e h t a M n o e c n e r e f n o C l a n o it a n r e t n I 7 1 0
2 maitcs,ModelilngandSimulaitonTechnologiesandAppilcaitons(MMSTA2017) 8
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21TianijnModernInnovaitveTCMTechnologyCo L d., t ., HuayuanIndustria lPark,Tianijn,China
2Schoo lofComputerScienceandTechnology,TianijnUniverstiy,Tianijn,China
*Correspondingauthor
: s d r o w y e
K Communtiydeteciton, Tempora lnetworks, NodeWeightMatrices.
.t c a r t s b
A Analyzing tempora l networks can uncover dynamic evolution and characterize the p
s i h T . s k r o w t e n e h t f o s e i t r e p o r
p aperproposesanove ltempora lcommunitydetectionmode lusing d
e c u d o r t n i e r a s e c i r t a m t h g i e w e d o N . n o i t a z i r o t c a f x i r t a m e v i t a g e n n o n e l p i r
t f or targeti ng centra l
s e i t i n u m m o c f o s e d o
n andreducingnumberofnodest ha thaveunobviouspropensitiesofbelonging s
e i t i n u m m o c o
t ,which improves thealgorithm performanceof community detection. Community s
s e n h t o o m s l a r o p m e t p i h s r e b m e
m constrain ti sadded to discover laten tstructure and evolutionary s
r o i v a h e
b oft empora lnetworks .Wet henproposeagradien tdescentalgorithmt ooptimizeobjective f o s s e n e v i t c e f f e e h t w o h s s k r o w t e n d e k r a m h c n e b l a e r d n a c i t e h t n y s n o s t l u s e r l a t n e m i r e p x E . n o i t c n u f
. s e g n a h c l a r o p m e t r i e h t g n i d n i f d n a s e i t i n u m m o c g n i t c e t e
d
n o it c u d o r t n I
t n a c i f i n g i s t s o m e h t f o e n o s i e r u t c u r t s y t i n u m m o
C propertiest ha treflec texactlyt heessencesofa o t n i r e t s u l c n e t f o s k r o w t e n n i s e c i t r e v e h t f i e r u t c u r t s y t i n u m m o c e v a h o t d i a s s i k r o w t e n A . k r o w t e n
n i h t i w f o y t i s n e d h g i h a h t i w s p u o r g t i n k y l t h g i
t -groupedgesand alowerdensityofbetween-group s
e g d
e [1] .Particularlyin thetempora lnetworks ,thestructureofarea lnetwork istheresul tofthe t
u l o v e s u o u n i t n o
c ionoft heforcest ha tformedi t [2] ,whichsimulatesnumberofresearcherst omake r
o p m e t d n a e r u t c u r t s y t i n u m m o c g n i z y l a n a n o s t r o f f e t a e r
g a ldynamics.
c i t a t s r o f d e s o p o r p n e e b e v a h s d o h t e m n o i t c e t e d y t i n u m m o c l u f r e w o p f o l a n e s r a e g r a l a e t i p s e D
s o m l a , s k r o w t e
n tal lnetworks change over time ,tha tmotivates a body of new work of dynamic
. n o i t c e t e d y t i n u m m o
c Comparedwithstatict echniques ,dynamiccommunitydetectionmethodsaim .
e m i t r e v o y a c e d d n a e n i b m o c , w o r g , e g r e m e s e i t i n u m m o c w o h g n i y f i t n e d i t
a Amongt hesemethods ,
f o e m o
s them study communities and their evolutions separately ,some of them are no table to u
m m o c f o y t i l a u q e h t n i a t n i a
m nitydetection ,orothersareno tcapableofguaranteei ngrobustnessand . s k r o w t e n l a r o p m e t y s i o n h t i w s n o i t a c i l p p a n i e t a i r p o r p p a n i e r a h c i h w , m h t i r o g l a e h t f o s s e n e v i t c e f f e
s e i t i n u m m o c g n i t c e t e d f o e l b a p a c s i h c i h w l e d o m l e v o n a e s o p o r p e w , r e p a p s i h t n
I and analyze
r i e h
t tempora levolutions using triple nonnegative matrix factorization ( iT -r NMF) .The main :
s w o l l o f s a e r a e k a m e w t a h t s n o i t u b i r t n o c
• Weproposeaunified TTNMF which can detec tand track dynamiccommunities .With the s
e n h t o o m s l a r o p m e t d e d d
a sconstrain tofcommunitymembership ,theinheren tstructureand c i m a n y d f o s r o i v a h e b y r a n o i t u l o v e e h T . d e r e v o c n u e r a s k r o w t e n l a r o p m e t f o s t i n u c i m a n y d
s e i t i n u m m o
c tha tcaptured byourmode lprovideusefu linformationforpredicting evolution f
o y c n e d n e
t tempora lnetworks.
