L e c ture I I ntro duc tio n to c o mple x ne two rks Sa nto - PowerPoint PPT Presentation

L e c ture I I ntro duc tio n to c o mple x ne two rks Sa nto F o rtuna to

Re fe re nc e s � E volution of networ ks S.N. Do ro g o vtse v, J.F .F . Me nde s, Ad v. Phys. 51, 1079 (2002), c o nd-ma t/ 0106144 � Statistic al mec hanic s of c omplex networ ks R. Alb e rt, A-L Ba ra b a si Re vie ws o f Mo de rn Physic s 74, 47 (2002), c o nd-ma t/ 0106096 � T he str uc tur e and func tion of c omplex networ ks M. E . J. Ne wma n, SI AM Re vie w 45, 167-256 (2003), c o nd-ma t/ 0303516 � Complex networ ks: str uc tur e and dynamic s S. Bo c c a le tti, V. L a to ra , Y. Mo re no , M. Cha ve z, D.-U. Hwa ng Physic s Re po rts 424, 175-308 (2006) � Community detec tion in gr aphs S. F o rtuna to a rXiv: 0906.0612

Pla n o f the c o urse Networ ks: definitions, c har ac ter istic s, basic c onc epts I. in gr aph theor y Re a l wo rld ne two rks: b a sic pro pe rtie s. Mo de ls I I I . I I I . Mo de ls I I Co mmunity struc ture I I V. Co mmunity struc ture I I V. Dyna mic pro c e sse s in ne two rks VI .

Wha t is a ne two rk? Ne two rk o r g ra ph=se t o f ve rtic e s jo ine d b y e dg e s ve ry a b stra c t re pre se nta tio n ve ry g e ne ra l c o nve nie nt to de sc rib e ma ny diffe re nt syste ms

So me e xa mple s No de s L inks So c ia l ne two rks I ndividua ls So c ia l re la tio ns I nte rne t Ro ute rs Ca b le s AS Co mme rc ia l a g re e me nts WWW We b pa g e s Hype rlinks Pro te in inte ra c tio n Pro te ins Che mic a l re a c tio ns ne two rks a nd ma ny mo re (e ma il, P2P, fo o dwe b s, tra nspo rt….)

I nte rdisc iplina ry sc ie nc e Sc ie nc e o f c o mple x ne two rks: -g ra ph the o ry -so c io lo g y -c o mmunic a tio n sc ie nc e -b io lo g y -physic s -c o mpute r sc ie nc e

I nte rdisc iplina ry sc ie nc e Sc ie nc e o f c o mple x ne two rks: � E mpiric s � Cha ra c te riza tio n � Mo de ling � Dyna mic a l pro c e sse s

he o ry Gra ph T ule r (1736) e o nha rd E Orig in: L

Gra ph the o ry: b a sic s Gra ph G=(V,E ) � V=se t o f no de s/ ve rtic e s i=1,…,n � E =se t o f links/ e dg e s (i,j), m Bidire c tio na l j i c o mmunic a tio n/ Undire c te d e dg e : inte ra c tio n i j Dire c te d e dg e :

Gra ph the o ry: b a sic s Ma ximum numb e r o f e dg e s � Undire c te d: n(n-1)/ 2 � Dire c te d: n(n-1) Co mple te g ra ph: (a ll to a ll inte ra c tio n/ c o mmunic a tio n)

Adja c e nc y ma trix n ve rtic e s i=1,…,n 1 if (i,j) E a ij = 0 if (i,j) E 1 0 1 2 3 0 0 0 1 1 1 1 1 0 1 1 2 2 1 1 0 1 3 3 1 1 1 0

Adja c e nc y ma trix n ve rtic e s i=1,…,n 1 if (i,j) E a ij = 0 if (i,j) E Symme tric fo r undire c te d ne two rks 0 1 2 3 1 0 0 0 1 0 0 1 1 0 1 1 2 0 1 0 1 2 3 0 1 1 0 3

