Identifying Influential Spreaders of Epidemics on Community Networks

1 / 22 T i t l e : I de n t i f yi ng I nfl ue n t i a l Spre ad ers o f E p i de mi cs o n C o mm uni t y N et w o rks A ut h o r s : S hi - L o ng L u o, K a i G o ng * a n d L i Kang A ff i l i ati o n s : Sc h oo l o f E co n o m ic I nf o r m a t i o n Engine er i ng , So u t hw es t er n U ni vers ity of F i na n ce a n d E co n o mi cs, C h e ng d u , Peop le s R ep u b l ic o f C hi n a C o r r e s p o nd e n ce au t h o r : K ai G o ng , P h D Sou t hw es t er n U ni v e r s ity o f F ina n ce a n d E co n o mi cs 555, Liut a i A v e nu e, W e n j ia ng Dis t r i c t , C h e ng d u , S i c hu a n , P . R . C hin a, 611 130 . E - m a il : g o ng ka i 1210@ sw uf e.ed u .c n A BS TR A CT A n e ff i c i e n t s t rat eg y fo r the i d e n t i fi ca t i o n o f i n f lue n t i a l s p r e ad ers t h a t co u l d b e u s ed t o c o n tro l ep ide mi cs w i t h i n p o p ulation s w o u l d b e o f co ns i d e rab l e i m p o rt an c e . G e n e ral l y , p o p u l a t i o n s a re ch a ract e ri z e d b y i t s co mm u n i t y s t ruc t u re s an d by t h e h e t ero g e n e o u s d i s t ri bu tion s o f weak t i e s a m on g n o d e s b ri d g i n g o v er c o mm u nit i e s . A s trat e g y f o r c o mm u n i t y n e t w o r ks c a p a b l e of i d en ti fy i ng i n f l u e n t i a l s p read er s t h a t ac ce l e r a t e t h e s p read o f d i s e as e i s h e re p r o p o s ed . In t h is s t rat eg y , i n f lue n t i a l s p read e rs s erv e as ta r g et n o d e s . Th i s i s ba s ed on t h e i d ea t h a t , i n k- s h e l l d ec o m p o s i t i o n , w e a k t i es an d s t ro n g ti e s are p roc es s e d s e p a rate l y . Th e s trat e g y was u s e d o n em pirica l n e t w o r ks c o n st ruc t e d f ro m o n li n e s o c i al n et wo rk s , an d r e s u l t s i n d i ca t e d th at th i s s trat e g y i s m o re ac c u ra te th a n o t h er s t rat e g i e s . Its eff ec t i v e n e ss s t em s fr o m t h e p a t t ern s o f co n n ec t i v ity am o ng ne i g hb o rs , an d i t s u cc ess fu l ly i d e n t i fi ed t h e i m p o rtan t n o d e s . I n a d d i t i o n , th e p er for m an ce o f th e s t rate g y re m a i n ed rob u s t ev en wh en th ere w e r e e rr o r s i n t h e s t ruc t u re o f t h e n et wo rk . 1 . I NTR ODUCT I ON E p i d e mi cs c an l e ad t o s eri o u s l o s s of li fe an d t h e y h av e h u g e an i m p ac t on t h e eco n o m y (K e e l i n g an d R o h an i 2 0 08 ), as wi tn e s s e d d u ri n g th e 200 3 o u t b reak o f s ev ere acu t e r e s p i rat o ry s y n d ro m e ( SAR S ) ( G o m e z - G a rd en es et al . 20 08 ; C o l i z z a et al . 2 0 0 6 ) , th e 2 0 0 9 ou t b r e ak o f H 1 N 1 i n f l u e n z a A v i rus ( F ras e r et al . 2 0 0 9 ), an d t h e 2 0 1 3 o u t b rea k of H7 N 9 In f l u en z a A v ir u s (Sa l a t h e e t al . 2 0 1 3 ) . Kno w l ed g e reg ard i n g t h e p a t h way s b y w h i ch d i s e a s e s s p r e ad i n g t h ro u g h n e t w o r ks a n d h o w t h i s n et wo rk m i g h t b e 2 / 22 u s e d to prev e n t ep i d emi cs i s o f g reat i m p o r t a n c e . Th i s i s s u e ha s at t ract e d a grea t de al o f at t e n t i o n fr o m res ea rch ers acro s s v a ri o u s f iel d s ( Ki ts ak e t a l . 2 0 1 0 ; Sa l at h e an d J o n es 2 010) . I d e n t i fy i n g t h e i n f lue n t i a l s p read er s t h at ca n h i n d er t h e sprea d o f d i s ea s e e ff ec t i v e l y s o a s t o s u p p ress out b rea k s r e m a i n s a n o p e n i s s u e ( G h o s h a l an d B arab a s i 2 011) . H ub s , i n divi d ua ls wh o h a v e h i g h c e n t ra l i t y i n n e t w o r ks , are co mm o n l y b el i e v e d to b e t h e m os t i n f l u e n t i a l no d e s in th e s p read ing p roc e ss b e ca u s e t h e y c an aff ec t m an y ne i g h bo r s (P a s t o r- Sat o r ras an d V es p i g n a n i 2 0 0 1 ; C o h en e t a l. 2 0 0 1 ; G o h et a l . 2 0 0 3 ) . In t h e c as e o f n et wo rk s w i t h b ro ad -de g ree d i s t ri b u t i o n ( B a rab a s i an d Albe rt 1999 ), t h e de g re e s t rat e g y for t h e w e l l -co n n ec t ed ind i v i d u a l s h a s b ee n s h o wn to b e a n eff ici en t m e t h o d o f i d e n t i fy i n g eff i c i e n t s p read ers ( C o h en e t al . 2 0 01 ; Pas t o r- Sat o rr a s a n d V es p i g n a n i 2 0 0 1) . Bet we e n n e ss i s a n o t h er ce n t ral i t y s t rate g y. It i nv o l v e s m e a s u ri n g t h e n u m b e r o f s h o rte st p at hs t h a t cro ss th e cu rr e n t n o d e . It h as be en us ed t o d et e r mi n e w h o ha s th e mo st i nf l u e n c e o n o t h e rs i n n e t works (F ree man 1 9 7 8 ; G o h e t al . 2 003 ). Ho w e v e r , K i t s a k et a l . p o i n t e d o u t t h a t t h e m o s t eff i c ien t s p read ers a re t h o s e l o c at e d w i t h i n t h e c o re o f a n et wo r k as t arg et e d by th e k -she l l d e co m p o s i t i o n s t rate gy rat h e r t h a n t a r g et e d b y d e g ree an d be t ween n e ss (K i t s ak et a l . 2 0 1 0 ) . Th i s m e t h o d i s b as ed o n i t erat i v e pru n i n g o f n o d es w i t h d eg r e e s m a l le r t h a n o r e q u a l to t h e k -co r e i n d e x o f t h e cu rr e n t l ay e r un t i l ea ch n o d e i s as s o c i at e d wi th k -co re i n d e x t h at re f l e ct s t h e c o re o r p er i p h ery l ay er i n ne t wo rk ( C a r mi e t a l . 