, n o i t c e t e d y t i n u m m o c f o y t i l a u q e h t n i a t n i a m l e d o m r u o t a h t e t a r t s n o m e
d track thetempora l
. t l u s e r a s a y c n e i c i f f e l a n o i t a t u p m o c e h t e v o r p m i y l b a r e d i s n o c d n a r e t t e b s n o i t u l o v e y t i n u m m o c c i m a n y d f o k r o w d e t a l e r w e i v e r e W . s w o l l o f s a d e z i n a g r o s i r e p a p e h t f o t s e r e h T n o i t c e t e
d methods of low rank approximation in Section of Related Work .In Section of Mode l d n a c i t e h t n y s n o d e m r o f r e p s t l u s e r l a t n e m i r e p x E . l e d o m d e s o p o r p e h t e b i r c s e d e w , n o i t a l u m r o F l a e
r -worlddataarepresentedi nSectionofExperiments andResults .Theconclusionsanddiscussions o i s u l c n o C f o n o i t c e S n i w o l l o
f nsandDiscussions.
k r o W d e t a l e R s d o h t e m n o i t c e t e d y t i n u m m o c c i m a n y d n o d n u o r g k c a b l a i t n e s s e e m o s e d i v o r p e w , n o i t c e s s i h t n
I of
n o i t a m i x o r p p a k n a r w o
l .Methodsconcernedconstructingstructuredl owrankapproximationi nclude l a r t c e p s , n o i t a z i r o t c a f x i r t a
m clustering and stochastic block mode l(SBM) ,and we wil lmainly . g n i w o l l o f e h t n i e r u t a r e t i l e h t n i s d o h t e m e e r h t e h t f o k r o w d e t a l e r w e i v e r e m i t d n a n o i t a r o l p x e r o f d e i l p p a y l e d i w e r a s n o i t a z i r o t c a f x i r t a m f o s d o h t e m e h
T -varying
i t n i n o i t c e t e d y t i n u m m o
c m -eevolving graph sequences .The mos tcommon factorization is the , n o i t c e t e d y t i n u m m o c o t s n o i t c e n n o c t n a t r o p m i s a h h c i h w , ) D V S ( n o i t i s o p m o c e D e u l a V r a l u g n i S g n i s s e c o r p l a n g i s d n a s c i t s i t a t s f o s a e r a d n a , g n i w a r d h p a r
g [3] .Forinstance in classica lspectra l
e b n a c d n a , s e c i r t a m d e t a l e r h p a r g f o D V S e h t y b n e v i g e r a e d o n h c a e f o s e t a n i d r o o c e h t , t u o y a l n i s m h t i r o g l a g n i s u y l t n e i c i f f e d e t a l u c l a
c [4] .Recently ,therehasbeenextensiveinteres tinspectra l g n i r e t s u l
c [5] ,which aimsto discovercommunity structurein eigenvectorsofthegraph Laplacian . x i r t a
m Low-rankapproximationswhichcomposedofnonnegativeentries ,referredt oasNMF ,have n o n f o n o i t a z i l a u s i v r o f s u o e g a t n a v d a e b o t n w o h s n e e
b -negativedata[6] .Non-negativityi st ypically t i w d e i f s i t a
s h networks ,asedgescommonlycorrespond toflows ,capacity ,orbinaryrelationships , n o n e r a h c i h w f o t s o
m -negative .In addition ,theoretica lconnections between NMF and importan t d e p o l e v e d n e e b e v a h g n i n i m a t a d n i s m e l b o r
p [7] ,and accordingly ,NMF has been proposed for
c i t a t s n o n o i t c e t e d y t i n u m m o c g n i p p a l r e v
o [8] anddynamic[9] networks.
