How can we detect communities when the social graphs is not available?

1. Influence'based,Network'oblivious,,
Community,Detec:on,
Nicola,Barbieri, ,Francesco,Bonchi,
,
Yahoo!,Labs,',Barcelona,,Spain,,
{barbieri,bonchi}@yahoo'inc.com
,
Giuseppe,Manco,
,
ICAR'CNR,,',Rende,,Italy,
,, manco@icar.cnr.it,

2. The,task,of,detec:ng,close&knit+communi.es,of,like&minded,
people,in,on'line,social,networks,has,plenty,of,applica:ons,in,
marke:ng,and,personaliza:on.,
If,a,user,responded,posi:vely,to,a,
certain,campaign:,
TARGET&USERS&IN&THE&SAME&
COMMUNITY.&
By,homophily,one,can,expect,similar,users,to,be,more,likely,to,
be,interested,in,the,same,product,than,random,users.,
If,more,users,in,the,same,community,adopt,the,same,
product,,this,might,eventually,create,a,word&of&mouth+buzz.,

3. The,companies,that,would,mostly,benefit,from,knowing,the,
structure,of,the,social,network,oSen,do+not+have+access+to+
the+network!,
How&can&we&detect&communi<es&when&the&social&graph&is&not&
available?&
A,company,adver:sing,or,developing,applica:ons,over,an,on'line,
social,network,owns,the,log+of+user+ac.vity,that,it,produces.,
Exploit,the,phenomenon,of,social+contagion+to+detect+
communi.es.,

4. Influence'driven,informa:on,propaga:on.,
Users,performs,ac:ons,(likes,,purchases,,shares,,tweets),and,those,
ac:ons,propagate,across,the,network.,
A,Propaga.on+model,governs,how,influence,propagates,across,a,
network.,
Independent+Cascade+Model:++
When,a,node,(v),become,ac:ve,it,is,considered,
contagious,and,it,has,a,single,chance,to,
ac:vate,each,inac:ve,neighbor,(u),with,
probability,pv,u.,
As,informa:on,spreads,over,social,connec:ons,,,the,network,
naturally,shapes,the,process,of,informa:on,diffusion.,,

5. where Nk =
Optimizing Q(
P
k ⇡k = 1, 0
Modeling maximum likelihood. We assume that each propagation trace is independent from the others, and we adopt
(Unobserved))Social)Network)
a maximum a-posteriori perspective. That is, we hypothesize
Propaga'on)Log)
that action probabilities adhere to a mathematical model
⇡k =
governed by a set of parameters ⇥.Our,framework,assumes,the,
The likelihood of the data
given the model parameters ⇥, canexistence,of,an,unobserved+
hence be expressed as:
Here, the prop
Y
Communi'es)
mation of
L(⇥; D) = social+network,having,a,modular+the
P (u|⇥)
u2V
structure., community no
suppressed. Th
where P (u|⇥) represents the likelihood to observe u’s behavnumber of com
ior relative to D. As a consequence, the corresponding learning
letting some o
ˆ
problem is finding the optimal ⇥ that maximizes L(⇥; D).
We,assume,that,user,ac:vi:es,are,governed,by,an,underlying, general
The
Following the standard mixture modeling approach [?], we
as explained i
stochas.c+diffusion+process,over,the,unobserved,social,network.,
assume that users’ actions can only happen relative to a
its robustness
community of membership. That is, we assume that a hidden
Figure 1: Topic-aware influence parameters are learnt from
arbitrarily
Each,user,is,associated,with,a,level,of,membership,and,influence+ larg
binary variable social network following
zu,k denotes the membership of user u to
the log of past propagations and the
PK
[?]. These are the community to build the INFLEX index
prerequisites k, with the constraints
in,each,community., zu,k = 1. Thus, pitfallstoof loca
k=1
likely be co
that we use to e ciently answer TIM queries. {⇡1 , . . . , ⇡K , ⇥1 , . . . , ⇥K }, where
⇥ can be partitioned into
P (⇥) is ma
⇥ represents the propagation model,
We,can,model,the,behavior,of,users,by,exploi:ng,the,standard, a of m
of these three paperskdefine an influenceparameter set relative to community k, and
same order
instead in a recent ⇡k =Barbieri et al. [?] extend the classic
work P (zu,k = 1). We can rewrite the likelihood as
mixture+modeling+approach:,
prior in [?] wo
IC model to be topic-aware: the resulting model is named
K
Topic-aware Independent Cascade (TIC) .
