
t
t
; it rec
o
ver to S s
t
ate if it
refuse to
accept the m
e
ssage with probability 1-
β
.
3) if the
nod
e
i is i
n
the S
state or
R
stat
e, it don't
do
any a
c
tion. T
he
sent m
e
ssage th
at
doe
s not con
t
ain virus do
es not affect
the pr
opo
ga
tion pro
c
e
ss
of virus and
therefo
r
e not
be
con
s
id
ere
d
in
the model.
At each time t, no matter whether a n
o
d
e
i is ac
tivated, it is
s
ubjec
t to the following rule:
4) if it is I state node, it will back into R
state with pro
b
ability
μ
.
In the se
con
d
point, we a
s
sume if a u
s
e
r
fi
r
s
t saw
the s
u
s
p
ic
ious
mess
ages
, suspic
ious
informatio
n or su
spi
c
iou
s
l
i
nks and refu
sed to
acce
p
t
them, then
he or she wil
l
never accpt
e
them. So, the co
rrespond
ing nod
e sta
t
e can
be ch
ange
d into S
state from
D state
at n
e
xt
moment.
In many types of empiri
cal
data, a wide rang
e
of pattern
s of huma
n
activity are kno
w
n to
exhibit long-t
a
iled dynami
cs[1
4
-1
6]. He
re, we mo
del
the node int
e
r-activation t
i
me heavy-tai
l
ed
distrib
u
tion
with the po
wer la
w di
stri
bution. Nod
e
inter-activa
tion time
τ
obey po
wer-l
a
w
distrib
u
tion wi
th lower b
oun
d [17]:
min
min
1
)
(
P
(1)
Whe
r
e
min
is a lowe
r bou
nd
of node inter-activation time
τ
and
α
is
the expone
nt or
scaling p
a
ra
meter of the
power-la
w
distribution.
3. Epidemic
Thresh
old
Key quantitie
s for epid
e
m
i
c dyna
mics
are th
e
so-called tran
smi
ssi
bility T an
d the
se
con
dary
re
prod
uctive n
u
mbe
r
R [1
8
]. T is the p
r
obability that
an infe
cted
individual
wo
uld
transmit viru
s to a susce
p
tible nei
ghb
or
before
it
re
co
vers, a
nd
R is the expe
cted
numb
e
r of n
e
w
node
s infe
cte
d
by an infect
ed nod
es.
An infected
n
ode resto
r
e in
to recovere
d
state
within
a
time step
with the proba
bility of
μ
,
whi
c
h o
bey the bin
o
mial
distrib
u
tion of
the mea
n
fo
r 1/
μ
. So the
averag
e tim
e
that a infe
cted
node of net
work
cha
nge
s i
n
to a re
covered nod
e is 1/
μ
. When an i
n
fected n
ode
is activated, it
will randomly
sel
e
ct
an its
neigh
borhood to
send an i
n
formation
containing virus.The neighbour
accept the i
n
formation at
some
futural time with probability of
β
and
will b
e
infecte
d
a
s
a
con
s
e
que
nce
if it is
previo
usly
S state. The
inte
r-acti
vation
time
τ
fo
r
e
a
c
h
n
ode
o
f
ne
tw
or
k
is
subj
ect to id
entically in
de
pend
ent di
stribution.
Acco
rding
to the
theory of u
pdate [19], t
he
transmissibilit
y T for the dynamics can be obtained as:
1
0
)
(
d
g
T
(2)
)
)
(
1
(
1
d
g
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046
TELKOM
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KA
Vol. 12, No. 10, Octobe
r 2014: 744
5
– 7451
7448
Whe
r
e
d
P
g
)
(
1
)
(
, g(
∆
) is to ge
nera
t
e time distri
bution [20], <
τ
> is t
h
e
mean
of n
o
de inte
r-activ
a
tion time,
P(
τ
) i
s
the
den
sity dist
ribution fu
ncti
on of
nod
e
inter-
ac
tivation time
τ
. Wh
ere the no
de
inter-activatio
n
time
τ
ob
ey power-la
w
di
stributio
n with
expone
nt
α
, given by Equation (1
), trans
missi
bility T can be written as:
(
3
)
The no
de th
at we arrive
at by followin
g
a ran
domly
cho
s
en
edg
e has th
e nu
mber of
remai
n
ing o
u
tgoing e
dge
s
excludi
ng we
along [21], de
noted by k
’
.
