The physical properties of agent based systems

LEABHARLANN CHOLAISTE NA TRIONOIDE, BAILE ATHA CLIATH TRINITY COLLEGE LIBRARY DUBLIN OUscoil Atha Cliath The University of Dublin

Terms and Conditions of Use of Digitised Theses from Trinity College Library Dublin

Copyright statement

All material supplied by Trinity College Library is protected by copyright (under the Copyright and Related Rights Act, 2000 as amended) and other relevant Intellectual Property Rights. By accessing and using a Digitised Thesis from Trinity College Library you acknowledge that all Intellectual Property Rights in any Works supplied are the sole and exclusive property of the copyright and/or other I PR holder. Specific copyright holders may not be explicitly identified. Use of materials from other sources within a thesis should not be construed as a claim over them.

A non-exclusive, non-transferable licence is hereby granted to those using or reproducing, in whole or in part, the material for valid purposes, providing the copyright owners are acknowledged using the normal conventions. Where specific permission to use material is required, this is identified and such permission must be sought from the copyright holder or agency cited.

Liability statement

By using a Digitised Thesis, I accept that Trinity College Dublin bears no legal responsibility for the accuracy, legality or comprehensiveness of materials contained within the thesis, and that Trinity College Dublin accepts no liability for indirect, consequential, or incidental, damages or losses arising from use of the thesis for whatever reason. Information located in a thesis may be subject to specific use constraints, details of which may not be explicitly described. It is the responsibility of potential and actual users to be aware of such constraints and to abide by them. By making use of material from a digitised thesis, you accept these copyright and disclaimer provisions. Where it is brought to the attention of Trinity College Library that there may be a breach of copyright or other restraint, it is the policy to withdraw or take down access to a thesis while the issue is being resolved.

Access Agreement

By using a Digitised Thesis from Trinity College Library you are bound by the following Terms & Conditions. Please read them carefully.

THE PHYS I CAL PROPERTIES OF A GENT- BAS ED

SYSTEMS

by

L o r e n z o S a b a t e l l i

A T h e s i s s u b m i t t e d to

T h e U n i v e r s i t y o f D u b l i n

f or t he d e g r e e o f

D o c t o r o f P h i l o s o p h y

D E P A R T M E N T O F P H Y S I C S T R I N I T Y C O L L E G E U N I V E R S I T Y OF D U B L I N

[ i ^ TRINITY C O L L E G ^

D E C L A R A T I O N

T h i s t h e s i s h a s n o t b e e n s u b m i t t e d as a n e x e r c i s e f o r a d e g r e e i n a n y o t h e r U n i v e r s i t y .

E x c e p t w h e r e o t h e r w i s e s t a t e d , t h e w o r k d e s c r i b e d h a s b e e n c a r r i e d o u t b y t h e a u t h o r a l o n e .

I a g r e e t h a t t h e L i b r a r y m a y l e n d or c o p y t h e t h e s i s u p o n r e q u e s t . T h i s p e r m i s s i o n c o v e r s o n l y s i n g l e c o p i e s m a d e f or s t u d y p u r p o s e s , s u b j e c t to n o r m a l c o n d i t i o n s o f a c k n o w l e d g m e n t .

S i g n a t u r e o f t h e a u t h o r

A C K N O W L E D G M E N T S

F i r s t l y , I w i s h t o t h a n k m y s u p e r v i s o r , P e t e r R i c h m o n d , f o r c o n c e i v i n g a n d s e t t i n g t h e r e s e a r c h p r o j e c t t h a t I h a v e b e e n w o r k i n g o n a n d f o r h i s s u p p o r t a n d a d v i c e d u r i n g t h e l a s t t h r e e y e a r s .

S e c o n d l y , I w o u l d l i k e t o e x p r e s s m y g r a t e f u l n e s s t o D i e t r i c h S t a u f f e r f o r t h e e x t r a o r d i n a r y a n d c o m p l e t e l y u n s e l f i s h h e l p h e o f f e r e d m e w h i l e w o r k i n g o n t h e c o n s e n s u s m o d e l s .