of communites
YX
Barbieri et al. also devise methods to learn,=
from a log P (u|⇥k )⇡k ,
of
L(⇥; D)
the computatio
past propagations, the model parameters, i.e., topic-aware
u k=1
without optimi
influence strength for each link and topic-distribution for
Stochas:c,Framework,for,Network,Oblivious,CD.,

6. u
v
Following [13] weinfluenced delaythe adoption ofto define
adopt to u in threshold
a define
ng [13]
meter set we adopt a delay threshold
i. Similarly we de
⇥)|D; ⇥0 ]
+
+
s.Community'Independent,Cascade,(C'IC)., 2 V |0 of users wh
Specifically, we define Fi,u = we define Fi,u
V |0 |t
ete-Data influencers. Specifically,{v =2{v 2 V u (i) = t{v > } 
Fi,u
v (i)
}tuas the tset ofusersfailedthe influencing u over i. potentially can speci
(i)
as potentially
v (i) 
v (i) (2) } who in set of users who Then, we
⇥k )The Generaliza:on,of,the,IC,Model:,each,new,ac:ve,user,v,exerts,her,
+ log P
hts. + the adoption of i. Similarlysurvives uninfected at set
analytical}optimization of that a user we define
d u in log ⇡kinfluenced(⇥)in the adoption of the least until time tu ) and hazardthe set
u
i. Similarly we define
as |t , ↵ ) (modeling instantaneous infections).
Y
We resort|t (i) explicit (i) > } of usersu who definitely weights. The analytical optimization of that a user surv
to the
modeling functions H(t v v,u
Learning influence
v 2 V u Fi,uv = {v 2 V |tu (i) tv (i) >)(u|⇥difficult. We resort + (i|u, ⇥modeling functions H(tk
t
influence,globally,,with,a,strength,that,depends,on,the,community+ ⇥ u
} of users P to the definitely
who
)
)·P
Q(⇥; 0
n data to(2) u the optimization we can specify theP is still )into= community-basedoptimization (i|u,
simplify over i. Then, We reformulate this P⇥
framework k hidden data to simplifyexplicit k
a
).
influencers
nfluencing failed in influencing u over of Then, k as can ispecify the (u|⇥k ) We reformu
k+of+the+targeted+node., be a binary variable such that scenario. The C
i. (u|⇥ we w
P
procedure. (C-Rate) propagation model
be
such that scenario. The analytical optimization of
(⇥) a binary variable Learning EM al- The Community-RateThat is, letthati,u,v survives uninfected at least until time tu) and ha
e done by means of influence weights.
u,v
a user
Y i Q(⇥; ⇥and still difficult. Wewherethe i,u,v (i|u,v ⇥k )functions H(t |t ,thethe iprobability that
triggered the adoption
as by u, 0) is is characterized by the explicit1modeling represents↵v,u) (modelingand is characterized
P, = if assumptions: u v item u, instantaneous infecti
+
Y to w followingthe set of all possible w of such thatby 2 F + .