Whe
n
a no
de
i infected by
its
neigh
bor n
o
d
e
j, node i se
lects rand
oml
y
one of
its neighb
or no
de
s as the
spre
ad obje
c
t and
the proba
bility the sele
cte
d
no
de i
s
n
o
t
node
j is k
’
/(
k
’
+1
). Th
us the rep
r
odu
ctive num
be
r R
equal T<k
’
>/
(
<k
’
>+1
)
in o
u
r mo
del
wh
ere
<k
’
> i
s
t
he ave
r
age
remainin
g de
gree
of net
work
node
s. It ca
n be expressed by n
ode
averag
e de
gree
<
k >
and the
se
cond o
r
de
r of
node
degree
s <k
2
> [18, 21], i.e., <
k
’
>=
(<
k
2
>
-
<
k
>)/<k
>
.
Henc
e the re
produ
ctive num
ber
R=T*(<k
2
>-
<k
>
)
/
<
k
2
>. A basi
c
conditi
on that virus
epidemi
c
in
n
e
twork i
s
that the rep
r
odu
ctive number
R
must be g
r
eat
er than on
e, combine
d
with
Equati
on (3
), we ca
n obtai
n the epidemi
c
thre
shol
d a
s
:
(
4
)
Whe
r
e
λ≡
β
/
μ
, whi
c
h is th
e
effictive transmissi
on
rate
of virus,
λ
c
i
s
epidemi
c
th
re
sthold,
C
=<k
2
>/
(
<
k
2
>-<k>
)
. Parameter C is
only
related to the s
t
ru
ctu
r
e of
the static n
e
twork G, a
nd
has
nothing to do
with the dyna
mic activation
prope
rtie
s of node
s.
4. Results a
nd Analy
s
is
Und
e
r the
co
ndition of n
o
des
dynami
c
activa
tion, the characte
ri
st
ics of the
epidemi
c
threshold
of
virus
are
ana
lyzed firstly. BA netwo
rk
[
22] is i
n
a typical
heterog
eneo
us
struct
ure
netwo
rk. Ea
ch new n
ode
conne
cts m ex
isting no
de
s
of the netwo
rk and th
e fina
l total number of
the network
node
s is
N.
For a limite
d
scale of
BA netwo
rk [2
3], the node d
egre
e
di
strib
u
tion
P(k)=
2
m
2
k
-3
/(1-N
-1
), the no
de average d
egre
e
<k>=
2m, the node
max degre
e
k
c
=m
N
(1/2)
. We
can g
e
t the param
eter C of
BA network as:
(
5
)
In Figure 1, the epide
mic t
h
re
shol
d of viru
s i
s
cal
c
ul
a
t
ed by Equation (4
) and E
quation
(5) a
c
cording
to the followin
g
con
d
ition
s
: the st
atic net
work G is the
BA network o
f
node avera
g
e
degree fo
r 10
, the total nu
mber
of nod
e
s
N =
500
0, the no
de inte
r-activation
time
τ
o
bey po
wer-
law di
strib
u
tion given
by Equat
ion
(1),
the minimu
m value of n
ode inte
r-acti
vation time
τ
min
=1,
node ave
r
ag
e
recove
ry time were sh
own in the illustration in Figu
re 1.
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TELKOM
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046
Burst
s
of Nod
e
Activation a
nd Asynchron
ous
Co
m
m
unication in Te
m
poral Net
w
orks (Yi
x
in Zh
u)
7449
Figure 1. The
Epidemic Th
reshold of Virus
λ
c
a
s
a Fu
nction of Exponent
α
of Power
-
la
w
Distri
bution which Nod
e
Inter-activation Time
τ
obey, for the Differe
nt Average Recove
ry Time
As we
ca
n see from Fi
gure 1 thre
e poi
nts:
First, the epidemi
c
thresh
old be
co
me larg
e
r
as the in
crea
se of the het
erog
ene
ou
s
of
node inte
r-activation tim
e
distri
bution
(i.e.,
α
de
cre
a
se
)
for different averag
e re
co
very time of
infect
ed no
d
e
. The small
e
r the po
wer-law exp
one
n
t
of
node inte
r-activation time
distrib
u
tion is, the great
er
the averag
e value of nod
e inter-activat
i
on
time derived
by Equation (1) is, i.e., the fewer
the av
erag
e times
of node activ
a
tion is in sa
me
time. That mean
s an infe
cted node h
a
s
less ch
an
ce t
o
sp
rea
d
viru
s to its adja
c
ent node
s bef
ore
it recove
r. Thus o
n
ly high effectiv
e trans
missi
on rate of
virus en
su
re its epid
e
mic
unde
r the ci
rcumstan
ce
s.