I w o u l d a l s o l i k e t o t h a n k :

S t e f a n o S a n v i t o f o r t h e v e r y i n t e r e s t i n g c o n v e r s a t i o n s w e h a d i n t h e l a s t t w o y e a r s a n d f o r h i s m o r a l s u p p o r t ;

S i m o n C o x , C h a r l e s P a t t e r s o n a n d S t e p h a n H u t z l e r f o r a d v i c i n g m e o n m a n y i s s u e s c o n c e r n i n g m y r e s e a r c h w o r k a n d t h e t h e s i s w r i t i n g ;

M a r c e l A u s l o o s , F r a n k S c h w e i t z e r a n d J a n u s z H o l y s t f o r g i v i n g m e t h e p o s s i b i l i t y o f p r e s e n t i n g m y w o r k i n s e v e r a l i n t e r n a t i o n a l m e e t i n g s ;

R o y A s h e r , M a r t i n N o l a n a n d H u g h L i n e h a n f o r s u p p o r t i n g ( f i n a n c i a l l y a n d m o r a l l y ) m y r e s e a r c h a c t i v i t i e s i n H i b e r n i a n I n v e s t m e n t M a n a g e r s .

I a c k n o w l e d g e s u p p o r t f r o m t h e E u r o p e a n C o m m i s s i o n v i a M a r i e C u r i e I n d u s t r i a l H o s t F e l l o w s h i p .

S U MM A R Y

T h i s t h e s i s d e a l s w i t h t h e p h y s i c a l p r o p e r t i e s o f a g e n t - b a s e d s y s t e m s . A g e n t m o d e l s a r e p o w e r f u l t o o l s f o r u n d e r s t a n d i n g ‘ s o c i a l ’ s e l f ­ o r g a n i z e d s y s t e m s a n d m a y p r o v i d e an e x p l a n a t i o n o f f e a t u r e s t y p i c a l l y f o u n d i n m a n y o f t h e s e s y s t e m s .

F o r e x a m p l e , e v e n i n q u i t e d i f f e r e n t s o c i e t i e s , t h e p e r s o n a l w e a l t h is p o w e r - l a w d i s t r i b u t e d w i t h an e x p o n e n t c l o s e t o 1.5. In t h e S t o c k M a r k e t , t h e d i s t r i b u t i o n o f r e t u r n s , v o l a t i l i t y a n d m a r k e t v o l u m e s a r e ‘ f at t a i l e d ’ . V o l a t i l i t y a n d v o l u m e s a r e a s s o c i a t e d w i t h l o n g m e m o r y s t o c h a s t i c p r o c e s s e s . T h e t i m e e l a p s i n g b e t w e e n t w o c o n s e c u t i v e t r a n s a c t i o n s i t s e l f is a n o n t r i v i a l s t o c h a s t i c p r o c e s s , s i g n i f i c a n t l y d i f f e r e n t f r o m t h e P o i s s o n p r o c e s s w e w o u l d o b s e r v e f o r i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d t i m e l a gs .

T h e r e l a t i o n b e t w e e n t h e m a c r o s c o p i c ( e . g. t h e s t o c k m a r k e t s t a t i s t i c a l p r o p e r t i e s ) a n d t h e m i c r o s c o p i c ( e . g. t h e i n t e r a c t i o n s b e t w e e n i n d i v i d u a l s ) f e a t u r e s o f s o c i o - e c o n o m i c s y s t e m s is g e n e r a l l y n o n - t r i v i a l a n d t h e o u t c o m e o f t h e s y s t e m d y n a m i c s m a y o f t e n s e e m c o u n t e r i n t u i t i v e . A t y p i c a l e x a m p l e is p r o v i d e d b y t h e s o c i a l d i l e m m a s . H e r e i n d i v i d u a l d e c i s i o n s t h a t m a k e s e n s e to i n d i v i d u a l a g e n t s c a n a g g r e g a t e i n t o o u t c o m e s i n w h i c h e v e r y o n e s u f f e r s .

T h e s i m p l e s t w a y o f t h i n k i n g o f an a g e n t , in p h y s i c a l t e r m s , is t o d r a w a c o m p a r i s o n w i t h B r o w n i a n M o t i o n . H o w e v e r , t h i s a p p r o a c h d o e s n o t a c c o u n t w e l l f o r t h e s t a t i s t i c a l p r o p e r t i e s o f f i n a n c i a l m a r k e t . A v e r y i m p o r t a n t e x t e n s i o n o f t h e r a n d o m w a l k t h e o r y is t h e ‘ C o n t i n u o u s T i m e R a n d o m W a l k ’ t h e o r y , in w h i c h t h e d i s t r i b u t i o n o f t h e w a i t i n g t i m e s b e t w e e n t w o c o n s e c u t i v e s t o c h a s t i c e v e n t s is t a k e n i n t o a c c o u n t . We s t u d y t h e w a i t i n g t i m e d i s t r i b u t i o n s o f t w o d a t a s e t s t a k e n f r o m t h e X I X c e n t u r y I r i s h s t o c k m a r k e t a n d f r o m t h e l a t e X X c e n t u r y w o r l d c u r r e n c y m a r k e t .

