Complex Networks - DiVA portal

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Y = ln X

2

f (y) ∼ e−(y−µ)

/2σ 2

f (x) f (ax) = , f (1) f (a)

a

F (ax) F (a)

F (x) =

=

f (ax) f (a)

= F (x)

f (x) f (1)

=

f (ax) f (a)

G(x) = log(F (x)) G(x) = α log(x)

F (ax) =

f (ax) f (1)

F (x) = xα

F (a) =

f (a) f (1)

G(ax) = G(x)+G(a) f (x) = xα · const

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new

new

'U

a)

b)

1

δE δ nn 1/s2

E ρE nn 1/s

10−1 P(>s)

10−2 10−3 10−4

c)

10−5 1

101 102 103 104 s

d) 1

101 102 103 104 δs

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AD

AD

kmax

link−end

0

0

0

k

0

0−γ

max

max

2−γ A γ−2 A

max

γ−2

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down

up

down

up

down

Nrand (lup , ldown )

Up

l up

a)

b)

6 5 4 3 2 1

5

1.4

4 1

3 2

0.6

1 0 1 2 3 4 5 6

l down

0.2 0

1

2

3

4

l down

5

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γ = 2.0

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γ=3

m

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Pi

Ii = − log2 (Pi )

Pi = 1/k

I = − log2 (1/k) = log2 (k)

'

t

P1(s → t)

P2(s → t)

!i ;K=?G5>24X5( m5a713,54%>24¨@p>B4A]6C9rq`4%=$4ED4>N@A24X2A242;)5GÀB;KE5=e!B9$D@$5;):K;KA]-rA29oJ9$:):K96 ;Ae P (s → t) 29?H A2,D;)NdN,Æ½PE`J9?>2H¨@AB;K9?Ee S(s → t) e C6 9?G5:uqÁ4Sg$4%>2- N@$djAB4>B;)8NdZ4%< Ht =

X

S(s → t)/N

X

S(s → t)/N

BA 4:):)3@p:):AN@p>B=$4ZAN< t a s

As =

^6 4r9$ÀN@p;)E@µÂ?G@pEA2;KA]-$e iLWYWXQh\Y\hkrhqVS_t UV^`PRqVkÁB,D@A AB4:):)< GDBq5;KE5=oA2,54%;K>¨ª8dd4%B4SE59`q`4B47@p>B4vE59S/];u29?LD@p=@A2;)9$EÁ9pJ2485( a!ªvED@$:K-`B9pAB4;)E¤;)EA24>N@$dZA2;)9$EE54ZA]6^9$>B_S;KEX-?4%@$B9pAB4;)E;KEA24%>B@?djAB;K9?ErE54ZA]6^9$>B_sd9$:)9$>\d9`q`4%qo@?dd9$>Nq`;KED=XA29 A e9$ E5 :)(wgÀ-µe139?;KE`,5ABAB44@$>N;KdZ@$ED9?dj4%>2ABq4;K9?EDq5.-w@B_.B9pAB4;)ED ,!/+T#))!/)/[[!BABP. ) 8c#,.+F)/!/0,#)/8 #a S70,& 0,)/ . #: ;+T)5&!/#*!/S

∼ log2 (N +1)−1−2/N +4log2 (N +1)/(N (N +1)) ≈ ∼ log2 (N/2)