2 0 0 7 ). C o mm u nity s t ruc t u re ( Fort u n a t o 2 0 1 0 ) i s u b i q u i t o u s i n c o m pl e x ne t wo rk s ( G i rva n an d N e w m an 2 0 0 2 ), s u ch a s Face b o o k (Trau d et al . 2011) an d Tw i t t e r ( G o n ca l v es et al . 2011) . It s erv es an i m p o rt an t fun c t i o n i n t h e d y n am i c s o f e p i d e m ic ( Li u an d Hu 2 0 0 5 ; Wu an d Li u 2 0 0 8 ; Hua n g et a l . 200 6) . In t h e p res en ce o f c o mm u n i t y s t ruc t u res , he t ero g e n e o u s d i s t ri b u t i o n of t h e nu m b er of w e a k t i es w a s ob s erv ed am o n g r ea l n et wo r ks , s u c h as air traff i c n et works ( G u i m era et a l . 2 005 ), s o c i al n e t w o r ks ( A r e n a s et a l . 3 / 22 2 0 1 0 ), an d c o mm u n i c at i o n ne t works (O n n el a et al . 2 007) . T h e w ea k t i e s ( G r a n o v et t e r 1 9 7 3 ) co n n ec t i n g a p air o f no d es be l o n g i n g t o dif fer e n t co mm u n i t i e s h av e be en fou nd to pro v i d e s h o rtcut s f ro m o n e co mm u n i t y t o an o th er ( G o n g et a l . 2011) . Th es e ti e s h av e be e n p ro v e n t o b e m o re ef f ici en t i n d iff u s i n g d i s e as e s th ro u g h n et wo rk ( Heb e rt-D u f res n e e t a l . 2 0 13 ) . I d e n t i fy i n g t h e i n f l u e n t i a l s p r e a d ers i n co mm u n i t y n e t wor ks i s q u ite c h all e n g i n g b ec aus e t h e m es o s c o p i c f e a t u res o f the c o mm u n i t y s t ruc t u re are co m p l e x ( G o n g et a l . 2 0 1 3). H e r e , a k -she l l w i t h co m mun ity s t r a t e g y i s p rop o s e d . Th i s s trat e g y , b a s ed o n t h e i d ea o f t h e k- s h el l d e co m p o s i t i o n p roc es s , i n v o l v es i d e n t i fy i n g t h e i n f l u e n t i a l s p read e rs u s i n g w e a k ties an d s t ro n g t i es . R e s u lts d e m o n s t rate d t h at th e p ropos e d s t rate g y p erf o rm s be t t e r i n e m p i r ica l n et wo rk s th a n d e g ree, be t ween n e s s , an d k -s h el l d e c o m p o s i t i o n s t r a t e g i e s do . Si m u l a t i o n a l s o s h o ws t ha t t h i s s t rat eg y h a s th e m eri t o f be i n g s i g nif i ca n t l y r o b u s t ag ainst n o i s e . 2. D A T AS A N D M ODE L Th e pres e n t p a p er co m pa res r e s u l t s b ase d o n t h e c urr e n t st rat eg y a nd i d en t i fy i n g eff i cien t s p read e rs i n e m p i r i c a l n et wor k s with i n ep i d e mi o l o g ical m od els . Fi rs t , d et a i l s are g i v en wi th res p e ct to th e fol l o w i n g i s s u e s : ne t work co n s t r u c t i o n a n d dyn am i c m o d e l . 2.1 N e tw o r k C o n s t ru ct i on Em p i r i c a l n et wo rk s are co n s t ru ct e d u s i n g o n li n e co l leg i a t e s o c i a l d at a f ro m Face b o o k (ht t p s : / / c o d e .g o o g l e.co m / p /s o c i alne t w o r ks i m ulat i o n ). D a t a f ro m u n i v e rs i ties i n U. S. wa s s t u d i e d h e r e . It i n c l u d e s a n ony m o u s d a t a fr o m s t u d en t s fr o m C a l t e ch , P r inc et o n , G e o rge t o wn , an d t h e U n i v ers i t y o f O k l ah o m a . Th e d at a co n cern th e d o rm i t o r i e s , m aj o r s, an d ye a r for eac h i n d i v idu a l , an d d o r m ito r i e s w ere fou n d t o b e key el e m e n t s i n t h e s o c i al o r g an i z a t i o n o f l arg e u n ive rs i t ies (Tr a u d et a l . 2 0 1 1) . B a s ed o n t h e s e d a t a, th e ne t work s wer e co n s t ruc t ed b y l i n k ing up pa i rs of ind iv idu a l s w h o ( i ) we re o n l i n e f r i en d s an d l i v e d i n t h e s a m e d o r mi t o ry , o r ( i i ) th e y l i v e d i n d i f f e ren t d o r mi t o r i e s but ha d t h e s am e m a j o r an d y ear . 4 / 22 Th e gi a n t n et wo rk wa s t h e n us e d f o r t h e pres e n t s tud y. B as i c s t at i s t ical p rop e rti e s o f n e t work s are g i v e n i n T a b l e 1 . A l l n et wo rk s e x h i b i t ed s m al l -w o r l d ch a ract e r i s t i c s w i t h h i g h c l u s te ri n g co e ff i c i e n t an d s h o rt av e rag e pa t h l en gth . Th ey a l s o h ad h i g h m o d u l ari t y . 2.2 D yn a m i c M od e l In th e r e al wor l d , ind i v i d u al s wil l b e ab l e to co n t ac t o n ly a l i mi t ed n u m b er of p eo p l e a t on ce d e s p i t e t h eir w ide ac q u a i n t a n c e, an d c o n s ide rab l e e x a m p l es h av e b e en o bs erve d : e p i d emi c c o n t a c t n e t w o r k s (Zho u et al . 2 0 06 ), p e e r- to -p ee r d i s t ri b u t e d s ys t e m s (J o v an ov i c 2 0 0 1 ) , a n d n e t wor k mar ket i n g ( Kim et a l. 2 0 0 6 ) . Her e , th e s u sce p t i b l e i n fec t ed r ec o v e red (S I R ) w ith i d en tica l ca p a b i l i t y o f ac t i v e co n t a ct s m o d el wa s u s e d t o co m p are th e pe rf o r m a n ce o f d i f fe ren t i d en t if i ca t i o n s t rate g i e s ( Z h o u et al . 20 0 6 ) . I n t h e S I R m o d e l , e ac h no de i n eac h ne t w o r k rep res en t ed an i n d i v i d u al w h o c o u ld be i n o n e of t h ree s t at e s : s u s c e p t i b le, i n f e ct e d , o r reco v ered , an d ea ch l i n k b e t w e e n no d es rep res en t ed o n e co n n ec t i o n t h a t co u l d s p read a n i n fect ion . I n i t iall y , a l l no d es we r e s u s c epti b l e. T o i n i t i a t e an infect i o n , on e n o d e wa s r a n d o m l y ch os en an d co n s i d e red i n fec t ed . E ac h s t ep an d ev e ry i n d i v i d u a l h a d s a m e i n fect iv ity A = 2, i n wh i ch ev e ry i n fec t ed i n divi d ua l g en erat e d i d en t i c a l co n t ac t s , m ult i p l e co n ta ct s to o n e ne i g h b o r we re a l l o w e d , an d th e pro b a b i l i t y t h at a g i v e n s u s ce p t i b l e n o d e w ou l d b e i n fect e d was   = 0. 