resort
he (u|⇥ )of= assignment ⇥ ) the (i|u, ⇥ ), W denote (3)
al randomthe itemP
⇥, ·
v
i,u,v
P adoption influence + (i|u, (u|⇥ P =data to simplify the optimization Psurvives uninfected atand until time• tUser’s infl
k
k analytical potential of ⇥ ) user We reformulate ), uframeworkP (i|u, ⇥ h
of the influencers
+
P
)
Pk let we cankrewrite the (i|u, ⇥ this relativei,u
(i|u, that a ·
Learning until thatweights. .The askhidden optimization influencers complete-dataklikelihoodleast (3) a community-b
activated
+Then,
u ) That i
k
to
possible wi,u,v i
v 2 Fi,u That is,• User’s be a binaryisvariable such the scenario. The Community-Rate into to. and )
wo steps such convergence: let wi,u,v influence W limited to that community she belongs (C-Rate) propagation m
procedure.
bility as functions the “out-of-react” influencers s
memb
Q(⇥; ⇥0 ) likelihoodu,i,u,v Wetoif v triggered theexplicit modeling none and is characterized by(modeling instantaneous infecti
still difficult. =
the adoption of the item by u, of H(tu |tv , ↵
w k as
complete-data iseach relative 1 resort toto. That i the that is i likely +to influence/bev,u ) the following assumptions: by influen
eEM⇥k ) representsWthe probability that some(D, Z,2W, ⇥) = P (D, W|⇥, Z) · P (Z|⇥) · PY
for
alu,k
is, user
influenced (⇥),
of
(i|u, the influencers as hiddendenote the ⇥ of)allrepresents theof Wei,ureformulatethat some ofto thea community-b
the
let
F
of
data rewrite possible wi,u,v suchPthat vprobability this framework the communityk ) bel
to simplify the optimization P . • ⇥ = 1
where P+ (i|u, set kthe complete-data likelihood relative to(i|u, User’s)influence is limited into (1 communit
p she
Then, u and
t (u|⇥k )⇡kactivated we can Pby(i|u, ⇥kof wheresame community,to.while is, the effectis likely to influence/be influe
⇥, the ,Probability,that,some,of,the,
+
k
members ) the proba- Y Y That the user
the ,Probability,that,none,of,the,the,“out'of'v
P procedure.
influencers That is, let w be a binary variable such that
W as
scenario. Community-Rate (C-Rate)
potential influencers activatedP (D,W|⇥, Z) = (i|u, ⇥apk ))z ofthe same + propagation e
u react”,influencers,succeeds,in,ac:va:ng,u,
and members (1 k the proba- • Informatio
i,u,v
poten:al,influencers,in,ac:va:ng,u, succeeded: P The by membersdifferentv2Fcommunity, while the m
of influence
of v
k none of ·the “out-of-react” influencers is marginal on
gence: Z) P(Z|⇥) · P(⇥),P (D, Z, W, ⇥) = P (D, W|⇥, Z) · P (Z|⇥) · P (⇥), i,u,k of influence is marginal on members communit
D, W|⇥,
i,u
of a ↵
P (u|⇥k )⇡k that adoption of the item i by u, and is characterized by the succeeded:
v2F
v triggered the none of the “out-of-react” influencers following assumptions: where diff
v,
k=1wi,u,v = 1 if bility
Y
Y community.
Y Y
community.
s
k tions that
w
·z
+
where
(i|u, Information pk (1 w )·z
)1
P maximizing all
let + denote the ) of 1 possible • (1 Y k ) v z2 Fi,u · P User’sv •to⇥kk is diffusion from the userpvvto)v of contagi
the⇥W (i|u, ⇥k set= Eq. 2. wi,u,v suchpthatdiffusion .fromYuserpkvinfluence =limited to (1 community she be
Y
v
Y
Information )k ) 1 i,u,k•v2F (1 community isthe k-th theby the density within the
the
vpv ) characterized
within v
P+
P (D,W|⇥, Z) + (i|u, ⇥ p =
=
(1
f(tu |tv , ↵
Then, to u,k the prior P i,ui,u,k v2F k relative to
likelihood
metric pk )zcan rewrite the complete-datacommunity isv characterized byto. That is, the v,k|tis ,v2F),to thetoexpected The parameter
v2F (⇥)
(1 we both
i,u
v
where ↵ user relatedin the influence/be influe
delay on
is 0
+the
v ↵v,k
As a consequence,the density f(tu Q(⇥; ⇥likely second influence:the ac
high
v2Fi,u contribution tov triggers)within community k. The proba
YYa Y k w ·z
tions
k (1
We W as
model the former in wayv k We is relatedYthe canby members that the activa- community, consequence de
row ofwEq. )·z expected delay on
2specify the complete-data likelihood
be rewritten as
2Fi,u
·
p
can to
of same
while
P (i|u, ⇥k ) =
(1 where ↵v,k 1 pv then 0 2 k of contagion depends on the time delayOn v.u.the
pv )
i,u,k v2F of
3zu,k 2 the basis
ow ·zu,kExpecta:on'Maximiza:on,algorithm,to,determine,the,parameters, diff
automatic estimation tions that k ) = within B of influenceThe probability on members of a term
P P(Z|⇥) triggers
(1 The parameter ↵v,k khas
pXk.
