Sandeep Krishna - PhD Thesis - August 2003 - Formation and Destruction of Autocatalytic Sets in an Evolving Network Mode

8/13/2019 Sandeep Krishna - PhD Thesis - August 2003 - Formation and Destruction of Autocatalytic Sets in an Evolving Net

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http://arxiv.org/abs/nlin/0403050v1


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Declaration


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Contents


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Contents


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Contents


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Contents


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Chapter 1

Introduction

1.1 Networks in chemical, biological and social systems


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Chapter 1. Introduction


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1.1. Networks in chemical, biological and social systems


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1.2. Graph representation of a network

1.2 Graph representation of a network

1.3 Difficulties of creating a graph representation


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1.4 Structure of networks

p

N

p


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1.4. Structure of networks

N

NN

pN

N

N


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1.5. Dynamical systems on networks

1.5 Dynamical systems on networks


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1.6. Evolution of networks

1.6 Evolution of networks


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1.7 Framework of a model in which the network co-evolves

with other variables

Cn, n= 1, 2, . . .

Cn1 Cn

C xi

i

xi

C xi

Cn1 n 1 xi T


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1.7. Framework of a model in which the network co-evolves with other variables

xi= fi(Cn1, x1, x2, . . .) fi

Cn1 xi

xi Cn1

xi Cn1

xi

xi

Cn1

n Cn

xi

fi

n 1

Cn1

n

Cn

xi

xi

T

xi

xi

xi

xi fi C


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xi fi

1.8 Extensions of the framework

1.9 The origin of life: evolution of a chemical network

C xi

xi


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1.9. The origin of life: evolution of a chemical network

C

xi

C

fi

xi

xi

xi


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1.10 Catastrophes and recoveries in evolving networks

1.10.1 Innovations


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1.11. A map of subsequent chapters

1.11 A map of subsequent chapters


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Chapter 2

Definitions and Terminology

2.1 Directed graphs and adjacency matrices

G= G(S, L) S L

S={1, 2, . . . , s} s

(j, i)

j

i

s

s s C= (cij)

G = G(S, L) s s s C= (cij) cij = 1 L (j, i) j

i cij = 0

cij = 1 i j

C

C

j i j

i i j


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Chapter 2. Definitions and Terminology

2

4

5

8

7

0(0)

3(1)1(0)

1(2)

1(2)

0(0)

2(2)

2(2)

1

3

6

4

5

3

2

14

3

5

6

7

C

CC

C

C

CC

a) b)

c) d)

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 1 0 0 0 0 00 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0

C =

S =

{3, 4, 5

}

C C7


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2.2. Degrees and dependency

G = G(S, L) G(S, L) S S L L

G = G(S, L) G(S, L)

G(S, L) S S S L L S

S ={3, 4, 5}

2.2 Degrees and dependency

2.2.1 Degree of a node and degree distribution of a graph

i

sj=1(cji + cji)

i s

j=1 cij

i s

j=1 cji

P(k)

k

Pout(k) Pin(k)

2.2.2 Dependency and interdependency

di i


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D(d)

d

d (1/s)si=1 di= d=0 d D(d)

di i

i d

2.3 Walks, paths and cycles

n i1 in+1

i1l1i2l2 . . . inlnin+1 l1 i1 i2 l1= (i1, i2)

l2 i2 i3 i1 in+1

7 8 7 8 7 8 . . .

C (Cn)ij

n j i (C2)ij = sk=1 cikckj j k k i

j

i

C j i i j

j i n 0 (Cn)ij >0


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2.4. Connected components of a graph

i j j i i

j i j i j j i

n

n n

n

1

n

n n

2.4 Connected components of a graph

C

C(s)

(j, i) L (i, j)

L

i

j

C

i

j

j

i

i j j i

C(s)

{1}, {2, 3, 4, 5}, {6} {7, 8}


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2.5 Partitioning a graph into its strong components

i i S1

i

i

S2

{i} (S1 S2) C1 C1

C1

C2

C = 1, 2, . . . , M C

C

{1}, {2}, {3}, {4}, {5}, {6}

{7, 8} C1 C2

C C1 C2

C

2.6 Condensation of a graph

C1, C2, . . . , C M

M C = 1, . . . , M

C C C

C C

C

C > C C

C

i j C C

> i > j


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= (x1, x2, . . . , xs)

i

sj=1 cjixj =xi C

CT

C

2.8.1 The Perron-Frobenius theorem

T s s r

r >0

r

r || =r r

B s s 0 B T B || r

|| =r

B= T

r

T

T s s r r 0 r

r ||

=r

B ss 0 B T B || r

C C

1(C)

C 1(C)


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2.8. Perron-Frobenius eigenvectors (PFEs)

x= (0, 0, 0, 0, 0, 0, 1, 1)T

C

1(C) = 0

1(C) 1

1

C

2.8.2 Basic subgraphs

C=

C1 0

C2

.