5 . Th e p rob ab i l ity th at an i n fect e d n o d e w ou l d r e c o v e r w as  = 1 . O n ce a n i n d i v i d u al w a s r e c o v e red , t h e re w ou ld b e n o f u r t h er ch a n g e. In th e s i m u l at i o n , s t at e s o f e v e ry n o d e we r e up d at e d s y n c hro n o us l y . Th e d y n a m i c s en d ed w h e n al l i n fec t ed r e co ve red . Th e a v erag e s i z e o f th e i n f e ct ed , M , a n d th e fr a ct i o n o f th e p o p u l at i o n e v e r i n fect e d at t h e end o f t h e ep i d em i c w e re r ec o rd ed , a l l o w i n g qu an t i fi ca t i o n of th e i n f l u e n c e o f g i v en n o d e o n s p rea d i n g p roc e ss . 5 / 22 3. M E T H ODS 3.1 Id e n t if i cat i o n St r at e gie s Th e i d ea s b e h i n d d e g ree , b e t ween n e ss , a n d k- s h e l l d ec o m po s i t i o n s t rate g y a re o u t l i n ed , an d t h e cu rren t s trat e g y i s d i s cuss e d . B ri e fl y , d e g ree s t rat eg y wa s ba s e d o n t h e i d e a t h a t m o s t i n fl u en c e n o d e s w o u l d b e th os e w i t h t h e l arge s t n u m b er o f co n n e ct ion s , an d i t i s on e m ea s u re o f l o c al i n f l u e n c e: o n l y th e s t ruc t u re arou nd th e node ha s t o b e co ns i d ered (L a t o ra an d M arch i o ri 20 0 7 ). B et ween n ess m ea s u res t h e n u m b er o f s h o rte s t p a t h s f ro m a l l n o d es t o ot h ers th a t c r o ss t h ro u g h th a t n o d e. K i t s a k e t a l . argu es t h a t t h e s t ruc t u re o f n e t w o r k o rga n i z at i o n s erv e s an i m p o r t a n t f u n c t i o n su ch t h a t the re a r e p l a u s i b l e ci rcu m s t an ces u n d er wh i ch t h e m o s t d e g ree o r h i g h e s t b e t ween n e ss a s i n fl u e n t i a l s p read ers h av e t h e l e as t p ron o u n c ed i m p a ct o n t h e s p r e ad i n g p r o cess ( Ki t s ak e t a l . 2 0 1 0 ) . K- s h el l d ec o m po s i t i o n i s o n e s trat e g y b as e d o n i t erat i v ely p run i n g o f n o d e s wi t h d eg r e e no m ore t h a n k -co re i n d e x o f t h e cu r ren t l a y er . Th e h i g h es t k -co r e i n d ex is cl o s el y re l a t ed t o t h e co n cep t o f m o s t i n f l u en t i a l n o d es o n s p read i n g p roc e s s . Th e y u s e d t h e st rate g y by i d en ti fy i n g th e co r e a n d p e r i p h e ry o f g i v e n n o d e i n rea l n et wo r k t o i d en t i fy key s p read er s a n d fou n d t h a t k - s h e l l de co m p o s i t i o n s t rat egy i s m o re acc u rat e . In fact , u n d er r e al - worl d co n d i t i o n s , su c h as t h o s e i n w o r l d w i d e a i r t raf f ic n et wo r k a n d e m p i r i c a l n et wor k s ( G o n g e t a l . 2 0 13 ; G u i m e ra et al . 2 005) , t h e he t ero g e n e o u s di st ri b u t i o n i n th e nu m b er of weak t i e s am o n g n o d e s i n d i c at e d a p ron o u n c ed d i ff eren c e i n th e s p read i n g p roc ess . U n f o r t u n at e l y, th i s i s s u e w as n o t t aken i n t o ac co un t i n t h e p ro ce s s o f k- s h el l d ec o m p o s i t ion . 3.2 K -S h e l l wi t h C o mm u ni ty St r ate gy Th e k - s h e l l w i t h co mm un i t y s t rate g y i s h e re pro p o s e d a s a m ea n s o f m o r e eff e ct i v e i d en ti f i c at i o n . It i s b as e d o n t h e i d e a t h at k -s h el l d e co m p o s i t i o n i nv o l v e s b o t h w e ak t ies an d s t ron g ti e s . H ere, s t ro n g ti e s are t h o s e f o r which t h e s o u rce a n d t arg e t o f t h e co n n ec t i o n l i e i n s i d e th e s am e co mm u n i t y. F i g u r e 1 s h o w s an ex a m p l e of w e a k t i e s an d s t ron g ties i n s i m p l e c o mm u nity n e t w o rk s th r o u g h v i s u a l i z at i o n . 6 / 22 Th e k -s he l l w ith co mm u n i t y s t rat e g y s t a rts w ith s u c ce ss i v e p run ing o f t h e n et wo r k b y re m o v i n g n o d e s w ith w e a k t i e s an d s t ro n g ties s e p a rat el y . Th is p roc e s s h as t h ree m a i n co m p o n e n t s : (i ) r e m ov al o f n o d e s w ith w e a k t i es , ( i i) r e m o v a l o f n o d es w i t h s t ron g t i es , an d (i i i) as s i g n me n t o f k - s co r e . (i) A ft er i n i t ia l i z a t i o n o f n e t w o r ks b y re m ov al of a l l node s w i t h s t ron g t i e s , al l n od es wi t h w e a k t i es k w = 1 a m o n g no d e s b r i d g i n g o v e r c o mm u n i t i es w e re t h en r e m o v ed , an d s o m e no d es w ith weak t i es m ay h av e r e m a i n e d , s o we co n tinu e d p r u n i n g t h e n e t w o r k r e p e at e d ly u n t i l t h e re w a s n o nod e l eft w i t h k w = 1 i n t h e n et work . Th es e n o d e s are as s oci at e d w ith an i n de x k W co r e = 1 . In a s imi l a r s t e p i n th e or i g i n al w o r k , t h e ne x t l e v e l , k w = 2, w a s i te rat i v e l y re m ov ed an d t h e s ys te m co n ti n u ed rem o v i n g h i g h er k w un ti l al l n o d es w e re as so c i at e d wi t h an i n d e x of k W co r e . (ii ) A fter i n i t i a liz a t i o n of ne t w o r ks b y r e m o v al of a l l weak t i es , a pro ce d u re an a l o g o u s to p r e v i o u s co m p o n e n t w a s rep ea t e d b y rem o v i n g n o d e s w ith s t ro n g t i es k s = 1 u n ti l t h e re we re n o n o d e s l eft wi t h k s = 1 . Th e s e n o d e s are ass o c i at e d w i t h an i n d e x of k S co re = 1 . Th e p r o ce d u re was r ep ea t ed u n t il ea ch no d e wa s ass o ci a t ed wi t h an i n d e x k S co re . (ii i ) Fi n a l l y , th e k -sco re w a s ass i g n e d t o e ac h n o d e . I t can be de fi ne d u s i n g t h e f o l l o w i n g e q u at i o n : k- s co re ( ) ( 1 ) ( 1 ) sw co re co re kk          ( 1) Th e pa ram et e r of  α  g i v i n g o n e c o e ff i c i e n t b e t w e en 0 a n d 1, w h ere 1 wa s prefer r e d to h u b n o d e s i n l o ca l s t ruc t u re an d 0 h a d a g reat e r fo cu s on b ri d g e hub s , was s h o w n i n a p rev i o u s w o r k ( K . G o n g et al . 