wi,u,v an P(D, Z, W, ⇥) =)·zu,k W|⇥, Z) (i|u, ⇥·vP(⇥),X X
(1 wi,u,v P(D,
community ) X is marginal a direct interpretation in
2.
v
NetRate model
0
log(1
v2F
@log
u,k Y ⇡k +
1 pk forAsthe i,u the·it
a consequence,
contribution to Q(⇥; ⇥ ) in the second influence:Y values )of ↵v,k cause short delays, and
Y
mmunities. v
As
latter, of contagion dependsk on the time delay v.u . pv k 7
high
6
u
v2Fi,u community.
that,maximize, 1 i v2F denote v as strongly influential6 P (u|⇥k
or P (⇥)
row of Eq. 2 can be rewritten as
(1 pv )5 1 4 k.
within
hen where
the specify the complete-data likelihood through: 4 consequence of the above observations,· we can adap
probability P (a) for0given P (D|Z, ⇥) =
Y then Theu,kX X v,k has a X i,u,k On thediffusion from the user v
+
in a way We can X Bspecify 2 ↵complete-data likelihood through: k C to v within i,u
•
XInformation
2
3zk parameter
3 u,k
X Y
v2Fin terms
thelog(1 pk ) + direct zinterpretationbasis to⌘i,u,v,k )of ofk )Sec. III, byv2F the
i,u
z
0 alternatives.⇡k + u,k
NetRatekmodel fit the scheme p A
plugging
e two different =
⌘
p
(1
u,k @
tribution to Q(⇥; ⇥Z) in the second (1 log pv )2 highY vof ↵v,k causei,u,v,k log2vis+characterizedY the z
P(D,W|⇥, )Y
3zu,k delays, andY log(1 3density f(tu |tv , ↵
mation of
influence: v2F values
short
as a by
Y6
i v2F community
u,k
u k
i 6
k 7
k 7
P (u|⇥k ) =
S(T |tv (i), ↵v,k )·
ten as 41
=
(1 i,u Y · 4 Y vwhere ⌘pv )5 where ↵ is Y the to the expected
i,u,k v2F pv )consequence denote (1 strongly influential within k. user v in triggering delay on the a
5
as i,u,v,k is71“responsibility” of i:u62C v2C k 7
latter, it
v,k related
6
6
k the
u,k
i,u
i,u,v
u,k
i,u,v
u,k
u,k
+
i,u
i,u
i,u,v
i,u,v
u,k
u,k
+
i,u
i,u
+
i,u
i,u
i,u,k
+
P (D|Z, ⇥)X X
=
1
(1
p )
·
Y
Y
(1 p )
i
i

7. X basis of
) in the depends influence: delay
the likelihood that values .of v,k On 0 short delays,
X Y Y
ofiscontagion secondthe user v in triggering= P(wi,u,vv.u1|u, i,↵u,k = causeadoption ) and as a
,; ⇥or“responsibility”X on the timehighthe consideredthemodel to(i),the above observations, we can ada
⌘i,u,v,k (i), ↵v,k ) =
z NetRate H(tu (i)|tv fit v,k scheme of Sec. III, u,kpluggin
1, ⇥ )
of
B the
↵ the
k
by
log(1 u k )
@log ⇡k + of the community k: S(tp(i)|tv denote v(5) strongly influential within k.