o t d e t a c i d e d s l e d o m k c o l b c i t s a h c o t s f o s n o i s n e t x e c i m a n y d f o h c r a e s e r t n a c i f i n g i s n e e b s a h e r e h T l a r e v e s t s a p e h t n i y l t s o m , s k r o w t e n c i m a n y d f o g n i l e d o m l a c i t s i t a t
s years .Xinge tal .[ 01 ] andHoe t
. l
a [ 11 ] proposeddynamicextensionsofamixed-membershipversionoft heSBM.I shiguroe tal .[12] e h t f o n o i s r e v c i r t e m a r a p n o n a s i h c i h w , l e d o m n o i t a l e r e t i n i f n i e h t f o n o i s n e t x e c i m a n y d a d e s o p o r p . l a t e g n a Y . M B
S [13] proposedanHM-SBMtha tpositsaMarkov mode lon theclassmembership r t a y b d e z i r e t e m a r a p s r o t c e
v ansitionmatrix .XuandHero [14] proposedanHM-SBMtha tplacesa e
t a t
s -spacemode lont heblockprobabilitymatrices.
n o it a l u m r o F l e d o M w , n o i t c e s s i h t n
I efirs tprovidesomenotationsanddefinitionson nonnegativematrixfactorization .l e d o m w e n e h t e b i r c s e d n e h t d n a , k r o w s i h t n i d e s u n o it a t o N . g . e , s e c i r t a m e t a n o d l l i w s r e t t e l e s a c r e p p u d l o b , r e p a p s i h t n
I ,boldlowercaseletterswil ldonate
e v n m u l o
c ctors ,e.g . ,whileoperators wil lstand for matrix transposition ,e.g . .Both and y r t n E e h t t n e s e r p e
r of the matrix . The Frobenius norms wil l be represented by r e d i s n o C
. i ng a dynamic -node network whose time-varying structures are captured by the e
m i
t -seriesadjacencymatrices , isonei ft herei sanedgefromnode tonode att ime . e .i , d e t c e r i d n u s i k r o w t e n c i m a n y d e h t t a h t e m u s s a e W . e s i w r e h t o o r e z s i d n a
, ,andt hereare
f l e s o
e h
T Uni ifedTTNMFMo F rdel o mula iton
e h t s e b i r c s e d n o i t c e s s i h
T Tempora lTripleNonnegativeMatrixFactorization (TTNMF) model .We s s e n h t o o m s f o t s o c l a r o p m e t d n a s e i g o l o p o t k r o w t e n g n i l e d o m f o t s o c t o h s p a n s e h t e c u d o r t n i
o i t u l o v e e t a r e n e g y l n o t o n s e i t i n u m m o c e h t ,l e d o m s i h t n I .t n i a r t s n o
c nsbu talsoarer egularizedsot ha t
. y l e k i l n u s i e g n a h c c i t a m a r d
r e d i s n o
C i ngtheobservednetworkatt imet ,denotedby ,t henonnegativedatamatrix canbe s
e c i r t a m o t n i d e z i r o t c a
f ,.ie. , ,with theconstraintstha t and ea r nonnegative . f
e h t n
I actorization, can be considered to be a centroid matrix as each column represen ta e
l i h w , e d o n l a r t n e c y t i n u m m o
c canbeconsideredt obeacommunitymembershipmatrixwith t
a h t y t i l i b a b o r p e h t g n i t o n e
d et h node belongst o et h community .Thenwe definet hecentra lnodes o
c l l a f
o mmunities in dynamicnetwork as ,and thecommunity memberships of al l s
a k r o w t e n c i m a n y d n i s e d o
n ,where representst henumberofcommunitiesofthe .