S
e
co
nd,
the
g
r
eater the
ave
r
age
recovery
time of i
n
fect
ed n
ode
1/
μ
is
,
whi
c
h mea
n
s infected no
des h
a
ve more chan
ce t
o
be activate
d and tran
smit virus to their
adja
c
ent nod
es. Hen
c
e t
he small
e
r the epide
mic
thresh
old is. Thirdly, as the power-l
aw
expone
nt
α
o
f
node inte
r-a
c
tivation time
τ
increa
se,
p
r
opa
gation th
reshold
is te
n
d
ing to a
sa
me
value no mat
t
er what valu
e node ave
r
a
ge re
cove
ry time 1/
μ
is. The increa
se
of the powe
r
-law
expone
nt
α
o
f
node inte
r-a
c
tivation time
τ
make the h
e
terog
eneity
and me
an of
τ
dimini
she
d
so
that there
are
a larg
e nu
m
ber of
node
s
of netwo
rk
a
c
tivated at every moment.
Until most of th
e
node
s
rem
a
in
active, dyn
a
m
ic a
c
tivation
network g
r
a
dually
clo
s
e t
o
the
static n
e
twork
G. In t
h
is
ca
se, epi
dem
ic thre
sh
old o
n
tempo
r
al n
e
t
work is
only related to the t
opolo
g
y of cu
mulative stati
c
netwo
rk
G, which
can b
e
p
r
oved from E
quation (4).
Figure 2. The
node de
nsity
infected by virus
a
s
a fun
c
t
i
on of virus transmi
ssion
rate
β
, for the
different expo
nent
α
of power-l
aw di
strib
u
tion whi
c
h n
ode inter-a
c
ti
vation time
τ
obey. Network
node n
u
mbe
r
N = 500
0, ne
w edg
e numb
e
r from e
a
ch
node m = 5, t
he re
cove
ry rate of the virus
spread
μ
= 0.
1
,
τ
min
=1
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046
TELKOM
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KA
Vol. 12, No. 10, Octobe
r 2014: 744
5
– 7451
7450
Figure 3. The
node de
nsity
infected by virus a
s
a fun
c
t
i
on of time t, for the differe
nt
expone
nt
α
of powe
r-l
aw di
stributio
n whi
c
h no
de inter-activation time
τ
obey. Network no
de
numbe
r N
= 5
000, ne
w edg
e numbe
r fro
m
each n
ode
m = 5, virus transmi
ssion
rate
β
=
0
.4, the
rec
o
v
e
ry
rate
of the v
i
rus s
p
rea
d
μ
= 0.1,
τ
min
=1
Model
simul
a
tion ba
sed
o
n
static BA n
e
twork,
n
e
twork scal
e is
5
000 n
ode
s. E
a
ch
new
node
con
nect
s
existing 5 n
ode
s of network. A ran
d
o
m
ly selecte
d
node i
s
set in
itially to infected
state, nam
ely se
ed n
ode.
The ave
r
ag
e
recovery p
r
o
b
ability
μ
of inf
e
cted
nod
es i
s
0.1. T
he n
o
d
e
inter-ac
tivation time
τ
obey the power l
a
w dist
ributio
n form
s of Equation (1) a
nd the minim
u
m
value of nod
e inter-activa
tion time
τ
min
=1. The
exponent
α
i
s
2.
1, 2.5 and 3
.
0, resp
ectiv
e
ly.
Figure 2
sho
w
the n
ode
d
ensity infe
cte
d
by vi
ru
s ch
ange
alon
g with virus tran
smissi
on
rate
β
.
It
is ob
serve
d
that the stron
ger the het
e
r
ogen
eity of n
ode inter-a
c
tivation time
τ
is, the greate
r
the
epidemi
c
th
re
shol
d of virus is a
nd th
e le
ss the final
sprea
d
scope
of virus is. T
he no
de
den
sity
infected
by v
i
rus chan
ge
along
with
time in
Fi
gu
re
3. As Figu
re 3
sh
own, the
stron
g
e
r
t
he
hetero
gen
eity of node inter-activation ti
me
τ
is, the sl
owe
r
the sp
re
ad sp
eed of the virus i
s
. That
the hetero
g
e
neity of node inter-a
ctivation time
τ
inhibits the prop
agation of virus is illu
strat
e
d
from two different aspe
cts
of the scale a
nd the sp
e
ed
of virus pro
p
a
gation re
sp
ectively in Figure
2 and Figu
re 3. Which de
monst
r
ate th
at the
data
si
mulation
resu
lts accords
with the theoret
ical
analysi
s
re
sul
t
s of Figure 1.