consequence
as terms v2C
v
Learning. Again, instead of directly optimizing the
k
X X
in the context
parameterwithin itime(i) T . This approach assumes that Y Y
has
of i,tu (i)
appen ↵iv,kv2F v2Ci,tu interpretation pin
i:u2C a direct
v
.
binary
observations, likelihood, we the
X On the = Q
nce: high values i,of ↵v,k⇥0cause basis of the+ above )as a P (u|⇥kwe can adaptintroduce|tthe latentv,k )· variable
=
S(T v (i), +
↵
= P(wi,u,v = 1|u, i,u between short delays,(1andk time of )the
zu,k = 1, H(t (i)|t (i), ↵ ) adoption
)
1
pw
u,v,kdependency
apk )
u
v the fit i,u scheme
v,k w2F
denoting the fact
NetRate modelLearning.the1 instead of Sec. III, by i:u62C abovethat u has been infected by v on i. Th
to
plugging
Again,
of directly optimizing thei v2Ci
quence denote v asv2C
strongly influential within k.
k
v
The,likelihood,of,an,ac:va:on,can,be,formulated,,by,applying,
pv
k
X
likelihood be
Finally, optimizing Q(⇥; 0 with adapt the
and thelog pkaboveu(i)the influenced. CIn NetRate [4], Y cani,u,v rewritten by defininghu,ii2D X
Q
. i,t
likelihood,k Y
the=⌘1i,u,v,k +the v pk )of observations, P ⇥ )canwerespect to pvthe latent binary variable w Y S(t (i)|t (i), ↵ )
basis of one+ (1 ⌘i,u,v,k ) log(1 Y)Aintroduce yields
we pk
X
(1 1w
P (u|⇥kdenoting the fact⌘that uS(Tbeen(i), ↵v,k )· on i. Then, the Y Y Y
) =hu,ii
|tv infected by v
u
v
v,k
survival+analysis:, by
u,k
ateLearning. w2Fi,u in Sec.directlythis dependency in modeled P(D, W|Z, ⇥) =
to fit
Sec. III, byabove · i,u,v,k
plugging
described instead of II,of optimizing the i:u62C v2C has
+model Again, the scheme
S(T|tv (i), ↵v,k )zu,k
+
2Fi,u
i:u2Ci v2Ci,tu (i)
v2F
i
i,u
likelihood can
mizing Q(⇥; ⇥0wewithY Ypklatent binarypkvariable wY i be rewritten by defining
respect to
Y,
(4)
C (tu |tv
likelihood, ) introducefthe v yields , ↵v,u ) of+ transmission, which
D k
v=
i,u,v
nal(u|⇥ ) = pk )A ofS(T user v in triggering Y Y S(t (i)|t (i), ↵ ) Xhu,ii62H(tv2Ci (i), ↵ )
P
)likelihood
log(1
is the khu,ii k · ⌘
the |tv (i), ↵v,k )·Sv,k + Sv,k
Yu
Y Y u (i)|tv hu,ii2D
Y
u,v,k “responsibility”
P
v,k
v
denoting the fact u,k ui,u,v,k infectedP v onP(D, W|Z, ⇥) = P
that has been
by
i. Then, the
zv,k
u,k
(5)
S(T|tv (i), ↵v,k ) v2C·
H(tu (i)|tv (i), ↵v,
+
n likelihoodv2Fi,u bei:u62thev2Cidefining= k: u,k and S i v2Ci,tu (i) u,k .