s k r o w t e n c i m a n y
d
c o n e r a e r e h t , n o i t a z i r o t c a f e v o b a e h t n i , e l i h
W onstraintsonthecentra lnodecolumnvectors .To ,
s e c i r t a m d e n i a t b o f o y t i l i b a t e r p r e t n i d o o g e v e i h c
a considert hedynamicnetworkatt ime ,wei mpose g
n i n i f e d s r o t c e v n m u l o c e h t t a h t t n i a r t s n o c e h
t liewithint hecolumnspaceofdatamatrix :
r o
, . ( 1) s n o i t a n i b m o c x e v n o c o t t c i r t s e r n e h t e w , y t i l i b a t e r p r e t n i d o o g g n i n i a t b o f o s n o s a e r r o f , r e v o e r o M f
o thecolumnsof . Theadvantageofimposing theconvex constrain tistha tthecolumnscan be l a r t n e c f o n o i t o n e h t e r u t p a c d l u o w s n m u l o c e s e h t ; s e d o n n i a t r e c f o s m u s d e t h g i e w s a d e t e r p r e t n i
; s e i t i n u m m o c n i s e d o
n inaddition ,numberofnodest ha thaveunobviouspropensitiesofbelongingt o s
e i t i n u m m o
c woulddecrease .Inpractice ,movementsofcentra lnodesofteni nfluencet hemovements s i y l e t a r u c c a s e i t i n u m m o c n i s e d o n l a r t n e c g n i t c e t e D . m e h t h t i w p i h s n o i t a l e r e s o l c e v a h o h w s e d o n f o
. s k r o w t e n l a r o p m e t f o s n o i t u l o v e c i m a n y d g n i z y l a n a r o f l a c i t i r
c Wedefinet henodeweigh tmatrices
s a k r o w t e n c i m a n y d n i s e d o n l l a f
o .As aresult ,wecanreconstruc tthetopologyof
e m i t t a k r o w t e
n as .
y l e k i l s s e l s i t i t a h t o s e r u t c u r t s y t i n u m m o c e h t e z i r a l u g e r o
T forunreasonablydramaticchanges in
o r f s p i h s r e b m e m y t i n u m m o c e h t f o s m r e
t m time to ,we impose the tempora lsmoothness
i h s r e b m e m y t i n u m m o c n o s t n i a r t s n o
c p matrices .We define the tempora lcos tas the difference e
m i t t a s e c i r t a m p i h s r e b m e m y t i n u m m o c e h t n e e w t e
b andt ha tatt ime .
g n i r e d i s n o
C thesnapsho tcos tofmodeling networktopologiesand tempora lcos tofsmoothness . t s o c l a c i r o t s i h d n a y t i l a u q n o i t c e t e d y t i n u m m o c f o m u s e h t s a n o i t c n u f t s o c e h t e n i f e d e w , t n i a r t s n o c
y t i n u m m o c e h t g n i z i m i x a m y b s i h t e v l o s e w , n o i t c e t e d y t i n u m m o c l a r o p m e t h t o o m s e v e i h c a o T
a u q n o i t c e t e
d lityofcurren ttime-stamp andminimizing thehistorica lcost, then wehavefollowing :
n o i t c n u f
. )( 2 where isat empora lsmoothnessparametert hatt radesoffbetweent hefirs tandsecondt ermofthe
. n o i t c n u f e v i t c e j b o
To solveobjectivefunctionin(2) ,weproposeagradien talgorithmusingfollowing updaterules .
s n o i t c n u f y r a i l i x u a y b d e n i a t b o
r o f e l u r e t a d p u e h
T isasfollows:
. )( 4 ,
r e v o e r o
M ourmodeli seasilyextendedt ot henetworkswhosenumberofnodesandcommunities , e c n e g r e m e e h T . s k r o w t e n l a r o p m e t n i n o n e m o n e h p n o m m o c y r e v a s i h c i h w , e m i t r e v o e g n a h c y a m
s e c i r t a m p i h s r e b m e m y t i n u m m o c g n i z y l a n a y b d e t c e t e d e b n a c s e i t i n u m m o c f o r e g r e m r o t i l p s , h t a e d
.l e d o m r u o y f i d o m e w r e t f a
s tl u s e R d n a s t n e m i r e p x E
Int hissection ,wepresentt heexperimen tresultsofourproposedalgorithmrunningononesynthetic l
a e r e n o d n a t e s a t a
d -world dataset .We compare the results of our methods with three popular ,
] 5 1 [ F M N S : s d o h t e
m Facetnet ][ 9 and genLouvain algorithm [16] .First ,we introduce the two r u o s A . s t e s a t a d t n e r e f f i d f o s t l u s e r t n e m i r e p x e t n e s e r p n e h t d n a d e s u e w s e r u s a e m n o i t a u l a v e
r e t e m a r a p s s e n h t o o m s l a r o p m e t e h t t e s e w , r e t e m a r a p e h t o t e v i t i s n e s t o n s i m h t i r o g l
a as0.2 .In
t 0 2 d o h t e m h c a e n a r e w , n o i t i d d
a imesandreportt heaverageresults.