Adja c e nc y ma trix n ve rtic e s i=1,…,n 1 if (i,j) E No n symme tric a ij = 0 if (i,j) E fo r dire c te d ne two rks 0 0 1 2 3 1 0 0 1 0 1 1 0 0 0 0 2 2 0 1 0 0 3 3 0 1 1 0

Spa rse g ra phs De nsity o f a g ra ph D=|E |/ (n(n-1)/ 2) Numb e r o f e dg e s D = Ma xima l numb e r o f e dg e s Spa rse g ra ph: D <<1 Spa rse a dja c e nc y ma trix Re pre se nta tio n: lists o f ne ig hb o urs o f e a c h no de l (i, V(i)) V(i)=ne ig hb o urho o d o f i

Pa ths G=(V,E ) Pa th o f le ng th l = o rde re d c o lle c tio n o f l ε V � l+1 ve rtic e s i 0 ,i 1 ,…,i l ) ε E � l e dg e s (i 0 ,i 1 ), (i 1 ,i 2 )…,(i l-1 ,i i i 4 3 i i 5 0 i i 1 2 Cyc le / lo o p = c lo se d pa th (i 0 =i l ) with a ll o the r ve rtic e s a nd e dg e s distinc t

Pa ths a nd c o nne c te dne ss G=(V,E ) is c o nne c te d if a nd o nly if the re e xists a pa th c o nne c ting a ny two no de s in G is c o nne c te d •is no t c o nne c te d •is fo rme d b y two c o mpo ne nts

T re e s A tre e is a c o nne c te d g ra ph witho ut lo o ps/ c yc le s � n no de s, n-1 links � Ma xima l lo o ple ss g ra ph � Minima l c o nne c te d g ra ph

Pa ths a nd c o nne c te dne ss G=(V,E )=> distrib utio n o f c o mpo ne nts’ size s Gia nt c o mpo ne nt= c o mpo ne nt who se size sc a le s with the numb e r o f ve rtic e s n E xiste nc e o f a g ia nt Ma c ro sc o pic fra c tio n o f the c o mpo ne nt g ra ph is c o nne c te d

Pa ths a nd c o nne c te dne ss: dire c te d g ra phs Pa ths a re dir e c te d Giant IN Giant SCC: Str ongly Giant OUT Component Connec ted Component Component Disc o nne c te d c o mpo ne nts T e ndrils T ub e T e ndril

Sho rte st pa ths Sho rte st pa th b e twe e n i a nd j: minimum numb e r o f tra ve rse d e dg e s j dista nc e l(i,j)=minimum numb e r o f e dg e s tra ve rse d o n a pa th b e twe e n i a nd j i Dia me te r o f the g ra ph= ma x[l(i,j)] Ave ra g e sho rte st pa th= ∑ ij l(i,j)/ (n(n-1)/ 2) Co mple te g ra ph: l(i,j)=1 fo r a ll i,j “Sma ll-wo rld” � “sma ll” dia me te r

Gra ph spe c tra Spe c trum o f a g ra ph: se t o f e ig e nva lue s o f a dja c e nc y ma trix A I f A is symme tric (undire c te d g ra ph), n re a l e ig e nva lue s with re a l o rtho g o na l e ig e nve c to rs I f A is a symme tric , so me e ig e nva lue s ma y b e c o mple x Pe rro n-F ro b e nius the o re m: a ny g ra ph ha s (a t le a st) o ne re a l e ig e nva lue μ n with o ne no n-ne g a tive e ig e nve c to r, suc h tha t | μ | ≤ μ n fo r a ny e ig e nva lue μ . I f the g ra ph is c o nne c te d, the multiplic ity o f μ n is o ne . Co nse q ue nc e : o n a n undire c te d g ra ph the re is o nly o ne e ig e nve c to r with po sitive c o mpo ne nts, the o the rs ha ve mixe d-sig ne d c o mpo ne nts

Gra ph spe c tra Spe c tra l de nsity Co ntinuo us func tio n in the limit k-th mo me nt o f spe c tra l de nsity