20 13 ) . Th e s trat egy i s i l l u s t rate d s c h em at i ca ll y in F igu re 2. 4. R E SU L T S 4.1 H e te r oge n e it y and R ol e s in E mpi r i cal Ne t w o r ks T o ill u s t rat e th e n ec e ss i t y o f a n i d e n t i fi cat i o n s t rat e g y t h at f o c us es o n t h e m o s t e f fi cien t s p r e ad ers in n et wo rk s us i n g h e t ero g e n e o us d i s tri b u t i o ns , t h e di s t ri b u t i o n s o f t h e cu m u l at i v e p rob ab i l ity d en s i t y we re 7 / 22 an a l y z e d a re b y w e a k t i e s an d st ro n g t i e s i n real n e t work s are d en o t ed ( Fi g u re 3 ) . Th e p h e n o m e n o n of h et e rog en e i t y i n dicat e s t h at n o d e s h ave s ign i fi ca n t ly d if fe ren t ac c o rdi n g t o t h eir p at t e rn . Th i s p h e n o m en o n al s o ind icat e s t h e stri k i n g i n fl u en c e t h a t h et e rog e n e i t y h a s on t h e s p r e a d i n g p ro ce ss . A s s h o wn i n F i g u re 4, r o l es o f n o d es b y sh a p e s i n Z- P p ara m et er s p ac e i n e m p i r i c a l ne t work s i n d icat e d t h at fun c t i o n al d i f fe ren c e s w ou l d em e rg e b e c au s e o f he t ero g e n e i t y (G u i m e ra a n d A m a ral 2 0 05 ). Her e , Z , th e in t ra-co mm u n i t y s co r e m e as u r e d h o w ma n y c o n n e c t i o n s n o d es h av e i n t h e s a m e c o mm u nity , an d t h e p art i cipa t i o n c o e ff i c i e n t P , o f t e n m eas u res h o w wel l d i s t ri b u t e d t h e w e a k t i e s o f n o d e s are a m o n g co mm u n i t i e s . Th es e ap p roa ch 1 i f co n n ec t i o n s ar e un i f o r ml y d i s t ri b u t e d am o n g al l co mm un i t i e s an d 0 i f are al l w i t h i n i ts v i c i n i t y . It ca n b e d efi n e d u s i n g th e f o l l o wi n g two eq u a t i o n s : () s s g g kk Z      (2) 2 1 ( ) g w k P k   (3) H e re , k s is th e n u m b e r of s t ron g ti e s , < k s > g i s t h e a v erag e s t ro n g t i e s i n s i d e co mm u n i t y g ,  g i s t h e s t a n d a rd d ev i at i o n o f < k s > g , an d k g W i s th e w ea k ti e s i n s i d e co mm u n i t y g . k is th e to ta l d eg r e e o f n o d e. Fi gu r e 4 s h ows th e f o l l o wi ng : ( i ) l o ca l h u b ( Z ≥ 2 . 5 ) i n d icat ed th a t n ea rl y al l co n n ect ion s were l o ca l ( P < 0. 2 ) o r a c ros s a few c o mm u n i t i e s ( P < 0 . 6 ); a n d ( ii ) s i z ea b l e n o r m a l n o d es ( Z < 2. 5 ) are wel l d i s t ri b u t e d i n t h e en tir e n et wor k ( 0 .6 ≤ P < 0. 8 ) . Th at w i l l m a k e it n e ce s s a ry for i d e n t i fy e ff i c i e n t s p read er s w i t h t h e h et e r o g en ei t y. 4.2 C o mp a r i s o n of S p r e ad i n g E ff i cie ncy T o c o m pa re th e eff ic i en cy o f d eg r e e, b e t w e en n e s s , a n d k - s h el l w i t h t h at o f o u r s t rate g y, t h e SI R w as p erfor m ed wi th i d e n t ical ac t i v i ty d y n a mi cs o n e m p i r ica l n et work s . Th e i mpre ci si o n fun c t i o n  , i n a p rev i o u s w o rk ( K i t s a k et al. 20 1 0 ), q u an t i f i e s th e d i ff eren c e b e t w e e n t he av e rag e i nfec t ed b et ween t h e f N 8 / 22 n o d e s (0 < f < 1) w i t h t h e high e s t d e g ree , b e t w e en n ess , k -co re i n d e x , k -s co r e , a n d t h e av e rag e i n f e c t ed o f t h e fN m o s t eff i c i en t s p read ers . Fo r a g i v e n fr ac t i o n f t h e s e t of th e fN m o s t eff i c i e n t s p reader s w a s f ir s t i d e n t i f i ed as m eas u red b y M ef f . (de s i g na t e d F ef f ). S i mi l a r ly , t h e fN no de s w i t h th e h i g h e st k - s c o re ( F k -s co re ) w e re i d en ti f i e d . In th i s w ay , t h e imp rec i s i o n o f k -s co re can be de fi ne d a s f o l l o ws : k-s core k-s core eff M ε =1 - M ( 4 ) H e re, M k -s co re a n d M ef f a re t h e a v e rag e i n f e ct e d p ercen t a g e s o v er t h e F k - sc o re a n d F ef f s et s o f n o des , res p ec t i v e l y.  k -s co re ap p ro ac h e s 0 , an i n d i c at ion t h a t t h e h i g h es t n o d e s ar e ch os e n b y t h e strat eg y an d ar e u s ual l y t h o s e t h a t c o n t ri b u t e t h e m o s t t o ep i d em i c s , an d v i ce v e rs a.  d eg ree ,  b et w ee n n e s s , a n d  k -co re are d efi n e d s imi l a rl y t o  k -s c o re . Here, t h e di f fer e n c es  ∆  i w o u l d i n d i c at e th e s p rea d i n g eff i ci e n c y o f s t r a t e g y rel at i v e to t h e p rop o s ed s t rat e g y i n th e fN s e t ca n b e d efi n e d u s i n g th e f o l l o wi n g eq u a t i o n : i k- score i i ε - ε Δε = ε (5) In m o s t c as e s , th e d i ff e ren ce s o f ∆  a re po s i t i v e o v e r a l m o s t a l l o f t h e s e t o f d i ff e ren t s t rat eg i es (F i g u re 5) . Fo r e x a m p l e , k -s co re wa s o n a v erag e 7 .6 6 % ( 4 0 . 