S t o c h a s t i c D y n a m i c a l e q u a t i o n s b a s e d on G e n e r a l i z e d L a n g e v i n a n d G e n e r a l i z e d L o t k a - V o l t e r r a e q u a t i o n s h a v e a l s o b e e n s t u d i e d . T h e p r o b a b i l i t y d i s t r i b u t i o n s o f w e a l t h o b t a i n e d f r o m t h e G e n e r a l i z e d L o t k a

V o l t e r r a a n d f r o m an e x p o n e n t i a l m a p p i n g o f a s i m p l e p e e r p r e s s u r e m o d e l h a v e b e e n s h o w n to be v e r y c l o s e . T h e r e s u l t s o b t a i n e d a r e c o m p a t i b l e w i t h t h e e m p i r i c a l o b s e r v a t i o n s o f t h e w e a l t h d i s t r i b u t i o n .

W e n e x t c o n s i d e r ‘ C o n s e n s u s M o d e l s ’ . In p a r t i c u l a r , w e m o d i f y t h e s o- c a l l e d S z n a j d c o n s e n s u s m o d e l w i t h s y n c h r o n o u s u p d a t i n g i n t r o d u c i n g an i n d i v i d u a l a g e n t m e m o r y a n d s t u d y i n g t h e p h a s e - t r a n s i t i o n p r o c e s s f r o m n o - c o n s e n s u s to c o n s e n s u s as a f u n c t i o n o f t h e m e m o r y l e n g t h . T h e e f f e c t o f m e m o r y h a s s i g n i f i c a n t l y r e d u c e d f r u s t r a t i o n . W e f i n d t h a t , i n t h e S z n a j d m o d e l , t h e i n t r o d u c t i o n o f a s m a l l a m o u n t o f n o i s e m a y i n c r e a s e t h e c o n s e n s u s w i t h i n t h e s y s t e m , w h e n i n t h e a b s e n c e o f n o i s e c o n s e n s u s is n o t a c h i e v e d . A l l o w i n g t h e n o i s e to d e p e n d o n t h e s i z e o f f l u c t u a t i o n s in t h e s y s t e m w e o b t a i n a s i m p l e m o d e l a b l e t o a c c o u n t f o r t h e s t a t i s t i c a l p r o p e r t i e s o f f i n a n c i a l m a r k e t v o l u m e s .

M o t i v a t e d b y t h e f a c t t h a t r e a l s o c i a l s y s t e m s d i s p l a y a c o m p l e x n e t w o r k - l i k e s t r u c t u r e , w e f i n a l l y s t u d y t h e s o - c a l l e d D e f f u a n t c o n s e n s u s m o d e l o n a s c a l e - f r e e n e t w o r k . We p r o p o s e t w o d i f f e r e n t a d a p t i v e v e r s i o n s o f t h e m o d e l b a s e d on t h e ‘r e l a t i v e e x p e c t a t i o n ’ o f i n d i v i d u a l a g e n t s . T h e f i r s t , l e a d s a g e n t s t o a d j u s t t h e i r e x p e c t a t i o n t o n e i g h b o u r i n g e n v i r o n m e n t a l s t a t e s . T h e s e c o n d , a l l o w s i n d i v i d u a l a g e n t s t o a d j u s t t h e i r e x p e c t a t i o n s t o t h e i r p a s t p e r f o r m a n c e s .

. . . t o t h o s e a l w a y s l o o k i n g f o r . . .

T A B L E O F C O N T E N T S

D e c l a r a t i o n _____________ i i

A c k n o w l e d g m e n t s ___________________________________________________________ i i i

S u m m a r y iv

D e d i c a t i o n vi

T a b l e o f c o n t e n t s vi i

L i s t o f f i g u r e s xi i

L i s t o f t a b l e s x i x

A b b r e v i a t i o n s XX

C h a p t e r 1. G e n e r a l I n t r o d u c t i o n 2

1.1 . C o m p l e x i t y 2

1.2. S t u d y i n g c o m m u n i t i e s 2

1.3. A g e n t M o d e l i n g . 6

1 . 4 . T h e p h y s i c a l p r o p e r t i e s o f a s e n t - b a s e d s y s t e m s 7