.

.R CM

.

C

|C I| = |C1 I| |C2 I| . . . |CM I|.

C C1, . . . , C M

1(C) = {1(C)} C 1 1

C 1(C)

C 1(C) = 1


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Chapter 3

Autocatalytic Sets

3.1 Autocatalytic sets (ACSs)

j i

j

i


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Chapter 3. Autocatalytic Sets

1

2

3 3

1

2

3

41

2

1

2

c) d) e)b)a)

1

3.2 Relationship between Perron-Frobenius eigenvectors and

autocatalytic sets

C 1(C) = 0

C 1(C) 1

i xi

1= 1

xi= 0

1(C) 1 C

C C

1(C) C 1(C) =

1(C) C

C

C


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3.3. Eigenvector profile theorem

x x1

1

1

1

=1

1

1

/ /

c)0 1 0

1 0 0

0 1 0

C/

0 0 0 0 0 0

1 0 1 0 0 00 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 0

0 0 0 0 1 0

0

11

1

0

0

=

0

11

1

0

0

C x x1a) 1

2

3

5

6

d)

b)

4

2

3

4

C 1= 1 C

C

C

C C

3.3 Eigenvector profile theorem


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C1

C1

C C D1, . . . , DK

Di

E1, . . . , E N

i= 1, . . . , N

Ei

C

N

3.4 Core and periphery of a simple PFE

C C

C Q

C


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3.5. Core and periphery of a non-simple PFE

=1.36 =1.001 1 1 =1.52

2

4

b)a)

1

5

4

3

2

1

3

5

1

2

3

4

5

1

2

3

5

4

d)c)

=1.191

1

1

1(Q)

1(Q) 1

3.5 Core and periphery of a non-simple PFE

Q

1(Q) = 1(C)

1(Q) = 1


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2

4

5

1

3

6

7

x =(1,0,0,0,0,0,0)

x =(0,0,0,1,0,0,0)

x =(0,0,0,0,0,1,0)

x =(0,0,0,0,0,0,1)4

3

2

1

T

T

T

T

2

5

1

3

6

7

4

x=(0,0,0,0,1,1,1) /3T

1 2 3

4

x=(0,0,1,1) /2T


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3.6. The profile of PFEs when there is no ACS

3.6 The profile of PFEs when there is no ACS

1(C) = 0

1 = 0

Q=

3.7 The profile of PFEs when there is an ACS but only one basic

subgraph

1 1

3.8 The profile of PFEs when there is an ACS and many basic

subgraphs

1 1

1= 1


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Chapter 4

Population Dynamics

4.1 The population dynamics equation

xi=s

j=1

cijxj xis

k,j=1

ckjxj ,

x J ={ (x1, x2, . . . , xs)T s|0 xi 1,s

i=1 xi = 1} s

xi

C= (cij)

xi

i {1, . . . , s}

s

s j i

j i C= (cij)

cij j i

i yi

j

A

B

i

A+B j i

yi =Vmaxab yj

KM+yj a, b

Vmax KM KM

yi yjab i

yi = k(1 +yj)ab yi k


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Chapter 4. Population Dynamics

yi= K yj

yi K i

yi=s

j=1 Kijyj yi Kij cij Kij =cij

yi=s

j=1

cijyj yi.

i xi yi/s

j=1 yj 0 xi 1,s

i=1 xi=

1 (x1, . . . , xs)T J xi xi

A j i

C

C

C

4.2 Attractors of the population dynamics equation

C

C J

J

J

J

C

C


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4.2. Attractors of the population dynamics equation

C

J

J C jcijxj = xi

=

=

k,jckjxj

= 0 = 0

(t) =eCt (0),

(0) (t) (0) C

(t) =et .