4 3 % , 5 5. 0 6% ) h i g he r t h a n k -co re (de g ree , b et wee n n e ss ) f o r t h e C al t ec h d at a s e t , 5 .2 8 % ( 1 2 .9 1 % , 4 2 . 1 9%) h i g h e r fo r t h e Pr i n cet o n se t , 5 .66% (17 . 8 8 %, 4 1. 6 5%) h i g h er f o r t h e G eo r g et o wn s et , an d 4 . 93 % ( 1 5 . 4 8 % , 40.3 4 % ) h i g he r fo r th e U n i v e rs ity o f O k l a h o m a s e t . 4.3 C o mp a r i s o n of Ave r age N u m b e r of Conn e cti o n s a m o n g N e i gh b o r s As s p readers , w h e t h er i t ha s i n f l u en c e on s p read i n g p roc e s s is clos e l y rel at e d t o i t s p a t t ern o f co n n ec t i o n s i n s p e ed i n g u p t h e t ran s m i s s i o n o f ep i d emi c s. Th e eff e ct i v en e ss of k -s co re is i llus t rat e d i n Fi gu r e 6 , wh i ch c o m p ares t h e w ea k ti e s a n d s tro n g ti e s am o n g n ei g h b o ri n g n o d e s ide n t i fi ed us ing d i ff e ren t s t rat eg ies i n n e t work s . H e re, t h e s et   i s th e u n i o n o f n e i g hbo r s o f n o d e s t ho s e i d e n t i fi ed b y 9 / 22 s t rate gy . L e t < k w >(  i ), w i t h i de n o t ing s t rat eg y, b e t h e av erag e n u m be r o f weak ties with i n th e u n i o n  afte r a p p l y i n g s ai d s t rat e g y . S i mi l a rl y , < k s >(  i ) m a y b e t h e av erag e n u m b e r o f s t ro n g ties . Th e dif fer e n c e ∆ < k w >(  i ) an d ∆ < k s >(  i ) are t h e m ea s u re o f h o w e ff e ct i v e k -s co re i d e n t i fy loc al h u bs an d b r i d g e h u bs w ith i n n et wor k s. Th i s can b e de fi n ed us i n g t h e f o ll o w ing t wo eq u a t i o ns : w i w k- sc ore w i Δ < k > Γ =< k > ( Γ ) < k >( Γ )  (6) s i k- sc ore i Δ < k > Γ =< k > ( Γ ) ( Γ ) ss  ( 7 ) A s sh o wn in F i g u re 6 , ∆ > 0 i n a l m os t al l th e ca s es i n d i ca t e s t h a t n o d e s i d en ti fi ed b y k- s co r e ha v e m o re e x t e n s ive co n ne c t i o n s o f n e i g h b o r s wi t h well-co n n e ct e d t h a n o t h e r st rat eg i es d o , i m p ro v i n g t h e eff e ct ive n e ss of ide n t i f i ca t i o n fo r e ff i c i en t s p read ers . 4.4 R ob u s t n e ss of th e S tr a t e gy A n o t h er im p o r t a n t as p ec t of a n eff e ct ive s trat e g y i s th e r o b u s t n e ss t o m i s s i n g o r n o i s e i n fo r m at i o n . Suc h er ro rs ar e co mm o n i n r ea l n et wor k s du e to , for e x a m p l e, t h e inc o n s is t e n c y wi th w h i c h t wo i n d i v i d u a l s de s c ri be th e i r rel at i o n s h i p ( L u et al . 20 1 1 ). Us i n g e m p i ri ca l ne t wor k s , t h e i nc o m p l e t e i n for m a t i o n by ran doml y re m o v i n g th e p erce n t a g e o f c o n n ec t i o n s f , t h e n h a v e t e s t e d t h e r o b u s t n e ss b y u s i n g K en d al l ran k c o r r e l a t i o n co e ff i c i e n t  ( -1 ≤ τ ≤ 1) (K en d al l 1 9 3 8) , wh i c h m ea s u res t h e s im i la ri t y of t h e o r d eri ng o f n o d e s w h en ran ked b y k -s co re on t h e i n co m p l et e a n d o ri g i n a l n e t wor k. T h i s c an b e d efi n e d u s i n g th e f o l l o w i n g eq u at i o n : nu mb er of c onc ord ant pair s nu mb er of d iscor dan t pa irs ( 1 ) / 2 NN     (8)  i s v ery s i mil a r to 1 d en o t e s an e x ac t ag ree m e n t b e t ween t wo ran k s for bo th e l e m e n t s , an d v i s e -ve rs e. Fi gu r e 7 s h o ws th a t the pro p o s ed s t rate gy p erf o r m ed s t ab l y at     ev e n f l arge r t h an 20 %. 5. C ONC L U S IO N S AND D ISC USSIO N Th e he t e rog e n e i t y d i s t ri b u t i o n o f w ea k t ies un d er r ea l - worl d co n d i t i o n s is i m p o rt an t t o i d en tif y i n g 10 / 22 t h o s e n o d e s w h ich are m o s t inf l u e n t i a l s p read e rs i n t h e t ran s m i s s i o n o f d i s ea s e s , b u t th e s e n o d e s a re d i ff i c u l t to ide n t i fy in c o mm un i t y n et wo r ks . In s u mmary , t h e k -s h el l wi t h co mm u n i ty s t rate g y p r o po s ed h ere w a s s t u d i e d for eff ec t i v e n e s s . Th ere i s on e s trat e g y for ide n t i fying n o d es . This s t rat e g y i s b as e d o n t h e i d e a t h a t k -s h el l d e co m p o s i t i o n i n v o l v e s b o t h w e a k an d s t ron g t ies . Th e cu rr e n t s t rate g y w as use d t o em p i r i ca l l y te s t ed n e t work s a m o n g s tu d en t s i n U. S . u n i v e rs i t i e s co n s t ruc t ed b y u s ing o n l i n e s o c i al n et wo rk s . In m o s t ca s e s , th e c u rr en t st rat eg y w e re f o u n d to b e m ore eff e c t i v e i n i d en ti fy i n g i n f l u e n t i a l s p read er s t h a n o t h er m e t h o d s , s u ch a s d e g ree, b et ween n e ss , an d k- s h e l l d e co m p o s i t ion s trat e g y for t h e ran g e o f t h e s et , and resu l t s wer e m o r e ac cu r a t e . I t s e ff e ct i v en e s s c a n b e a t t ri b u t e d t o t he n eigh bors i d e n t i f i ed b y o u r s t rat e g y b e ca us e t h e p a t t ern o f co n n e ct i o ns a m o n g n ei g h b o r s was m o re s t rik i n g th an t h o s e p ro d u c e d u s i n g o t h e r s t rat e g i e s . For t h i s reas o n , i t c an s p read t h ro u g h t h e en ti re n e t wor k m o re q u i c k ly an d ef fec t i v el y . The cu r ren t m et h o d k e p t i ts s t a b i li t y w e l l t h rou g h o ut t h e m i