1. 4. 1. T h e t o p i c o f t h i s t h e s i s 7

C h a p t e r 2. P h e n o m e n o l o g i c a l d e s c r i p t i o n

o f S o c i o - E c o n o m i c S y s t e m s _______________

2. 1 I n t r o d u c t i o n ___________ ______

10

10

2 . 2 S t a t i s t i c a l q u a n t i t i e s ________________________________________________ 1_0

2 . 3 D i s t r i b u t i o n o f w e a l t h ______________________________________________ H

2 . 4 F i n a n c i a l M a r k e t s _________________________________________________ \

6

2 . 3 . 1 I n t r o d u c t i o n ________________________________________________ 1_6

2 . 3 . 2 C o r r e l a t i o n F u n c t i o n s _____________________________________T7

2 . 3 . 3 P r o b a b i l i t y D i s t r i b u t i o n F u n c t i o n s _____________________ j_8

2 . 3 . 4 V o l a t i l i t y __________________________________________________ 2_3

2 . 3 . 5 M a r k e t V o l u m e s __________________________________________ 2_8

2 . 3 . 6 C r o s s c o r r e l a t i o n b e t w e e n

s t o c k s a n d d a t a c l u s t e r i n g _________________________________ 2_8

2 . 4 V o t e d i s t r i b u t i o n s ___________________________________________________2 ^

2 . 5 M i c r o s c o p i c f e a t u r e s ________________________________________________3_0

2 . 5 . 1 I n t r o d u c t i o n ________________________________________________ 3_0

2 . 5 . 2 N e t w o r k s ___________________________________________________ 3 £

2 . 5 . 3 S o c i a l d i l e m m a s ^______________________________________3J_

2 . 5 . 4 P l a y i n g w i t h r a t i o n a l i t y : a g u e s s i n g g a m e ______________ ^ 4

C h a p t e r 3. D i f f u s i v e A g e n t s 36

3. 1. I n t r o d u c t i o n 36

3. 2. L a n g e v i n a p p r o a c h 36

3. 3. C o n t i n u o u s t i m e r a n d o m w a l k 38

3. 3. 1 T h e o r y 38

3 . 3 . 2 T e s t i n g a c o n t i n u o u s - t i m e

r a n d o m w a l k m o d e l 40

C h a p t e r 4. S t o c h a s t i c D y n a m i c a l A g e n t s 45

4 . 1 . I n t r o d u c t i o n 45

4 . 2 . G e n e r a l i z e d L o t k a - V o l t e r r a ( G L V ) M o d e l s 45

4. 3 . P e e r p r e s s u r e a n d d i f f u s i o n 47

4. 4 . P e e r p r e s s u r e e f f e c t s in t h e G e n e r a l i z e d

L o t k a - V o l t e r r a M o d e l s 50

C h a p t e r 5. C e l l u l a r a u t o m a t a ; t h e S z n a j d M o d e l 55

5. 1. I n t r o d u c t i o n 55

5 . 1 . 2 . C o n s e n s u s M o d e l s _______________________________________ ^ 5 . 1 . 3 . P e r c o l a t i o n t h e o r y _______________________________________ ^ 5. 2. T h e S z n a j d M o d e l ________________________________________________ ^ 5. 3. S z n a j d M o d e l a n d V o t e D i s t r i b u t i o n s ____________________________ ^ 5. 4. S y n c h r o n o u s u p d a t i n g m e c h a n i s m ________________________________^ 5. 5. M e m o r y ____________________________________________________________M 5. 6. P h a s e t r a n s i t i o n s __________________________________________________ ^ 5. 7. S y n c h r o n o u s u p d a t i n g in

t h e S z n a j d M o d e l w i t h a d d e d n o i s e _______________________________ 73. 5. 8. M i c r o s t r u c t u r e ____________________________________________________ 7 ^ 5. 9. A t o y m o d e l f o r m a r k e t v o l u m e s _________________________________ ^

C h a p t e r 6. C o m p l e x N e t w o r k s a n d t h e D e f f u a n t M o d e l ____________________ ^ 6. 1. I n t r o d u c t i o n ______________________________________________________ ^ 6. 2. N e t w o r k s _________________________________________________________ ^ 6 . 2 . 1 . D e f i n i t i o n s _______________________________________________ ^ 6 . 2 . 2 . S c a l e f r e e n e t w o r k s ______________________________________ ^ 6. 3. T h e D e f f u a n t M o d e l o n s c a l e f r e e n e t w o r k s _____________________ 90

6 . 3 . 1 . G e n e r a l o v e r v i e w o f t h e D e f f u a n t M o d e l ________________ ^