5. Conclusio
n
Different f
r
o
m
previou
s
studies that th
e het
e
r
og
ene
ous of inte
r-e
vent time di
stribution
affect the spread of the virus,
this wo
rk is ba
sed on th
e hetero
gen
e
ous di
strib
u
tion of node inter-
activation ti
me and
est
ablishe
s the
asyn
chrono
us
comm
uni
cation m
odel
, which is
more
obviou
s
ly uni
versality th
an
the fo
rme
r
.
Asynch
ro
nou
s interaction
style is
suitable for the
case
that the two
sides
of interaction are
not
alway
s
ac
tive at the same t
i
me, whi
c
h i
s
prevailing in t
he
appli
c
ation
s
f
r
om i
n
tern
et
and m
obile
in
ternet.
Whe
r
e no
de inte
r-activation tim
e
follo
ws po
wer-
law
distri
butio
n, epid
e
mic thre
shol
d of th
e mod
e
l i
s
d
e
duced
by me
ans of the
the
o
ry of
update
s
.
Simulating in
BA network,
it is conclu
d
ed that
the stronge
r the h
e
terog
eneity of node inter-
activation tim
e
is,
the
gre
a
t
er the
epi
de
mic th
re
shol
d
of viru
s i
s
a
nd the
small
e
r th
e
scale
and
spe
ed of virus p
r
op
agati
on is,
whi
c
h
con
s
i
s
tent
s with the
re
sults
of thre
shol
d theo
ret
i
cal
derivation.
In this work, asyn
chrono
u
s
com
m
uni
ca
tion is
elabo
rated by mea
n
s of the example of
sen
d
ing
and
receiving E
-
mails
and
m
e
ssag
es,
and
epid
e
mic thresh
old i
s
d
e
ri
ved by u
s
ing
the
power-la
w
distributio
n as the het
eroge
n
eou
s distri
but
ion of node inter-activatio
n
time. But
time
statistics of h
u
man
be
havior i
s
fa
r f
r
om
so
simple.
Different
dat
a sets,
su
ch
as th
e
data
sets
from mo
bile
pho
ne tex
t
messag
es,
blog,
BBS
, online
se
rvices,
etc., have diffe
rent
hetero
gen
eo
us time
dist
ribution of in
di
vidual be
hav
i
o
r [13], so th
e time di
strib
u
tion of indivi
dual
behavio
r itsel
f
is a compli
cat
ed and
wort
h studying i
s
sue.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Burst
s
of Nod
e
Activation a
nd Asynchron
ous
Co
m
m
unication in Te
m
poral Net
w
orks (Yi
x
in Zh
u)
7451
Ackn
o
w
l
e
dg
ements
This
work
wa
s su
ppo
rted i
n
part by the
National
Nat
u
ral Sci
e
n
c
e
Found
ation o
f
China
(Grant Nos. 6
1133
016, 61
1630
66 an
d
6090
2074
), a
nd
in pa
rt by the Nation
al
High T
e
chnol
ogy
Joint Research Program of Chin
a (86
3
Prog
ram, Gran
t No. 2011AA
0107
06).
Referen
ces
[1
]
Ke
e
l
i
n
g MJ, KT Ea
me
s. Ne
t
w
orks a
n
d
ep
i
demi
c
mo
de
l
s
.
J
o
urna
l of th
e R
o
yal S
o
ciety Inte
rface
. 20
05
;
2(4): 295-
30
7.
[2]
Anders
on RM,
RM Ma
y
,
B Anders
on. Infecti
ous d
i
seas
es of humans: d
y
namics a
nd co
ntrol.
Wile
y
Onlin
e Libr
ary
. 199
2; 28.
[3]
Ne
w
m
a
n
M. Net
w
o
r
ks: an intr
oducti
on.
20
09
: Oxford Univ
er
sit
y
Press.
[4]
Holme P, J Sar
a
mäki. T
e
mporal net
w
o
rks
.
Physics reports
. 201
2; 519(
3): 97-12
5.
[5]
Masud
a
N, P Holme. Pre
d
i
cting a
nd c
o
ntrolli
ng i
n
fecti
ous dis
eas
e
epi
demics
usi
ng temp
oral
net
w
o
rks
.
F
1
0
00pr
ime rep
o
rts
.2013; 5.
[6]
Holme
P, J
Saramäk
i
. T
e
mporal
Net
w
o
r
ks as a M
o
d
e
lin
g F
r
ame
w
ork, in T
e
mpo
r
al N
e
t
w
orks.
Sprin
ger. 20
13
: 1-14.
[7]
T
a
kaguchi T
,
N Masu
da, P
Holm
e. Burst
y
Commu
nicat
i
on P
a
tterns
Facilitate
Spre
adi
ng
in
a
T
h
reshold-Bas
ed Epi
demic D
y
n
a
mics.
PLoS
ONE
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