s s i n g pro ce ss e s , a n d h as al s o b e en stron g ly p e r form ed rob u s t n e ss . Th e pe rf o r m a n ce o f a n y s t rate g y is i n fl ue n c ed by t h e s t r u ct u re o f n et work s . In th e cu rr e n t m e t h o d , w h en α ex cee d e d th e th r e sh o l d lev el , s uc h a s w h e n i t ap pro ac h ed 1 , t h i s w as t a k e n t o i n d i c at e the hi g h e s t p rob ab i l ity o f i d e n t i fy i n g th e h u b n o d e s i n a s i n g l e area. W h e n i t appro ac h e d 0 , th a t w as t aken to i n dicat e t h at the m e t h o d o n ly p referr e d t o c h o o se b ri d g e - h u b s ove r c o mm u n i ties . B o th t h es e c o n d i t ions were u n a cc e p t a b l e . H en c e , f i n d i n g t h e o p t i m al va l u e s o f t h e va ri ab l e i s s till an u n res o l v ed i s s u e . F I G U R E LE G E N D S FIGU R E 1 . V i s ua l i zi ng t he w e ak ti es a n d s t r o ng ties in t he n e t w ork . N o d es fr o m t wo co mm u n i t ies as i l l u s t rat e d by dif fer e n t l o g o s , al s o il l u s t rate d ar e s t ro n g ti e s ( s o l id) t h a t s o u rce a n d ta rge t o f th e co n n ec t i o n l ie i ns i d e s am e co mm u n i t y an d wea k t i es ( d a s h ) be t ween c o mm u n it i e s . FIGU R E 2 . S chemat i c i l l us trat i o n o f s tra teg i e s b e t wee n k -s hell d e c o m po s i tio n a n d k -s hell w i th 11 / 22 co m m u ni ty . Th e s ch e m a t i c rep res en ta t i o n o f a n e t work un d er t h e k- s h el l d ec o m po s i t i o n s t rat eg y ; al l n o d e s i n i n n erm o s t co re h a v e s a m e k -co re i n d e x , e v e n th o se wh i ch h a v e weak ti e s ( d a s h e d l ine ). H o w e v e r, i n t h e p r o c e ss o f t h e k -s h el l w ith co mm u n i t y s t rat e g y, t h e s e n o d e s h a v e s a m e i n d e x fo r k S co re an d dif fer e n t i n d e x fo r k W co re , 2 fo r A , 3 for B, 1 f o r C , an d 0 fo r D. F i n a l l y , t h e n o d e B l ies at th e m o s t i m p o rt an t l o c at ion o f t h e en t i re ne t wor k , th a t i s , h a s h i g h e st k -s c o re = 3 . 5 co m p a re w i t h o t h e r s ( 3 for A , 2 . 5 f o r C an d 2 f o r D ), ac co r d i n g to eq u at ion (1 ) . H e r e ,   = 0 .5. FIGU R E 3 . C u mulat i v e d i s tri bu ti o n o f the n u m b er o f w e ak t i es a n d s tro ng t i es i n e m p i ri c a l net wo rk s . Fo r e v ery n et work , c o mm un i ty s t ruc t u re w as de t e c te d us i n g th e m et h o d pro p os e d b y V i n cen t et al . ( V i n ce n t et a l. 20 0 8 ) . Th e wea k t i e s an d s t ro n g t ies we re th e n i d e n t i f i ed , an d t h e n u m b e r of w ea k t i es k w an d st ro n g t ies k s em a n a t i n g fr o m e ac h n o d e w e re r e c o rde d t o g i v e t h e c u m u l at i v e d i s t ri b u t i o n . Em p i r i c a l r es u l t s a re s h o w n for C a l t e ch (s q u a res ), P ri n ce t o n (ci rc l es ), Geo rge t o wn ( u p w a r d - fac i n g t ri a n g l es ), a n d t h e U n i v e rs i ty o f O kl ah o m a (d o wnward- faci n g tr i an g l es ). FIGU R E 4 . R o l es a n d reg ion s i n the Z - P s pa ce f o r e mp i r i c a l net wor k s . H e r e , w e c l a s s i fi ed n o d e w ith Z ≥ 2 . 5 as l o c a l hu b an d n o d e w i t h Z < 2.5 as n o r m al n o d e ac c o rdi n g to a p rev i o u s w o rk ( G u i m era an d A m a ra l 2 005 ) , no d es ca n b e n at u ral l y as s i g n e d i n t o reg i o ns : ( 1 ) P < 0.2 ; ( 2 ) 0 . 2 ≤ P < 0 .6 ; ( 3 ) 0 . 6 ≤ P < 0 . 8 ; ( 4) P ≥ 0 .8. FIGU R E 5. Com pa ri s on of s p re a ding e ff i ci e ncy o f i d e nt i fi c at i on in e mp i r i c a l n e t w ork s . Th e d i ff e ren ce i n im p rec i s i o n ∆  i , i d en o t ing d e g ree (l eft p an e l ), b e t ween ne ss (m i d d l e p a n e l ) , an d k -co re (r i g h t p a n e l ), a re s h o wn f o r e ac h n e t wor k s a s fun ct i o n of f . Th e p o s i t i v e p ercen t a g e s s h o ws th a t k- s c o re i s m o re ac cu r a t e . R e s u l t s are o b t a i n e d b y av erag i n g o ver 2 0 0 0 f o r ea ch n o d e . Her e,  Cal t ec h = 0 . 9 5 ,  P ri n ce t on = 0 . 9 5 ,  G eo rg e t o w n = 0 . 8 8 , an d  Okla h o ma = 0 .96 for ne t wor k s , r e s p e ct i v el y. FIGU R E 6 . Compa r i s on o f a v e rage n u m b er o f wea k t i e s a n d s tron g tie s a mon g nei g h bo rs o f no des 12 / 22 i d en tifie d by k -s c o r e w i t h t ho s e o th e r s . T h e d i ff e ren ce ∆ < k w >(  i ) i n w e a k t i e s ( s i mi l a r to ∆ < k s >(  i ) i n s t rong t i es ) wi th i d e n o t i n g d eg r e e ( cir cles ), b et ween n ess ( up ward-f a c i n g t r i a n g l e s ) an d k -co re ( s q u a res ) o f n e i g h b o r s e t  are s h o wn for di f fer e n t f i n e m p i r ica l n e t w o r ks . H e re,  Cal t ech = 0.9 5,  P ri n cet o n = 0 . 9 5 ,  G eo r g e t o w n = 0 . 8 8 , an d  O k l ah o m a = 0.9 6 for n et wor k s . FIGU R E 7 . R o bu s tnes s o f th e s tra teg y i n net w ork s w ith n o i s e a s m o d e l e d by ra n do m r em o va l of con nec tio ns .   i s s ho w n as th e f u n ct i o n f of nu m b e r o f co n n e c t i o n s r an d o m l y r e m o v ed fr o m n e t works . R es u l ts are o b t ai n ed b y av e rag i n g o v er 2 0 0 0 rea l i z a t i o n s f o r eac h va lue of r e m o ve d co nn ec t i o n s . H e re,  Cal t ec h = 0 . 9 5 ,  P ri n ce t o n = 0 .9 5 ,  G eo rg e t o w n = 0 . 8 8 , an d  O k l a h o m a = 0 . 