= /

sj=1 y

j C

Rs

(0)

(0) =

a

t

Re()

(t)t e1t 1 ,

1 C

1

=

1

C

C

Rs

1 t (t) t

1 1

1


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4.3 Attractor profile theorem

C

J C

C

C1, . . . , C M

D1, . . . , DK

D K Di

j

i

Dj

Di

Di D

Fi, i= 1, . . . , N

i = 1, . . . , N

Fi

J

4.4 The attractor for a graph with no ACS

1 = 0

1(C) = 0

1 = 0

yi

1 = 0

(0, 0, 0, 1, 0, 0, 0)T

4.5 The dominant ACS of a graph

1 1


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4.6. Examples of the attractor for specific graphs

2

4

5

1

3

6

7

C

i Xi > 0

C

C 1(C) 1

1 1

Fi C

4.6 Examples of the attractor for specific graphs

1 = 0

= (0, 0, 1)T

= 0 y1 = 0 y1(t) = y1(0)

t

y2 = y1 = y1(0) y2(t) = y2(0) +y1(0)t

y3 = y2 y3(t) = (1/2)y1(0)t2 +y2(0)t+y3(0) t y1 = y2 t

y3 t2 y3 Xi = limt xi(t) X1 = 0, X2 = 0, X3= 1


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1 2 4

1 2 3

1

e)

g)

5

4 5

32 4

2 3a)

1

b)

1

2

3

1 2

d)1 2

c)

f)

3


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1 = 1 2 = 0 = (1, 0)T

y1 =y1, y2 = 0 y1(t) =y1(0)et, y2(t) =y2(0) t

= (1, 0)T =

y1= y1

1 = 1, 2 =1

= (1, 1)T/2

y1 = y2, y2= y1.

y1= y1

y1(t) =Aet + Bet, y2(t) =Aet Bet.

t

y1 Aet, y2 Aet = (1, 1)T/2 =

1 = 1 = (1, 1, 1)T/3

y1 y2

y3 = y2 y3(t) = Aet +Bet +

t

y1, y2, y3

Aet

= (1, 1, 1)T

/3 =

C 1 C

1 > 1

C

C

C

C

1= 0 C

1 = 1

t y1 t0, y2 t1, y3 t2, y4 et, y5 et

t

= (0, 0, 0, 1, 1)T/2

t


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2 = (0, 0, 1,

2, 1)T/(2 +

2)

2

y1 =y2, y2 =y1, y3 =y4, y4 =y3+y5, y5=y4

y1 y2

y4= y3+ y5= 2y4

y4(t) =Ae2t + Be

2t, y3(t) =

12

(Ae2t + Be

2t) + C,

y5(t) = 1

2(Ae

2t + Be

2t)

C.

=

1 1

1(C)

1 1(C)

1 = 1 = (0, 0, 1, 1)T/2

y1 = y2, y2 = y1, y3 =

y4+ y2, y4 = y3

y1(t) =Aet + Bet, y2(t) =Ae

t Bet,

y3(t) = t

2

(Aet

Bet) + Cet + Det,

y4(t) = t

2(Aet + Bet) + (C A

2)et + (

B

2 D)et.

t

y1 et, y2 et, y3 tet, y4 tet 1 t y3 y4 y1 y2 = (0, 0, 1, 1)T/2 =


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1

1

7 12

65432

8 9 10 11

e = ( 1 0 0 0 0 0 0 )

e = ( 0 0 0 0 0 0 1 )

e = ( 0 0 0 1 0 0 0 )

e = ( 0 0 0 0 0 0 0 0 0 0 1 1 ) / 2

e = ( 0 0 0 0 1 1 0 0 0 0 0 0 ) / 2

e = ( 1 1 0 0 0 0 0 0 0 0 0 0 ) / 2

1

1

2

3

2

3

2 4

3

b)

a)

6 75

T

T

T

T

T

T

1, 2, 3 1= 0 3 1, 2, 3 1 = 1 3

A,B,C,D

1 = 0

= 3

yi tk i k t t t2

et

tet

t2et

= 3


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4.7 Timescale for reaching the attractor

C y1

y2 y1 et

11 = 1

t (t) limt (t) i |xi(t) Xi| et/t y3 y1 y2

t

1 = 0 t

1 1 (1 2)1 2 C 1

1 1

t

4.8 Core and periphery of a graph

C Q(C) C

C C

1(C) = 0 Q(C) =

1(Q(C)) = 1(C)

Xi


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4.9. Keystone nodes

=1.36 =1.001 1 1 =1.52

2

4

b)a)

1

5

4

3

2

1

3

5

1

2

3

4

5

1

2

3

5

4

d)c)

=1.191

4.9 Keystone nodes

i C

C i i C

C C

Ov(C, C) C C

(j, i)