the delay community hu,ii i:u2C of a
n the context of Ci v,u . SThe likelihood = hu,ii propagation
i,tu (i)
with +
v,k X2D k v2Ci
pk = can rewritten by v,k
,Y
(4) + 0
Y
hu,ii6
hu,ii2D k v2Ci,tu (i)
v
+
v2F
1 Similar formulations can
i,u
i,u
Y
mulatedSi,u,v S= 1|u, Yzu,kstandard survivalv2FYH(tu (i)|t[14], v,k ) wi,u,v zu,k
by Y Y i, = u ⇥ )z
= P (w v,k + applying S(t1,(i)|tv (i), ↵v,k ) Y analysis v (i), ↵
,v,k v in triggering
v,k
user
zu,k
(5)
P P(D, W|Z, ⇥) = kP
· YNAMICS
H(tu (i)|t (i), ↵v,k ) omitted · S(tu for
S(T|tV.(i), ↵v,k ) u,k T EMPORAL DLearning. Again, v instead of directlyhere(i)|tv (i), ↵v,kthe
optimizing
Modeling,the,probability,that,a,user,survives, lack) of
v M ODELING
are
nitysurvival v,k pvi v2Ci,tu (i)
k:
v2C the probability
of hu,ii u,k and Si:u2CD uX i,u,k . .v,u ) (modeling i,tu (i) v2Ci,tu(i) we introduce the latent binary variable
S(t khu,iiv ↵
= |t
hu,ii2D k
v2C
Q hu,ii62 v2F C-IC does not explicitly model temporal dynamics, as it
+
=
likelihood,
v2F
i,u
Y w)
+Y
1
= 1, ⇥0 )user Yi,u (1 i,u pkuninfected wi,u,vleast uninfected,at,least,un:l,:me,tu,, theby v on i. Th
z
w2F
that a TEMPORAL DYNAMICS H(t(i)|tv (i),v↵v,kbinary zactivations by employingfactu↵v,k )uu,khas P(D|Z, ⇥) with above component
survives on modeling just ) at u,k denotingS(tu(i)|tav t and replacing been infected
) and
· the
focuses H(tu u (i)|t (i), ↵v,k ) of until time (i),that the hazard we adopt the exponential d
M ODELING ·
likelihood. In the following
Learning. Again, instead
directly optimizing
above
i,tu
likelihood alterdiscrete-time pv yields
functions 0hu,ii2Dv2Clikelihood,propagation model. HereModeling,instantaneous,infec:ons, ↵v,k v,u}, which
H(tkuv2Cdynamics, as kwe introduce the latent anbinary tioninfections).v,k exp {
|ti,t(i)↵to ) (modeling we present can be variable , ↵wi,u,v ↵
,(i) v,u it
instantaneousrewritten v,k ) =
f(tu |t by defining
sizing Q(⇥;model with respect
. not explicitly ⇥ ) temporal vu
and replacing P(D|Z,to characterizeabove component vin theY
⇥) with the the
P
exploits
z
)
· S(tmodeling thatlikelihood. delays
(i)|t
modeling just binary activationsnativeemploying a↵fact u,k timethe following wea community-based ↵v,k v,u } and H(tuu,k, ↵
v,
·by directly
denoting (i), v,k ) that u the P (D, W|Z, the = on|ti. distribu- exp
hu,ii
We reformulate udiffusion optimizingInhas been infected⇥)exponential↵Then, the {S(T |tv (i), ↵v,k )z |tv
this process.
into adopt by vS(tuY v,kY=
u,k of⌘i,u,v,k vthe framework above
ning. Again, instead overall
+
v2F Here
propagation model.i,u we likelihood can tion rewritten )by ↵v,k exp { ↵v,k v,u↵v,k . 1 Then,
an
k
k yields
which enables
toand = introduce with the above componentf(tu the, (C-Rate) propagation model
p
pv we
hood,vreplacing P(D|Z, ⇥) Community-Rate|tv[0, T], thei,u,vis to explicitly },hu,ii62D k v2Ci
binary in ↵v,k
scenario.time+ topresent an observation window (4) w idea
The the latentalter- be variable = defining
Given , the
ing that exploits S delays v,k
characterize
X XX
Yv ↵v,k X
↵ i.