s e r u s a e M n o it a u l a v E
e h t e t a u l a v e o t d e s u e r a ) I M N ( n o i t a m r o f n I l a u t u M l a m r o N d n a e r o c s F g n i d u l c n i s c i r t e m o w T
. s c i r t e m o w t e h t f o s s e c o r p l a n o i t a t u p m o c e h t e c u d o r t n i y l f e i r b e W . m h t i r o g l a r u o f o e c n a m r o f r e p
b m o c e r o c s
F inest hei nformationofprecisionandrecal lwhichi sextensivelyappliedi nevaluating t
l u s e r n o i t c e t e d y t i n u m m o c e h
t [17] .Theprecisionandrecal larecalculatedas:
. )( 5
. )( 6 s
F e h t n e h
T coreoft hedetectedcommunity andt herea lcommunity canbecomputedas:
. )( 7 z
i l a m r o
N ed mutua linformation (NMI) isan increasingly popular measure ofclustering quality [ 71 ], whichcanbeformulatedas:
. )( 8
w o r G :t e s a t a D c it e h t n y
S -shrinkbenchmark
l e d o m k r a m h c n e b e h t o t g n i d r o c c a s p e t s e m i t 1 2 h t i w s k r o w t e n o w t e t a r e n e g e
W [18]and evaluate
o s t l u s e r r u
o nt heset wodynamicnetworks .Thisbenchmarkmodelst hemovemen tofnodesfromone .
r e h t o n a o t y t i n u m m o
c istheration of each community’snodes switching to othercommunities , r
o w t e n h c a e e t a r e n e g e W . s k r o w t e n r o f s c i m a n y d f o e e r g e d e h t s e c n e u l f n i h c i h
w kwithNnodesi n10
r o f d n a s n u
r and ,r espectively ; .I tcanbeknownf romf igure1andf igure2t ha t .
g i
F u 1re .AverageNMIresultsofsyntheticdataset : , .
g i
F u 2re .AverageNMIresultsofsyntheticdataset : , .
l a e
R -worldDatase:t KIT-Ema li
e e h t n o m h t i r o g l a r u o n a r e w , d o h t e m d e s o p o r p r u o f o e c n a m r o f r e p e h t e t a u l a v e o
T -mai l
e e d u l c n i k r o w t e n e h T . T I K t a s c i t a m r o f n I f o t n e m t r a p e D e h t n i k r o w t e n n o i t a c i n u m m o
c -mai l
e h t f o s t c a t n o
c departmen twhichchangesfromSeptember2006t oAugus t2010 .Theedgesaresparse , e
v i t c a 0 1 2 g n o m a s e c i r t a m y c n e c a j d a e h t t c u r t s n o c e w o
s members .Int hisnetwork,t hegroundt ruth
s s e c o r p e W . T I K t a e c n e i c s r e t u p m o c f o s t n e m t r a p e d e h t e r a s e i t i n u m m o c f
o thesnapshotsandge t
, s h t n o m 6 d n a 4 f o s t o h s p a n s e h t r o f , 7 2 d n a 5 2 s i s e i t i n u m m o c f o r e b m u n e s o h w s k r o w t e n o w t
w o h s 3 e r u g i F . y l e v i t c e p s e
r s theaveragevaluesofFscorefort hesnapshotsof4months ,andfigure4 t
r o f e r o c s F f o s e u l a v e g a r e v a e h t s w o h
s hesnapshotsof6months.
g i
s n o is s u c si D d n a s n o is u l c n o C
a t n e s e r p e w , r e p a p s i h t n
I unified mode lnamed TTNMF which can detec tcommunitiesand track s s e n h t o o m s l a r o p m e t g n i s o p m i d n a s e c i r t a m t h g i e w e d o n g n i c u d o r t n i f o e s u a c e B . s n o i t u l o v e r i e h t
h t i w d e r a p m o C . d e v o r p m i y l t n a c i f i n g i s s i n o i t c e t e d y t i n u m m o c f o e c n a m r o f r e p e h t , t n i a r t s n o c
t a t
s e- fo -ar tmethods ,ourmethodemploysthehistoryinformationmoreeffectivelyin theprocessof F M N T T t a h t w o h s a t a d l a e r d n a c i t e h t n y s e h t n o s t l u s e r l a t n e m i r e p x e e h T . s c i m a n y d g n i z y l a n a
a m o t u a e h T . n o i t u l o v e y t i n u m m o c n i s d o h t e m r a l u p o p r e h t o s m r o f r e p t u
o tic determination of the
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