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

• 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
• rtuna to

a rXiv: 0906.0612

Pla n o f the c o urse

I.

Networ ks: definitions, c har ac ter istic s, basic c onc epts in gr aph theor y

I I .

Re a l wo rld ne two rks: b a sic pro pe rtie s. Mo de ls I

I I I .

Mo de ls I I

I V.

Co mmunity struc ture I

V.

Co mmunity struc ture I I

VI .

Dyna mic pro c e sse s in ne two rks

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 AS Ca b le s 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 ne two rks Pro te ins Che mic a l re a c tio ns 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

Gra ph T he o ry

Orig in: L e o nha rd E ule r (1736)

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 Undire c te d e dg e : Dire c te d e dg e :

i

Bidire c tio na l c o mmunic a tio n/ inte ra c tio n

j i j

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

0 1 2 3 0 0 1 1 1 1 1 0 1 1 2 1 1 0 1 3 1 1 1 0

3 1 2

n ve rtic e s i=1,…,n a ij=

1 if (i,j) E 0 if (i,j) E

Adja c e nc y ma trix

0 1 2 3 0 0 1 0 0 1 1 0 1 1 2 0 1 0 1 3 0 1 1 0

Symme tric fo r undire c te d ne two rks

n ve rtic e s i=1,…,n a ij=

1 if (i,j) E 0 if (i,j) E 3 1 2

Adja c e nc y ma trix

0 1 2 3 0 0 1 0 1 1 0 0 0 0 2 0 1 0 0 3 0 1 1 0

3 1 2

No n symme tric fo r dire c te d ne two rks

n ve rtic e s i=1,…,n a ij=

1 if (i,j) E 0 if (i,j) E

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 Ma xima l numb e r o f e dg e s

D =

Re pre se nta tio n: lists o f ne ig hb o urs o f e a c h no de Spa rse g ra ph: D <<1 Spa rse a dja c e nc y ma trix

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+1 ve rtic e s i

0,i 1,…,i l ε V

l e dg e s (i

0,i 1), (i 1,i 2)…,(i l-1,i l) ε E

i

2

i i

1

i

5

i

4

i

3 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

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 A tre e is a c o nne c te d g ra ph witho ut lo o ps/ c yc le s

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 c o mpo ne nt Ma c ro sc o pic fra c tio n o f the 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

T ub e T e ndril T e ndrils

Giant SCC: Str

• ngly

Connec ted Component Giant OUT Component Giant IN Component

Disc o nne c te d c o mpo ne nts

Pa ths a re dir

e c te d

Sho rte st pa ths

i j

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 dista nc e l(i,j)=minimum numb e r

• f e dg e s tra ve rse d o n a pa th

b e twe e n i a nd j 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 μn with o ne no n-ne g a tive e ig e nve c to r, suc h tha t |μ|≤ μn 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 μn 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

• 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

• 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

i ki=5

• Clo se ne ss c e ntra lity

g i= 1 / ∑j l(i,j)

F

• r dire c te d g ra phs: kin, ko ut

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 slm sho rte st pa ths b e twe e n l a nd m si

lm sho rte st pa ths g o ing thro ug h i

b i is the sum o f si

lm / slm o ve r a ll pa irs (l,m)

NB: simila r q ua ntity= load li=∑ σi

lm

NB: g e ne ra liza tio n to e dge be twe e nne ss c e ntrality

Pa th-b a se d q ua ntity

j

b i is la rg e b j is sma ll

i

E ig e nve c to r c e ntra lity

i x3 x4 x1 xi x5 x2

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

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)

k

C(i) =

# of links between 1,2,…n neighbors k(k-1)/2

Cluste ring c o e ffic ie nt o f a no de

i

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 k1,k2,…,kn No t ve ry use ful!

• Histo g ra m:

nk= numb e r o f no de s with de g re e k

• Distrib utio n:

P(k)=nk/ 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)=nk/ 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

Aver age=< k > = ∑i ki/ n = ∑k k P(k)=2|E

|/ n

F luc tuations: < k2 > - < k > 2

< k2 > = ∑i k2

i/ n = ∑k k2 P(k)

< kn > = ∑k kn P(k)

Sparse graphs: < k > << n

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

Sta tistic a l c ha ra c te riza tio n

Multipo int de g re e c o rre la tio ns Me a sure o f c o rre la tio ns: P(k’ ,k’ ’ ,…k(n)|k): c o nditio na l pro b a b ility tha t a no de o f de g re e k is c o nne c te d to no de s o f de g re e k’ , k’ ’ ,… Simple st c a se : P(k’ |k): c o nditio na l pro b a b ility tha t a no de o f de g re e k is c o nne c te d to a no de o f de g re e k’

• fte n inc o nve nie nt (sta tistic a l fluc tua tio ns)

Sta tistic a l c ha ra c te riza tio n

Multipo int de g re e c o rre la tio ns Pra c tic a l me a sure o f c o rre la tio ns:

Aver age degr ee of near est neighbor s

ki=4 knn,i=(3+4+4+7)/ 4=4.5

Sta tistic a l c ha ra c te riza tio n

Ave ra g e de g re e o f ne a re st ne ig hb o rs Co rre la tio n spe c trum: putting to g e the r ve rtic e s ha ving the sa me de g re e

c la ss o f de g re e k

Sta tistic a l c ha ra c te riza tio n

Ca se o f ra ndo m unc o rre la te d ne two rks

P(k’|k)

• inde pe nde nt o f k
• pro b . tha t a n e dg e po ints to a ve rte x o f de g re e k’

pro po rtio na l to k’ itse lf

Punc(k’|k)=k’P(k’)/< k >

numb e r o f e dg e s fro m no de s o f de g re e k’ numb e r o f e dg e s fro m no de s o f a ny de g re e

T ypic a l c o rre la tio ns

Asso rta tive b e ha vio ur: g ro wing knn(k)

E xa mple : so c ia l ne two rks L a rg e site s a re c o nne c te d with la rg e site s

Disa sso rta tive b e ha vio ur: de c re a sing knn(k)

E xa mple : inte rne t L a rg e site s c o nne c te d with sma ll site s, hie ra rc hic a l struc ture

Co rre la tio ns: Cluste ring spe c trum

• P(k’ ,k’ ’ |k): c umb e rso me , diffic ult to e stima te fro m da ta
• Ave ra g e c luste ring c o e ffic ie nt C=a ve ra g e o ve r no de s with

ve ry diffe re nt c ha ra c te ristic s

Cluste ring spe c trum: putting to g e the r no de s whic h ha ve the sa me de g re e

c la ss o f de g re e k

(link with hie ra rc hic a l struc ture s)

Mo tifs

Mo tifs: sub g ra phs o c c urring mo re o fte n tha n o n ra ndo m ve rsio ns o f the g ra ph

Sig nific a nc e o f mo tifs: Z-sc o re !

We ig hte d ne two rks

Re a l wo rld ne two rks: links

c a rry tra ffic (tra nspo rt ne two rks, I

nte rne t…)

ha ve diffe re nt inte nsitie s (so c ia l ne two rks…)

Ge ne ra l de sc riptio n: we ig hts i

j w ij

a ij: 0 o r 1 wij: c o ntinuo us va ria b le

• Sc ie ntific c o lla b o ra tio ns: numb e r o f c o mmo n pa pe rs
• I

nte rne t, e ma ils: tra ffic , numb e r o f e xc ha ng e d e ma ils

• Airpo rts: numb e r o f pa sse ng e rs
• Me ta b o lic ne two rks: fluxe s
• F

ina nc ia l ne two rks: sha re s

• …

We ig hts: e xa mple s

usua lly wii=0 symme tric : wij=wji

We ig hte d ne two rks

We ig hts: o n the links Stre ng th o f a ve rte x:

si = ∑j ε V(i) w ij

=>Na tura lly g e ne ra lize s the de g re e to we ig hte d ne two rks =>Qua ntifie s fo r e xa mple the to ta l tra ffic a t a no de

We ig hte d c luste ring c o e ffic ie nt I

si=16 c i

w=0.625 > c i

ki=4 c i=0.5 si=8 c i

w=0.25 < c i

w ij=1 w ij=5

i i

• A. Ba rra t, M. Ba rthé le my, R. Pa sto r-Sa to rra s, A. Ve spig na ni, PNAS 101, 3747 (2004)

We ig hte d c luste ring c o e ffic ie nt I I

• J. Sa ra mä ki, M. K

ive la , J.-P. Onne la , K . K a ski, J. K e rté sz, Phys. Re v. E 75, 027105 (2007)

De finitio n b a se d o n sub g ra ph inte nsity

We ig hte d c luste ring c o e ffic ie nt

Ra ndo m(ize d) we ig hts: C = C w C < C w : mo re we ig hts o n c liq ue s C > C w : le ss we ig hts o n c liq ue s

w ik

Ave ra g e c luste ring c o e ffic ie nt C=∑i C(i)/ n C w=∑i C w(i)/ n Cluste ring spe c tra

We ig hte d a sso rta tivity

ki=5; knn,i=1.8

We ig hte d a sso rta tivity

ki=5; knn,i=1.8

5 1 1 1 1

i

We ig hte d a sso rta tivity

ki=5; si=21; knn,i=1.8 ; knn,i

w=1.2: knn,i> knn,i

w 1 5 5 5 5

i

ki=5; si=9; knn,i=1.8 ; knn,i

w=3.2: knn,i < knn,i

w 5 1 1 1 1

i

Weighted assortativity

Pa rtic ipa tio n ra tio

1/ ki if a ll we ig hts e q ua l c lo se to 1 if fe w we ig hts do mina te

Pla n o f the c o urse

I .

Ne two rks: de finitio ns, c ha ra c te ristic s, b a sic c o nc e pts in g ra ph the o ry

II.

R eal wor ld networ ks: basic pr

• per
• ties. Models I

I I I .

Mo de ls I I

I V.

Co mmunity struc ture I

V.

Co mmunity struc ture I I

VI .

Dyna mic pro c e sse s in ne two rks