Wig ne r’ s se mic irc le la w F o r re a l symme tric unc o rre la te d ra ndo m ma tric e s who se e le me nts ha ve finite mo me nts in the limit

Ce ntra lity me a sure s Ho w to q ua ntify the impo rta nc e o f a no de ? � De g re e =numb e r o f ne ig hb o urs= ∑ j a ij k i =5 i F o r dire c te d g ra phs: k in , k o ut • Clo se ne ss c e ntra lity g i = 1 / ∑ j l(i,j)

Be twe e nne ss c e ntra lity fo r e a c h pa ir o f no de s (l,m) in the g ra ph, the re a re s lm sho rte st pa ths b e twe e n l a nd m lm sho rte st pa ths g o ing thro ug h i s i lm / s lm o ve r a ll pa irs (l,m) b i is the sum o f s i Pa th-b a se d q ua ntity b i is la rg e i j b j is sma ll NB: simila r q ua ntity= load l i = ∑ σ i lm NB: g e ne ra liza tio n to e dge be twe e nne ss c e ntrality

E ig e nve c to r c e ntra lity x 5 x 1 x i x 2 i x 4 x 3 Ba sic princ iple = the impo rta nc e o f a ve rte x is pro po rtio na l to the sum o f the impo rta nc e s o f its ne ig hb o rs So lutio n: e ig e nve c to rs o f a dja c e nc y ma trix!

E ig e nve c to r c e ntra lity No t a ll e ig e nve c to rs a re g o o d so lutio ns! Re q uire me nt: the va lue s o f the c e ntra lity me a sure ha ve to b e po sitive Be c a use o f Pe rro n-F ro b e nius the o re m o nly the e ig e nve c to r with la rg e st e ig e nva lue (princ ipa l e ig e nve c to r) is a g o o d so lutio n! T he princ ipa l e ig e nve c to r c a n b e q uic kly c o mpute d with the po we r me tho d!

Struc ture o f ne ig hb o rho o ds k Cluste ring c o e ffic ie nt o f a no de # of links between 1,2,…n neighbors C(i) = i k(k-1)/2 Cluste ring : My frie nds will kno w e a c h o the r with hig h pro b a b ility! (typic a l e xa mple : so c ia l ne two rks)

Struc ture o f ne ig hb o rho o ds Ave ra g e c luste ring c o e ffic ie nt o f a g ra ph C= ∑ i C(i)/ n

Sta tistic a l c ha ra c te riza tio n De g re e distrib utio n •L ist o f de g re e s k 1 ,k 2 ,…,k n No t ve ry use ful! •Histo g ra m: n k = numb e r o f no de s with de g re e k •Distrib utio n : P(k)=n k / n=pro b a b ility tha t a ra ndo mly c ho se n no de ha s de g re e k •Cumula tive distrib utio n : P > (k)=pro b a b ility tha t a ra ndo mly c ho se n no de ha s de g re e a t le a st k

Sta tistic a l c ha ra c te riza tio n Cumula tive de g re e distrib utio n Co nc lusio n: po we r la ws a nd e xpo ne ntia ls c a n b e e a sily re c o g nize d

Sta tistic a l c ha ra c te riza tio n De g re e distrib utio n P(k)=n k / n=pro b a b ility tha t a ra ndo mly c ho se n no de ha s de g re e k age =< k > = ∑ i k i / n = ∑ k k P(k)=2|E |/ n Aver Sparse graphs: < k > << n luc tuations : < k 2 > - < k > 2 F < k 2 > = ∑ i k 2 i / n = ∑ k k 2 P(k) < k n > = ∑ k k n P(k)

Sta tistic a l c ha ra c te riza tio n Multipo int de g re e c o rre la tio ns P(k): no t e no ug h to c ha ra c te rize a ne two rk L a rg e de g re e no de s te nd to c o nne c t to la rg e de g re e no de s E x: so c ia l ne two rks L a rg e de g re e no de s te nd to c o nne c t to sma ll de g re e no de s E x: te c hno lo g ic a l ne two rks