Qij Qij C C

Ov(C, C)

i C C

Ov(C, C i) = 0

n n

1


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Chapter 5

Graph Dynamics

5.1 Graph dynamics rules

(i, j)

i= j

i, jS ={1, 2, . . . , s} cij p

1 p cii

i S

Gps xi

[0, 1] xi s

i=1 xi= 1

C

L Xi L = {i S|Xi= minjSXj} k L

k k

i =k cik cki p 1p ckk

C xk x0 xi

Xi xi s

i=1 xi= 1


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Chapter 5. Graph Dynamics

New node

Selection

Population

Dynamics

Node with least Xi

Novelty

(step 3)

(step 2)

(step 1)


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5.2. Features of the graph dynamics

5.2 Features of the graph dynamics

5.2.1 Evolution in a prebiotic pool

5.2.2 Coupling of population and graph dynamics: two timescales

xi C

C xi

xi

5.2.3 Absence of self-replicators

cii = 0


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0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

20

40

60

80

100

120

140

160

180

200

n

numberoflinks

p=0.0025

p=0.005

p=0.001

no selection

n

s= 100

p p= 0.001 p= 0.0025

p= 0.005

5.2.4 Selection and novelty


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5.3. Implementation

5.3 Implementation

5.4 Results of graph evolution

n

s = 100

p

L s

s1

Xi > 0 1

d

s= 100

s

s ps


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0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

10

20

30

40

50

60

70

80

90

100

110

n

s1

100 1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

20

40

60

80

100

120

140

160

n

s1

100 1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

50

100

150

200

250

n

s1

100 1

Xi> 0 s1 1 n s= 100 p= 0.001 p= 0.0025 p= 0.005 1 s1


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5.4. Results of graph evolution

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000

interdependency

n

n s = 100 p = 0.0025

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

10

20

30

40

50

60

70

80

90

100

s1 n s= 100, p= 0.0025 n= 50000


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s1 n = 1

n = 50, 000 s1

s

Xi

s1 s

p= 0.0025

n= 1

n= 2854

n = 3022

n= 3386

n = 3387

s1

n= 3402

n = 3403

n = 3488

n = 3489

n= 3880

n= 4448

n= 4695

n= 4696

1


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Chapter 6

Formation and Growth of

Autocatalytic Sets

6.1 The random phase

s= 100, p= 0.0025 n= 1

1= 0

Xi > 0

Xi= 0

Xi

Xi = 0

L Xi

n= 1 n= 2853 Gp

s

Gps s = 100, p = 0.0025

ps(s1) = 24.75

Gps

107 Gps


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6.1. The random phase

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

-1 0 1 2 3 4 5 6

P

(k)

in

k

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

-1 0 1 2 3 4 5 6

P

(k)

out

k

n= 1 n= 2853 Bs1p (k) s1Ckpk(1p)s1k s= 100, p= 0.0025

Bs1p (k)[1Bs1p (k)]

285300


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Chapter 6. Formation and Growth of Autocatalytic Sets

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14

D(d)

d

n= 1 n= 2853 107 s= 100, p= 0.0025

D(d)[1D(d)]

285300

D(d)

Gps

p ps 1 p

6.2 The growth phase


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6.2. The growth phase

p2s

C

a = 1/p2s

P(na) = p2s(1 p2s)na1 p

n = 2854

Xi

1 n s1< s

L p

p

p

6.2.1 Timescale for growth of the dominant ACS

s1(n) n

ps1 n


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6.3.1 Probability of a random graph being fully autocatalytic

s

p m = p(s 1)

C

P

P =

=

]s

= [1

(

)]s

= [1 (1 p)s1]s= [1 (1 m/(s 1))s1]s

O(s)

m O(1) ps

O(1)

s m O(1) P (1 em)s es O(1)

Gp

s

a= 1/p2s

gln s

m = ps 0.25 P 3 1066 m n= 3880 m = 1.24, P 3 1015 1015


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6.3. Fully autocatalytic graphs

6.3.2 Clustering coefficient

s= 100, p= 0.0025

p = 1.27/(s 1)

6.3.3 Degree and dependency distributions

Bs1p (k)

Gp

s p= 1.27/(s 1) Bs1p (k 1)

s= 100

s

s

p =

1.27/(s1) 1.5

106

Gps p = 127/(s 1), s = 100


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1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 5 10 15 20 25 30 35 40

P(k)

k

s = 100, p = 0.0025

Bs1p (k) Gps , p = 1.27/(s 1)

Bs1p

(k)1Bs1

p (k)