x 9pJ@pEr@$qDq`;AB;K9?ED@p:kG5EDq`4%>2:)-.;KE5=¨BGDdjABG5>B4~N@pL5,D;)d%@p:Àe4ABd$( Ä2,D@pAvL5>29g.;uq`4%BHX@pA2;)9$Ee$GD29`d4%B;AB,5H¨85;K= N@A24?e ρ → ∞ e ek68,54%>24 ;)MAN@pEDd4¨9pJ3A2,54ÁE54;)=$,.9$>Ne?;)EÁJ9?>84ZY5@pHXL5:)4SB_.NdN,Ád@$E4FBHX@pA2;)9$EE544q`4%q A29µNdN,Æ;KJ79$E54r@p:)>B4%@$q5-µ_ED96vB@$:K:ÂB>B@ ¨d¾D96S( ½PE A2,5;u< d%@$B4:u@A2;)g$4 6C4%;K=?,A\9pJc4Y.;KA:);)E5_`B9$H @ E59`q`4 ;)!@$:K:.;)Eg?4%B_`N@p=?4=?@p;)EDBdN,FN@AB4=?-$(k©\96C4%g$4>e%A2,54^dZ9$EAB>2;)5GÀB;K9?EF;) l ≤ 3 @pEDqFA2,.GD2ED< 9$GÀvAB9 4F;KED4 dZ;)4EAvAB9¨J9$:):K96 A2,D4S¾D96É68,54EAB,54SAN@p>B=$4ZAv;uB.-aC.- J9$:):)968;KE5=XAB,54 ¾D96-$9?G 68;K:):E54%@$>2:)-X@p:)6^@}-`B:K9.9?_B9?@$qDCED4;)=$,.9?>2,D99`q(#½PE AB4>BHX2EB9$H -?9$G5>h4EDq0AN@p>B=$4ZA(!13,D4E¨;KAhED@ABG5>N@p:):K-LD@}-`NqÁAB,540HX@$;KEw24@pH(13,D40q`;uB4 4dZ9$HX424J9$>B40q54ZtDE54N@$djAB4>B;)B;K£%9$Ee l S (l) − S(l) e@A 68,5;)dN,9?E54S2405( 5a\ª\ED@p:)-`B:)qoE54ZA]6^9$>B_`B_ 9pJkno@$:KHeVx$xHgÀ(¥½PEA24%>2E54AC;uB_9$Jk@pGÀB9$E59?H09?GDB9?24< S(l) 68;KA2,wA2,54 BdN, S (l) HX4%@$B4%q ;)E @ >B@$EDq`9?H0;)£4qµg$4%>BB_( E54s9$2g?4%2_`B4 68;KA2, A2,54XNdN,r9$5AB@p;)E54qo68,54%E¦9$ED4GD2_(#13,D4v=$:)9$D@$:À2>N@ d8;)E`J9?>2H¨@AB;K9?EX,54:)LD
i

i

ki

i

ij

j=1

2

ij

S = 10.5 T = 0.14 R = 0.245

S = 7.9 T = 0.55 R = 0.72

S = 6.8 T = 0.81 R = 0.82

a

b

c 5

S = 14.5 T = 0.16 R = 0.20

S = 7.9 T = 0.55 R = 0.72

S = 6.6 T = 0.73 R = 0.84

d

e

f 0

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i

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3.5 T (ER) 3

T (SF)

2.5 2 T 1.5 1 0.5 0

γ

2.0 8.8

2.1 6.4

2.2 5.0

2.3 3.9

2.4 3.2

2.5 2.9

2.6 2.6

2.7 2.4

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B

Bacteriophage 186 (30.6 kbp)

Bacteriophage

λ

(48.5 kbp)

E. coli proteins Regulatory proteins Early lytic proteins Late lytic proteins Lysogenic proteins N

B

RecA

A

ClpXP Q

RecA

O P

Tum CI

CI

LexA

LexA Lon

Apl

Int

Cro

CP69 Xis

CII

Int

RexA

RexB

CIII HflB

CII

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Signaling Distance: DS (A, B) = ∑i, j |ASij − BSij | λ

186

DS = 43 186r

λr

DE = 32

Random, P(k) preserved DS = 108 ± 34 λrr

DE = 32 ± 1

186rr Random, P(k) – Poisson

DS = 1050 ± 90 DE = 1060 ± 200

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1

2

2

σ12 N1

σ22 N2

1

[\4ZA]6^9$>B_

iE 36 17 2

λ 186

iS 17 0.7 11

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