6 . 3 . 2 . D e f f u a n t m o d e l o n B a r a b a s i N e t w o r k w i t h l o c a l

r e l a t i v e c o n f i d e n c e b o u n d a n d t i m e - a d a p t i v e

a g e n t s ______________________________________________________ 9 1

C h a p t e r 7. C o n c l u s i o n s _______________________________________________________ 103

7. 1 S u m m a r i z i n g _______________________________________________________ 1 03

7 . 2 O u t l o o k _________________________________________ _____ _____________1 06

B i b l i o g r a p h y ___________________________________________________________________ 1 0 7

A P P E N D I X______________________________________________________________________ 113

A P P E N D I X I A U T O - C O V A R I A N C E_____________________________________________ 1 13

A P P E N D I X ! V O L A T I L I T Y______________________________________________________ 119

L I S T O F F I G U R E S

F i g u r e 2.1; C u m u l a t i v e w e a l t h d i s t r i b u t i o n s i n t h e U n i t e d K i n g d o m in 19 96 ( c o u r t e s y o f D r a g u l e s c u & al.; P h y s i c a A 2 9 9 2 1 3 - 2 2 1 ( 2 0 0 1 ) ) .

F i g u r e 2.2: I n d e x S t a n d a r d a n d P o o r 5 0 0 , da il y c l o s i n g v a l u e d a t a t a k e n f r o m 1 9 6 0 t o 2 0 0 1 .

F i g u r e 2.3: S t a n d a r d a n d P o o r 5 00 , da il y r e t u r n s .

F i g u r e 2.4: S t a n d a r d a n d P o o r 5 00 , da il y l o g a r i t h m i c r e t u r n s .

F i g u r e 2 . 5 : S t a n d a r d and P o o r 5 0 0 , a u t o c o r r e l a t i o n f u n c t i o n o f t he l o g a r i t h m i c r e t ur ns . T h e da t a s h o w t he a b s e n c e o f a n y s i g n i f i c a n t c o r r e l a t i o n .

F i g u r e 2 . 6 : S t a n d a r d and P o o r 5 0 0 R e t u r n s . T h e c u r v e d i s p l a y s t he p r o b a b i l i t y d e n s i t y f or t he n o r m a l i z e d l o g a r i t h m i c r e t u r n s ( r e t u r n s d i v i d e d by t he s a m p l e s t a n d a r d d e v i a t i o n ) . T h e f i g u r e is t a k e n f r o m h t t p : / / s o m a . u c h i c a g o . e d u / ~ c o w a n / f i n a n c e / P a p e r s / M a t a c z . p d f

F i g u r e 2.7: S t a n d a r d a n d P o o r 5 0 0 , a u t o c o r r e l a t i o n f u n c t i o n o f t h e l o g a r i t h m i c r e t u r n s . T h e d a t a s h o w t h e a b s e n c e o f a n y s i g n i f i c a n t c o r r e l a t i o n .

F i g u r e 2.8: S t a n d a r d a n d P o o r 500. A u t o c o r r e l a t i o n f u n c t i o n o f t h e a b s o l u t e v a l u e o f t h e l o g a r i t h m i c r e t u r n s .

Fi gu r e 3.1: Av er age Survival Ti me P r ob a b i l i t y D i s t r i b u t i o n f u n c t i o n for I r is h St oc k M a r k e t d ata b e t w e e n 1850 a nd 1854. Fit p a r a m e t e r s for Mittag- L ef f l e r f u nc t i o n: ;k = 0.025 ,/? = 0.4. T h e relative e r r o r for the e x p o n e n t is a b o u t 0 . 1 . Th e e r r o r bars are o f the same size as t he p o i n t - m a r k e r s .

Fi gu r e 3.2: Survival T i m e P r ob a b i l i t y D i s t r i b u t i o n f u n c t i o n for Yen C u r r e n c y M a r k e t data b e t w e e n 1989 a nd 1998. T h e relative e r r o r for the e x p o n e n t s is a b o u t 0.1. T he e r r o r bars are o f t he same size as the p o i n t - m a rk e rs .