16065900

Bs1p (k 1)

Bs1p (k1)[1Bs1p (k1)]

16065900


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6.3. Fully autocatalytic graphs

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

1 10

P

(k)

out

k

s= 100, p=0.0025

0

0.005

0.01

0.015

0.02

0.025

0.03

0 20 40 60 80 100

D(d)

d

s= 100, p= 0.0025 1.5 106 Gp

s s = 100, p = 1.27/(s 1)


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Chapter 7

Destruction of Autocatalytic Sets

7.1 Catastrophes and recoveries in the organized and growth

phases

Xi

s1 s s 1

s1 l

s1

s1

s1 P(s1)

Xi > 0 s1(n) s1(n) s1(n 1) p


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7.1. Catastrophes and recoveries in the organized and growth phases

0 1 2 3 4 5 6 7 8 9

1

10

100

1000612

55

18

8

3 3

1 1

core overlap

s1 s/2 s = 100 p = 0.0025


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Chapter 7. Destruction of Autocatalytic Sets

1

2

3

4

6

7

9

10

8

5

0.170.09

0.110.14

0.11

0.09

0.09

0.08

0.06

0.06

1= 1.22 Xi Xi 1/1 Xi X6 = X5/1 Xi Xi

7.2 Crashes and core-shifts

n s1(n)< s/2

s = 100, p = 0.0025

s1

n Ov(Cn1, Cn) = 0

Ov(Cn1, Cn)

s1

(n) 0k1070 events

f

(This implies that Q = Q

i

f

f

1 23

4

f

(This implies that Q = )f

i

i

Cor

Coretransforming innov

existing ACS:

Core enhancing innovations:

created:

Incremental innovations:

N= created:

Innovations:

Random phase innovations:

Non innovations:


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7.3. Addition and deletion of nodes from a graph

n = 1 n = 10000

Xk

N N=

Qi

Qf

N

7.3.2 Addition of a node: innovations

k Xk

Xk

L


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n= 79

n = 3022

n = 4696

1 n= 3489

n= 6062

n = 2853

Cn Cn1 k s 1 k Cn1 Qn C

n

Nn

n Nn

Qn1

1(Nn)> 1(Qn)

1(Nn) =1(Qn) Nn Q

n


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7.4. Classification of core-shifts

0.8

1

1.2

1.4

1.6

00.5

11.5

100

101

102

136

1

(Cn)

f

194

13

2

1(C

n1)

f 612 s= 100 p= 0.0025 1 1(Cn1) 1(Cn) 1(Cn1) =1 1(Cn) = 0 1(Cn) 1(Cn1) 1 1(Cn1) > 1(Cn) 1

f

Qn1 Nn Qn1 Nn Qn1

7.4 Classification of core-shifts

136 241

235


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99/169


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1(Qn) =1(Cn1) k

Cn1 1(Cn) =1(Nn)

1(Q

n) =1(Cn

1)

1

7.4.3 Takeovers by dormant innovations

n= 4696

n= 5041 1= 1.24

n= 5041

85

n= 5042

36

74 26 90 n= 5042 36 74

11

s1 97

1 1(Cn1) > 1(Cn) 1

85

36

74

85

26 90

7.5 Timescale of crashes

ns

ns

s = 100, p = 0.0025

s ns 1088.3 ns 1581.0 log10ns s

ns s


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7.5. Timescale of crashes

0

50

100

150

200

250

300

0 2000 4000 6000 8000 10000 12000 14000 16000

0

10

20

30

40

50

60

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

ns s = 100, p = 0.0025 min{ns} = 1, max{ns} = 16625, s ns 1088.3 ns 1581.0 log10ns


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1e-06

1e-05

0.0001

0.001

0.01

0.1

1

-1 0 1 2 3 4 5 6 7

P

(k)

in

k

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 5 10 15 20 25 30 35 40

P

(k)

out

k


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 1.2 1.4 1.6 1.8 2 2.2


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Chapter 8

Robustness of the ACS Growth

Mechanism

8.1 Variants of the model

cij

cij

cii

Xi

Xi

yi

cij

[0, 1]

cij


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Chapter 8. Robustness of the ACS Growth Mechanism

cii

f Xi

1/Xi

Xi= 0

q

Xi q

1 q

xt

Xi < xt

xi

8.2 Variable link strengths

cij [0, 1]

(i, j)

i =j cij p cij = 0 1 p cij [0, 1] cii

i

a

= 1/p2s

g = 1/p

1

a b

[0, 1] 1=

ab


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21

a

b

a, b[0, 1]

ab

1 1 1> 0 C 1 > 0

C 1>0

1

p

1

8.3 Negative links: emergence of cooperation

cij [1, 1] i =j [1, 0] i= j yi

yi

yi=

ri ifyi> 0 orri 00 ifyi= 0 andri< 0

where ri=s

j=1

cijyj yi.

yi yi

xi=yi/s

j=1 yj

xi=

fi ifxi>0 orfi 00 ifxi = 0 andfi< 0

where fi=s

j=1

cijxj xis

k,j=1

ckjxj.