9 6 for n et wor k s . R E FE R E N CE S A ren a s , A ., B o rg e - H o lth o e fer , J ., G ó m e z , S., & Za m o ra - L ó p ez , G. (20 1 0 ). O p ti mal m a p o f th e m o d u l ar s t ruc t u re o f co m p l e x n et work s . N e w Jo u rn a l o f Phy si c s, 12 (5 ) , 0 5 3 0 0 9 . B arab a s i , A . L . , & A l b ert, R . ( 1 9 9 9 ). Em e rg en ce o f s ca l i n g i n r a n d om n e t w o r ks . S c i en ce , 2 8 6 (5 4 3 9 ) , 5 0 9- 5 12. C ar mi, S ., Hav l i n , S . , K i rk p a t r i ck , S ., Sh av i tt , Y . , & Sh i r , E. ( 2 0 0 7 ) . A mo d el o f I n t e rne t to p o l o g y us i n g k -s h el l d e co m p o s i t i o n .Proc e ed ing s o f t h e N a t i o n a l Aca d emy of S c i e n ce s o f th e U n it e d S t a t e s o f Am e ri c a , 10 4 ( 2 7 ), 1 1 1 5 0 -1 1 1 5 4 . C o h e n , R . , E r e z , K. , B e n -A v rah a m, D ., & H av l i n , S . ( 2 0 0 1 ). B rea kdo w n o f t h e i n t ern et u n d er i n te n t i o n al at t a ck. Ph y si ca l Rev i e w Let t ers, 86 (1 6 ), 3 6 8 2 -36 85. C o l i zz a , V ., B a r rat , A ., B a rth el e m y, M., & V e s p i g n an i, A . ( 2 0 0 6 ) . Th e r o l e o f th e a i r line tran s p o r t a t i o n n et wo rk i n t h e p r e d i c t i o n an d p r e d i c t a b i l i t y o f g l o b al e p i d e mi c s . Pro c ee d i n gs o f th e N a t i o n a l Aca d emy of S c i en ces o f th e U n i t e d S t a t e s of Am e ri c a , 10 3 (7 ) , 201 5 - 2 0 2 0 . Fo r t u n at o , S . ( 2 0 1 0 ) . C o mm u n i ty d et e c t i o n i n g rap h s . Ph y si cs Rep o rts, 48 6 (3 - 5) , 75- 1 7 4 . Fr a s er , C ., Do n n e l l y, C . A . , C au c h e m e z , S ., H an ag e , W . P ., V an K e rk h o v e , M. D ., H o ll ing s wo rth , T . D . , e t al . ( 2 009) . Pan d e m i c p o t e n t i a l of a s t rai n o f i n f l u en z a A ( H 1 N 1 ) : ear l y f i n d i n g s . S c i e n ce , 3 2 4 (5 9 3 4 ), 1 5 5 7 - 1 5 6 1. Fr e e m an , L . C . ( 1 9 7 8 ). C e n t ral i t y i n s o ci a l n et work s co n c e p t u a l cl a r i f i c at i o n . So ci a l Ne t w o r k s,1 (1 9 7 8/7 9) , 2 1 5- 2 3 9. G ho s h al, G. , & B a rab á s i , A . L. ( 2 011) . R a n k ing s t a b i l ity an d s u p e r -s t a b l e n o d e s i n c o m p l e x n e t work s . Na tu re C omm u n ic a t i o n s, 2 (7 ) , pp .3 9 4. G i rv an , M., & N e w m an , M . E . J. (20 0 2 ). C o mm u n i t y s t ru ct u r e i n so c i a l a n d biol o gical ne t w o r ks . Pro ce e d i n g s o f th e N a ti o n a l Ac a d emy o f Sc i en ce s o f th e U n i t ed S t a te s o f A m e ri ca , 99 ( 12), 7 8 2 1 - 7 8 2 6. G oh , K .-I ., Oh, E., K a h n g , B ., & K im , D . ( 2 003) . B et wee n n e s s ce n t ral i t y co r re l a t i o n in s o cial n e t w o r ks . Ph y sic a l Rev i e w E, 6 7 (1 P t 2) , p p .01 710 1 . G ó me z - Gar d eñ es , J., L at o ra, V . , M oren o , Y ., & Profu m o , E . ( 2 0 0 8) . Sp read i n g o f s ex u a ll y t ran s mi t t e d 13 / 22 d i s e as e s i n h e t ero s e x u al p o p u l at i o n s . Pro ce e d i n g s o f t h e N a t i o nal Ac a d e my o f S c i en ces o f t h e U n it e d S t a t e s of Am eri c a , 10 5 (5 ), 1 3 9 9 -14 0 4 . G on ca l v e s , B . , Pe rr a , N ., & V es p i g n a n i , A . ( 2 01 1 ). M o d el i n g U s e rs ' A c t i v i t y o n Twi t t e r Netwo r ks : V a li d at i o n o f Du n b a r' s N u m b e r . Pl o s One , 6 ( 8) , e2 2 656. G on g , K. , T an g , M ., Hu i , P. M. , Zh a n g , H. F ., Y o u n gh ae , D ., & L ai , Y . - C . ( 2 0 1 3) . A n eff i c i en t i mm u n i z a t i o n s trat e g y f o r co mm un i t y ne t wor k s . Pl o s O n e , 8 (12 ) , e 8 3 4 8 9 . G on g , K. , T an g , M . , Y a n g , H. , & Sha n g , M . ( 2 0 1 1 ). V a ri ab i l i t y o f c o n t a ct p ro ce ss i n c o m p l e x ne t work s . Ch a o s, 2 1 (4 ) , 0 4 3 1 3 0 . G ran o v et t e r , M . S. ( 1 9 7 3 ). Th e S t ren g t h o f Weak Ti e s . Ameri ca n Jo u rn a l o f S o c i o l o g y , 78 (2 ) , 1 0 5 - 1 3 0 . G u i m e rà , R ., & Nu n e s A m aral, L . A . ( 2 0 0 5 ). Fu n c t i o n al ca rto g rap h y o f co m p l e x met ab o l i c n et wo rk s . N a t u re, 433 (7 0 2 8), 8 9 5 – 9 0 0. G u i m e rà , R . , Mo s sa, S ., Tu r t s ch i , A ., & A m a ral , L . A . N . (2 0 0 5 ). Th e w o rl d wi d e a i r t ran s p o rta t i o n n et wo rk : A no m a l o u s cen tral i t y, co mm u n i t y s t ruc t u re, a n d ci t ies ' g lob a l r o l es . Pro ce ed ings o f t h e Na ti ona l Acade my of S ci e n c es o f th e U n i t e d S t a t e s o f Am e ri c a , 1 0 2 (2 2 ), 7 7 9 4 -77 9 9 . H é b e rt-D u f res n e, L . , A l l a rd , A . , Y o u n g , J . - G ., & Dub é , L . J . (20 13). G lob a l ef fi c ien cy o f l o c a l i mm u n i z a t i o n on co m p l e x n et wo r ks . S c i en ti fic Rep o rts , 3 , 2 171. H u an g , L ., Par k, K ., & L a i , Y . C. (2 0 06). Info r m a t i o n p ro p a g a t i o n o n m o d u l ar n e t w o r k s . Ph y si ca l Re vi e w E, 7 3 (3), 0 351 03. Jo v a n o v i c, M . A . ( 2 0 0 1) . M o d e l i n g La r g e - sc a l e Pee r- to - Pe e r N e t w o r k s a n d a C a se S t u d y o f G n u t el l a . Cinc i n n a t i : U n i v e r s i t y o f C i n c i n n a t i . K e e l i n g , M. J ., & R o h an i, P . (2 0 08). Mo de l i n g i n f ec t i o u s d i sea ses in h uma n s a n d a nimal s. P r i n c et o n : Pr i n c et o n U n i v e rs i ty Pr e s s . K e n d al l , M . G. ( 1 938) . A Ne w M e a s u re of R a n k C o r rel at i o n . Bi o metri k a , 3 0 (1 /2) , 8 1- 9 3. K i