ing process. In thatv,k hasSbeen thethe exponentialuthev ,Y )atYexpeach↵v,k v,u } and H(tu |tzu,k) /) = Ylog ⇡k
the factBy,adop:ng,the,exponen.al+distribu.on,as,density,for,the,
u + weby infected by v|t1on v,k Then, the user adopted Q(⇥; ,⇥0Y u,k
model the following assumptions:
likelihoodS(t distribu- = Y
of time which {
,k
is
ion characterized adopt
P likelihood. the following P (D, W|Z, ⇥)↵= . Then,
P
H(tu (i)|t (i), u,k
S(T |tv (i), ·↵ )
v,k
condi:onal,transmission,likelihood,and,by,introducing,hidden,vv2Ci ↵v,k
each item, or }, u,k .
hood can |tbe↵rewrittenidea {definingthe which enables the considered adoption v,k
byis to v,k v,u likelihood that
hu,ii
tion f(twindowand ↵v,k =
u,k
hu,ii62D k
bservation u u,k v,k ) T],Sv,kexp ↵ explicitly
, hu,ii v , [0, = the
(4)
k
hu,ii62DX v2Ci
+
Xthat Xshe belongs X
X hu,ii2D k v2Ci,tu (i)
X X
v2F
did userv2Fi,u within ,time )T. Y approach
is
community
S(t• v User’s which↵each not }happenH(tu |tvQ(⇥; ⇥0 )=ThistheY ⇡assumes
↵v,k /
Yz
ui,u ↵v,k time Yinfluenceand limited to
v,k v,u
kelihood|tof,variables,for,modeling,the,iden:ty,of,the,influencer,,we,obtain:, logu,k
the ) = atexp { Y Y adopted
+
u,k log ktime of the
u,k v ↵v,k · S(t
i,u,v,k u,k z
u,k
there is a dependency
W|Z,1⇥) = T EMPORAL DYNAMICS · ↵v,k ) the adoption H(tu (i)|tv (i), ↵v,k )wi,u,v zu,k(i)|tv⌘(i), ↵v,k ) ↵v,k
M ODELING That considered S(T |tv (i), between
↵v,k .likelihood that the is, theadoption is likely to influence/be influenced u
Then,
or the to.
user of the u,k
hu,ii6
hu,ii2D k v2Ci,tu (i)
hu,ii
u,k . hu,ii62D k influencer and the onehu,ii2Dinfluenced. In NetRate2D k v2Ci
[4],
v2Ci X X
v2Ci,t X
k X X u (i)
X
X
en i,u explicitly model temporal dynamics, as it and replacing P (D|Z, ⇥) with X X
not by X Y Y the same community, while the effect above component
2F within time T. This approach assumes that
Y log ⇡ previously described in Sec.↵II, this dependency in modeled by u,k log ↵z
0
the
⌘
Q(⇥; between the adoption time of the u,k v,k
) / members of
u,k u,v ↵v,k ,
u,k
u,k
k
i,u,v zu,k
ependency⇥just binary activations H(tu (i)|tv (i),+ a )w· S(tu (i)|ti,u,v,k ↵v,k ) v,k
odeling ·
by employing↵v,k likelihood. In(i), following we adopt the exponential di
v the
v
NAMICS
aIn NetRate [4], i
conditional v2C
hu,ii2D k
hu,ii2D k
u,k
D
d the oneof theinfluencehu,ii6is kpresent f(tu |tv , ↵v,uX members of a differentv2Ci,tu (i)
of hu,ii2D k Herei,twe marginal on of transmission, which
influenced.v2C 2(i) likelihood an alter- ) X v2Ci,tu (i)
propagation model. X uon the delay
X u |tv , ↵v,k ) = ↵v,k exp { ↵v,k v,u }, which e
X X depends
tion a
ral dynamics, as it and modeled by logP (D|Z, ⇥) with of f (tabove component in the
v,u
replacing ↵v,k . Thezlikelihood thepropagationu,v ↵v,k ,
scribed in Sec. + this dependency in ⌘i,u,v,k characterize the
II,