Fi gu r e 4.1: G L V P P vs. GLV. T h e curve displays the c u m u l a t i v e p r o b a b i l i t y de n si t y (the p r o b a b i l i t y o f fi ndi ng a ra ti o W e a l t h / A v e r a g e W ea l th larger t h a n the value r e p o r t e d on the x-axes) o f the data o b t a i n e d b o t h G L V P P a nd G L V m o de l s, with the f oll owi ng p a r a m e t e r choice: ^'=0, aIW=0.00023,

D = 0 . 0 0083,

=

s t u d ie d . T h e tail exh ib it a p o w er -l a w b e h a v i o u r w it h e x p o n e n t b e t w e e n 1.3 a nd 1.6. T h e relative e r r o r for each data p o i n t is a b o u t 5%. T h e e r r o r bars are o f the same size as the p o i n t - m a r k e r s in the g r a ph .

F igur e 5.1. An e x amp le o f b o n d p e r c o l a t i o n .

F igur e 5.2: T h e p ha se t r a n s i t i o n f r o m a n o - c o n s e n s u s s tat e to a t otal c o n s e n s u s stat e is d ri ve n by the value M(0) o f the m a g n e t i z a t i o n at time z er o. T h e lattice linear d i m e n s i o n is

L —17. T h e m e m o r y l e n g t h T = 0 , 2, 8.

T

F i g u r e 5.4; T h e p h a s e t r a n s i t i o n f r o m a n o - c o n s e n s u s s t a t e t o a t o t a l c o n s e n s u s s t a t e is d r i v e n by t h e v a l u e M ( 0 ) o f t h e m a g n e t i z a t i o n at t i m e z e r o . T h e l a t t i c e l i n e a r d i m e n s i o n is JL = 3 0 1 . T h e m e m o r y l e n g t h T = 0 , 2, 8. T h e t r a n s i t i o n p o i n t is s h i f t e d t o w a r d s z e r o as t h e a g e n t m e m o r y l e n g t h T i n c r e a s e s . T h e e r r o r b a r s a r e o f t h e s a m e s i z e as t h e p o i n t - m a r k e r s .

F i g u r e 5.5: T h e p h a s e t r a n s i t i o n f r o m a n o - c o n s e n s u s s t a t e t o a t o t a l c o n s e n s u s s t a t e is d r i v e n by t h e v a l u e M{ 0 ) o f t h e m a g n e t i z a t i o n at t i m e z e r o . T h e l a t t i c e l i n e a r d i m e n s i o n is L = 1 0 0 0 . T h e m e m o r y l e n g t h T = 0 , 2, 8. T h e t r a n s i t i o n p o i n t is s h i f t e d t o w a r d s z e r o as t h e a g e n t m e m o r y l e n g t h T i n c r e a s e s . T h e e r r o r b a r s a r e o f t h e s a m e s i z e as t h e p o i n t - m a r k e r s .

F i g u r e 5.6: T h e p h a s e t r a n s i t i o n f r o m a n o - c o n s e n s u s s t a t e t o a t o t a l c o n s e n s u s s t a t e is d r i v e n by t h e v a l u e A4(0) o f t h e m a g n e t i z a t i o n at t i m e z e r o . T h e l a t t i c e l i n e a r d i m e n s i o n is L = 5 0 . T h e m e m o r y l e n g t h T = 0 , 2, 8, 100, 5 0 0 , 100 0 . T h e t r a n s i t i o n p o i n t is s h i f t e d t o w a r d s z e r o as t h e a g e n t m e m o r y l e n g t h I' i n c r e a s e s . T h e e r r o r b a r s a r e o f t h e s a m e s i z e as t h e p o i n t - m a r k e r s .

F i g u r e 5. 7: V a r i a t i o n w i t h L o f \ - M ^ ( o n e m i n u s t h e a b s o l u t e v a l u e o f t h e d i f f e r e n c e i n t h e i n i t i a l p r o b a b i l i t i e s f o r + 1 a n d - 1 ) f o r w h i c h i n h a l f o f t h e c a s e s a c o n s e n s u s w a s r e a c h e d . T h a t m a y b e s e e n as t h e p h a s e t r a n s i t i o n p o i n t f r o m t h e s t a t e w i t h o u t c o n s e n s u s t o t h e s t a t e w i t h c o n s e n s u s . T h e e s t i m a t e d s l o p e is - - 0 . 3 9 f o r T = 0 , - 0. 21 f o r T = 2 , - 0. 11 f o r T = 8 . T h e e r r o r b a r s a r e o f t h e s a m e s i z e as t h e p o i n t - m a r k e r s .