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8.3. Negative links: emergence of cooperation

1 2

1

3

11

11

1

xi 1/3 2

3 xi

xi 1/3

2

3

1/3 xi

xi

C

s= 100 p= 0.005

Xi

= 0 s1

nth cij

>0

l+ cij


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8.3. Negative links: emergence of cooperation

2

31

1

1

1

1

X= (0, 0, 1)T

n = na = 1904

Xi = 0

Xi = 0 Xi= 0 L Xi

s1 l+ l

s1 l+ n= na

p a

p/2 a 4/p2s(= 1600 p= 0.005 s= 100 P(na) = p2s4 (1 p2s4 )

na1

n = 1904 n = 3643 s1 s

Xi= 0

s1 < s Xi = 0 L

s1

Xi= 0


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8.4. Self-replicators

0 500 1000 1500 2000 2500 30000

10

20

30

40

50

60

70

80

90

100

n

s1

cii [

1, 1] s= 100, p= 0.005

8.4 Self-replicators

cii

[1, 1]

p/2

p2s/4 p 2/s p

1 = 1


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8.5 Non-extremal selection

Xi

Xi Xi

Xi

f Xi

1/Xi

Xi= 0

q

Xi

q

1 q

s= 100, p= 0.005

f = 0.02

s = 100, p = 0.005

s

s 2 s

10

s

q= 1

q= 0

q

q

z

(1 q)z/s ne = s/{z(1 q)} p

s z

q

ne s


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8.5. Non-extremal selection

0 1000 2000 3000 4000 5000 6000 70000

10

20

30

40

50

60

70

80

90

100

n

s1

0 500 1000 15000

10

20

30

40

50

60

70

80

90

100

n

s1

s = 100, p = 0.005


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0 500 1000 1500 2000 2500 3000 3500 4000 45000

10

20

30

40

50

60

70

80

90

100

n

s1

s= 100, p= 0.005 1/Xi Xi= 0


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8.5. Non-extremal selection

500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20

30

40

50

60

70

80

90

100

n

s1

500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20

30

40

50

60

70

80

90

100

n

s1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20

30

40

50

60

70

80

90

100

n

s1

s= 100, p = 0.0025 q= 0.95 q= 0.99 q= 0.99999


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8.6 Variable number of nodes

Xi

xt

xt

Xi

Xi = 0

si=1 xi = 1 1/xt

xt = 0.005

200 = 1/0.005


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8.6. Variable number of nodes

0 2000 4000 6000 8000 10000 120000

20

40

60

80

100

120

n

s1

0 2000 4000 6000 8000 10000 120000

20

40

60

80

100

120

140

160

180

200

n

links

Xi


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8.7 Different population dynamics

xi= xi

s

j=1

cijxjs

k,j=1

xkckjxj

.

xi xi i

xi = 0

cij

Xi

s1

a > 1 X = (0, 0, 1)T

Xi


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8.7. Different population dynamics

0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000 3500 4000

n

Xi > 0 s1 n

s= 100, p= 0.0025

1

2

3

4

a>121 3

a) b)

a > 1 X = (0, 0, 1)T

X= (1, 1, 1, 0)T/3


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xi=

sj,k=1

Aijkxjxk xis

l,j,k=1

Aljk xjxk.

xi s

i=1 xi= 0

Aijk = ijcjk ij = 1 i= j i =j


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Chapter 9

Concluding Remarks

9.1 Interesting features of the model

p

a g

s

s

ln s


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Chapter 9. Concluding Remarks

s1

s1

p

s1


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9.1. Interesting features of the model

1

1= 1

yi

yi


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Chapter 9. Concluding Remarks

cij

yi

yi

1 s

p

s

p

s1


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9.1. Interest

Documents

Sandeep Krishna - PhD Thesis - August 2003 - Formation and Destruction of Autocatalytic Sets in an Evolving Network Mode