m, B. J ., Ju n , T . , K i m , J. - Y ., & C h o i , M . Y . ( 2 0 0 6 ) . N e t w o r k ma r ket i n g o n a s m a l l- wor l d n et wo r k. Ph y sic a A: S t a ti st i c a l Mec han i c s a n d i t s A p p l i c a t i ons , 3 60 (2 ) , 4 9 3 - 5 0 4 . K its ak, M . , G a l l o s , L . K. , Hav l i n , S . , Li l j e ros , F . , Mu c h n ik , L., Sta n l e y , H . E. , et al . (2 0 1 0 ) . Id en tif i ca t ion o f i n f l u e n t i a l s p read e rs in co m ple x n e t w o r k s . Na tu re Phy si cs, 6 (1 1 ) , 8 8 8- 8 93. L a t o ra, V . , & M arch i o r i , M . ( 2 0 0 7 ) . A mea s u re o f c en t ral i t y ba s e d on n et work ef fi c i e n cy. Ne w Jo u rn a l o f Ph y sic s, 9 (6), 1 88. Li u , Z . H ., & H u , B . ( 2 005). E p i d e m ic s p read i n g i n co mm u n ity n e t w o r ks . Eu ro p h ysi c s Let t e rs, 7 2 ( 2) , 3 1 5- 3 21. L u , L . Y . , Zh an g , Y . C., Y eu n g , C . H. , & Zh o u , T . ( 2 0 11). L ea d e rs i n s o ci a l n e t w o r k s , t h e Delici o u s ca s e . Plo s O n e, 6 (6 ) , e 2 1 202. O n n el a , J.-P ., Sara mä ki , J ., H y v ö ne n , J. , S z ab ó , G. , L a z e r , D. , K a s ki , K ., et a l . ( 2 0 0 7 ) . Stru ct u r e an d t i e s t ren g t hs i n m o b i l e co mm u n i ca t i o n n e t w o r ks . Pro ce e d i n g s o f th e Na t i o nal Aca d emy of Sc i en ce s of t h e Uni t e d S t a t e s of Am e ri ca , 1 0 4 (18) , 7 3 32- 7 3 3 6. Pas t o r- Sat o r ras , R ., & V e s p i g n an i , A . ( 200 1 ) . E p i d e mi c s p r e ad i n g i n s ca l e -f ree ne t wor k s . Phy si ca l Rev i ew Let t ers, 8 6 (1 4 ), 3 2 00- 3 2 0 3 . Sal at h é , M . , & Jo n es , J. H . ( 2 010 ). Dy n a m i c s an d C o n t rol o f D i s e as e s i n Net work s wi th C o mm u n i ty Struc t u re. Plo s Co m p u ta ti ona l B i o l o g y , 6 (4 ) , 3 8 7 - 3 9 5 . Sal at h é , M . , Fr e if e l d , C. C . , M e kar u , S. R . , T o m a s u lo, A . F . , & B ro wns t ei n , J. S . ( 2 0 1 3) . In fl ue n z a a (h7 n 9 ) an d t h e i m p o r t a n c e o f d igi t a l e p i d e mi o log y. Ne w En g l a n d Jo u rnal o f M ed i c i ne , 369 ( 5) , 4 0 1- 4 04. Trau d , A . L . , Ke l s i c, E. D ., M u ch a , P . J ., & P o r t e r, M . A . (2 0 1 1) . C o m p a ri n g C o mm u n i t y Stru ct u re t o 14 / 22 C h arac t er i s t i c s i n O n l i n e C o l l e g i a t e Soc i a l N et w o rk s . S I AM Rev i ew , 53 ( 3) , 52 6 - 5 4 3 . V i n ce n t , D . B . A . , G u i l l a u m e, J. L., & L e feb v r e , E . ( 2 0 0 8 ). Fas t u n fo l d i n g o f c o mm u n i t i e s i n lar g e n et w o rk s . Jo u rn a l o f S t a t i s ti c a l Me ch a n i cs The o ry & Exp eri m e n t , 30 ( 2 ), 1 55- 1 6 8. Wu, X . Y . , & Li u , Z . H . ( 2 0 0 8 ). Ho w c o mm un i t y s t ru ct u re i n f l u e n c es ep i d emi c s p read in s o c i a l n et wo rk s . Ph y sic a A: S t a ti st i c a l Mec han i c s a n d i t s A p p l i c a t i ons , 3 87 (2 - 3 ), 6 2 3 - 6 3 0 . Zh o u , T ., L iu, J . G ., B a i , W . J. , C h e n , G. , & Wan g , B . H . (20 0 6 ). B e h a v i o r s o f s us c e p t i b l e - i n f e c t ed ep ide mi cs o n s ca l e -f ree ne t wor k s w i t h i d en t i c a l i n fect i v ity . Ph y si ca l Rev i e w E, 7 4 ( 5) , 1 2 1 - 1 3 7 . 15 / 22 T A B L E S N e t wor k s N E k m a x C r H G Q C a ltec h 5 7 1 7 1 2 7 2 4 .9 6 3 96 2 . 9 6 5 0 . 5 7 4 0 . 0 0 1 0 . 3 6 3 9 0 . 7 9 4 Pr i n c et o n 3 9 7 5 2 3 4 5 7 1 1. 8 0 2 1 2 9 4 . 7 2 1 0 . 3 2 1 0 . 4 1 2 0 . 4 3 3 30 0 . 7 4 0 G e o rg et o w n 6 3 0 9 7 3 0 2 2 2 3 .1 4 8 3 1 1 4 . 2 1 2 0 . 2 5 8 0 . 2 4 2 0 . 4 5 7 16 0 . 6 8 3 U. O kl ah o m a 6 8 5 0 1 5 2 9 8 5 4 4 .6 6 7 2 4 7 4 . 3 6 1 0 . 5 2 4 0 . 4 8 8 0 . 5 1 1 42 0 . 9 2 6 T A B L E 1 . S truc tu r al p r o p e r ties o f net w o rk s . St ruc t u ral p ro p e r t ies i n cl u d i n g t h e n et wo r k s iz e ( N ), n u m b er o f e d g e s ( E ), a v e rag e d eg ree ( < k > ) , ma x d eg ree ( k max ), av erag e s h o rt e s t p at h l e n g t h ( < d > ) , cl u s te ri n g co e ff i c i en t ( C ), d eg ree as s o rta t i v ity ( r ) , de g ree h et e r o g en e i t y ( H ), t h e n u m b e r o f c o mm un i t i es ( G ) , a n d m o d ular i t y ( Q ) a re t a b u late d f o r e ac h ne t wor k s . Th es e ne t work s inc l u d e da t a fr o m th e C a liforn i a Ins t i t u t e o f T e c h n o l o g y, Pr i n c et o n U n i v ers i ty, G eo r g et o wn Uni v ers ity , an d t h e U n i v ers i t y o f O k l ah o m a. 16 / 22 FIGU R E 1 . Vis ua l i z i ng th e w e a k tie s a n d s tro ng tie s in t he net w o r k . 17 / 22 FIGU R E 2 . S c hemat i c il l us tra tio n o f s t r a teg i e s b e t wee n k -s hel l d e c o m po s i t i on a n d k -s h e l l w i th co m m u ni ty . 18 / 22 FIGU R E 3 . C u mulat i v e d i s tri bu ti o n o f th e n u mb er o f w e a k t i es an d s tro ng ti es i n e mp i r i c a l net wo rk s . 19 / 22 FIGU R E 4. R o l e s a n d r eg i on s i n t he Z - P s p a c e f o r e m p i ri c a l net w o r k s . 20 / 22 FIGU R E 5. C o mpa r i s on o f s p re a d ing e ff i ci e ncy o f i d e n t i fic a tio n i n e m pir i cal n e t wo rk s . 21 / 22 FIGU R E 6. C o m pa ri s on o f a v er a g e n u m b er o f wea k t i e s a n d s tron g tie s a mon g nei g h bo rs o f no des i d en tifie d by k -s co r e w i t h t h o s e o t her s . 22 / 22 FIGU R E 7. R o bus tnes s o f t h e s t r a teg y i n n e t w o r k s w i t h no i s e a s m o del e d by ra n do m r e m o v a l of con nec tio ns .