F i g u r e 5. 8: V a r i a t i o n w i t h T o f t h e d i f f e r e n c e i n t h e i n i t i a l p r o b a b i l i t i e s f o r w h i c h i n h a l f o f t h e c a s e s a c o n s e n s u s w a s r e a c h e d . T h a t m a y b e s e e n as t h e p h a s e t r a n s i t i o n p o i n t f r o m t h e s t a t e w i t h o u t c o n s e n s u s t o t h e s t a t e w i t h c o n s e n s u s . T h e e s t i m a t e d s l o p e is 0 . 4 6 f o r jL = 17 a n d 0 . 4 7 f o r JL=50. T h e e r r o r b a r s a r e o f t h e s a m e s i z e as t h e p o i n t - m a r k e r s .

F i g u r e 5.9: M a g n e t i z a t i o n at e q u i l i b r i u m as a f u n c t i o n o f q, f o r L = 5 0 a n d i n i t i a l m a g n e t i z a t i o n v a l u e s 0. 9, 0. 2, 0 . 0 2 a n d z e r o . T h e e r r o r b a r s a r e o f t h e s a m e s i z e as t h e p o i n t - m a r k e r s .

F i g u r e 5. 10: M a g n e t i z a t i o n a t e q u i l i b r i u m as a f u n c t i o n o f q, f o r L = 1 0 0 a n d i n i t i a l m a g n e t i z a t i o n v a l u e s 0. 9, 0. 2, 0 . 0 2 a n d z e r o . T h e e r r o r b a r s a r e o f t h e s a m e s i z e as t h e p o i n t - m a r k e r s .

F i g u r e 5. 11: M a g n e t i z a t i o n at e q u i l i b r i u m as a f u n c t i o n o f q, f o r L = 2 5 0 a n d i n i t i a l m a g n e t i z a t i o n v a l u e s 0. 9, 0. 2, 0 . 0 2 a n d z e r o . T h e e r r o r b a r s a r e o f t h e s a m e s i z e as t h e p o i n t - m a r k e r s .

F i g u r e 5. 12: T h i s s h o w s t h e m i c r o s c o p i c d e t a i l o f a c a l c u l a t i o n f o r a l a t t i c e o f l i n e a r d i m e n s i o n L = 5 0 , a p r o b a b i l i t y o f r a n d o m f l i p p i n g ^ = 0 . 0 0 1 a n d a n i n i t i a l m a g n e t i z a t i o n Af(0) = 0 . 2 . T h e a r r o w p o i n t s t o t h e ' c r o s s - s h a p e d ' i s l a n d ( p o s i t i v e s p i n s , b l a c k ) c e n t r e d i n t h e s i t e { 3 3 , 1 7 } a n d s u r r o u n d e d by t h e sea ( n e g a t i v e s p i n s , w h i t e ) t h a t h a s f o r m e d a f t e r 3 9 6 t i m e - s t e p s .

F i g u r e 5. 13: T h i s s h o w s t h e m i c r o s c o p i c d e t a i l f o r t h e c a l c u l a t i o n s h o w n in f i g u r e 11 a t t h e s u b s e q u e n t t i m e s t e p . T h e r a n d o m f l i p p i n g o f s p i n {33, 16} ( see t h e a r r o w ) h a s n o w e a s e d t h e f r u s t r a t i o n , it w o u l d o t h e r w i s e e x h i b i t at z e r o t e m p e r a t u r e , d e s t a b i l i s i n g t h e ' c r o s s s h a p e d ' i s l a n d .

F i g u r e 5. 14: C u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n f o r t h e s i m u l a t e d d e m a n d ( v o l u m e o f s h a r e s ) . T h e d e c a y f o l l o w s a p o w e r - l a w b e h a v i o u r w i t h an e x p o n e n t ( b e t w e e n 1. 4 a n d 1.6) c l o s e t o t h e v a l u e o f 1. 5 f o u n d i n N Y S E v o l u m e d i s t r i b u t i o n s . N u m b e r o f t r a n s a c t i o n s ( d a t a p o i n t s ) = 2 0 0 0 0 , { a = 0 . 9 8 , a = 0 . 0 0 5 , N = 2 5 0 0 } . T h e e r r o r b a r s a r e o f t h e s a m e s i z e as t h e p o i n t - m a r k e r s .