u,k
ng that community. S(tu (i)|t by applying ) u,k S(tu analysis u,k
exploits time delays to
exp { ↵v,k be } and H(t different den
[14],
ns by employing a can ·be formulated v (i), ↵v,kstandard survival|tv , ↵v,k ) = 1 Similar formulations canv,uobtained by adopting u |tv , ↵v
Modeling,temporal,dynamics,with,C'Rate.,

8. Evalua:on,on,Synthe:c,Data.,
We,use,a,generator,of,benchmark,
graphs[1],,which,generates,directed+
unweighted+graphs,with,possibly+
overlapping+communi.es.,
• Number,of,nodes,=,1000;,
• Average,in'degree,=,10;,
• Maximum,in'degree,=,150;,,
• Min/max,of,the,community,sizes,=,50/750.,
,The,four,networks,differ,on,the,percentage,μ,
of,overlapping,memberships.,
• Propaga:on,cascades,are,generated,
according,to,the,Net'Rate,propaga:on,
model.,
• The,transmission,rate,for,each,link,is,
sampled,from,a,Gamma,distribu:on,
(shape=2,,scale=0.3).,
) µ = 0.001
(b) µ = 0.01
TABLE I: Statistics for the synthetic data: four networks
corresponding to four values of µ as in Figure ??.
# of communities (K)
avg # of adoptions
avg trace length
avg % of communities
traversed by a trace
S1
9
56k
38
17%
S2
7
59k
38
24%
S3
11
82k
54
24%
S4
6
370k
256
82%
The strength of each link is determined by considering both
the outdegree (out ) of the source and the indegree (in ) of
[1],A.,Lancichineh,and,S.,Fortunato.,Benchmarks,for,tes:ng,community,detec:on,algorithms,on,directed,and,weighted,
·
·
graphs,with,overlapping,communi:es.,Physical,Review,E,,80,,2009.,
the destination:

9. Results.,
Baseline+Models+
• Based,on,network,reconstruc:on,
(assuming,a,dense,graph):,
• Inference,for,the,IC,Model[2];,
• Net'Rate[3];,
• Communi:es,are,detected,by,
applying,METIS[4],on,the,
reconstructed,graph.,
• Mul:nomial,EM,
[2],K.,Saito,,R.,Nakano,,and,M.,Kimura,,Predic:on,of,informa:on,diffusion,probabili:es,for,independent,cascade,model.,
KES’08.,
[3],M.,Gomez'Rodriguez,,D.,Balduzzi,,B.,Schölkopf.,Uncovering,the,Temporal,Dynamics,of,Diffusion,Networks.,ICML,2011.,
[4],G.,Karypis,and,V.,Kumar,,A,fast,and,high,quality,mul:level,scheme,for,par::oning,irregular,graph.;,SIAM,Journal,on,
Scien:fic,Compu:ng,,vol.,20,,no.,1,,pp.,359–392,,1999.,

10. Evalua:on,on,real,data.,
TwiEer+data+
•
•
•
•
Number,of,nodes,=,28,185;,
Number,of,links,=,1,636,4511;,
Number,of,propaga:ons,(urls),=,8,541;,,
Tweets,=,516,412.,
TABLE II: Summary of the evaluation on real data.
Communities
Community size
(min/max/median)
QG
Conductance
Internal Density
Cut Ratio
Time (mins)
C-IC
20
C-Rate
64
156/3651/1319
97/1758/328
0.3274
0, 5849
0, 031
0, 001
105
0.2424
0, 6791
0, 051
0.0009
122
Internal,density,is,an,order,of,magnitude,higher,than,the,density,of,
the,whole,graph,(0.0041).,
Modularity,and,the,diagonal,block,structure,of,the,incidence,matrix,,
confirm,the,existence,of,a,good,community,structure.,

11. THANKS!