F i g u r e 6.1: N e t w o r k s . A b o v e are d i s p l a y e d g r a p h i c e x a m p l e s of: a r e g u l a r l a t t i ce, a s m a l l - w o r l d n e t w o r k a n d a r a n d o m n e t w o r k . S m a l l - w o r l d n e t w o r k s ma y be s e e n as a t r a d e - o f f b e t w e e n t he r e g u l a r t o p o l o g i c a l s t r u c t u r e o f a l a t t i ce a n d t h e r a n d o m t o p o l o g i c a l s t r u c t u r e o f a r a n d o m g r a p h . B e l o w , we c a n a p p r e c i a t e t he p i c t o r i a l d i f f e r e n c e e x i s t i n g b e t w e e n R a n d o m a n d Scal e- f r ee g r a p h s M o r e o v e r , in t he r a n d o m g r a p h , t he 5 m o s t c o n n e c t e d n o d e s ar e c o n n e c t e d t o 2 7 % o f all n o d e s . In t he s c a l e - f r e e g r a p h , t he 5 m o s t c o n n e c t e d n o d e s are c o n n e c t e d to 6 0 % o f all n o d e s .

F i g u r e 6.2: N u m b e r o f s u r v i v i n g o p i n i o n as a f u n c t i o n o f t h e i ni t i al n u m b e r o f o p i n i o n s in an a d a p t i v e D e f f u a n t - l i k e mo d e l . T h e c o n s e n s u s t h r e s h o l d is a f r a c t i o n (1, 0.2, 0. 04 a n d 0. 008) o f t he a v e r a g e l ocal d i s t a n c e . T h e a v e r a g e s are p e r f o r m e d o v e r t he di r e c t n e i g h b o u r h o o d . I ' h e n u m b e r o f a g e n t s is e q u a l t o N = 1 0 0 0 . E a c h a g e n t has m —\ d i r e c t c o n n e c t i o n s . T h e s i m u l a t i o n s h a v e b e e n c a r r i e d o u t o v e r 1000 s a mp l e s . T h e e r r o r ba r s are o f t h e s a me size as t he p o i n t - m a r k e r s

F i g u r e 6.3: N u m b e r o f s u r v i v i n g o p i n i o n s d i v i d e d by t h e n u m b e r N o f a g e n t s , as a f u n c t i o n o f t he i ni t i al n u m b e r o f o p i n i o n s in an a d a p t i v e D e f f u a n t - l i k e mo d e l . T h e c o n s e n s u s t h r e s h o l d is 0. 008 t i me s t he a ve r a g e l ocal d i s t a n c e . T h e a ve r a g e s are p e r f o r m e d o v e r t h e d i r e c t n e i g h b o u r h o o d . T h e n u m b e r o f a g e n t s is e qual t o N = 1 0 , 100, 1000, 10000. E a c h a g e n t has w = 4 d i r e c t c o n n e c t i o n s . T h e s i m u l a t i o n s h a v e b e e n c a r r i e d o u t o v e r 1000 s a mp l e s . T h e e r r o r ba r s are o f t he s a me size as t he p o i n t - m a r k e r s .

F i g u r e 6.4: N u m b e r o f s u r v i v i n g o p i n i o n s d i v i d e d by t h e n u m b e r N o f a g e n t s , as a f u n c t i o n o f t he ini t i al n u m b e r o f o p i n i o n s in an a d a p t i v e D e f f u a n t - l i k e m o d e l . T h e c o n s e n s u s t h r e s h o l d is 0. 02 t i me s t h e a v e r a g e l oc al d i s t a n c e . T h e a ve r a g e s are p e r f o r m e d o v e r t h e d i r e c t n e i g h b o u r h o o d ■ T h e n u m b e r o f a g e n t s is e q u a l to AT= 100, 1000, 10000. E a c h a g e n t has m —\ d i r e c t c o n n e c t i o n s . T h e s i m u l a t i o n s h a v e b e e n c a r r i e d o u t o v e r 1000 s a mp l e s . T h e e r r o r ba r s are o f t he s a me size as t h e p o i n t - m a r k e r s .

Fi gu r e 6.5; N u m b e r o f su rv ivi ng o p i n i o n s d iv id e d by the n u m b e r N o f a ge nt s , as a f u n c t i o n o f the initial n u m b e r o f o p i n i o n s in an a d ap ti v e D e f f u a n t - l i k e mo de l. Th e c o n s e n s u s t h r e s h o l d is equal to the average local d is ta nce . T h e averages are p e r f o r m e d o ve r the d i r e c t n e i g h b o u r h o o d . The n u m b e r o f a gents is equal to N = 100, 1000, 10000. E a c h a g en t has m=4 d i r e c t c o n n e c t i o n s . Th e s i m u la t i o ns h ave b e e n c ar r ie d o u t ov e r 1000 sampl es . T h e e r r o r bars are o f the same size as the p o i n t - m a r k e r s .

Fig ur e 6.6: N u m b e r o f su rv ivi ng o p i n i o n s d iv id e d by t he n u m b e r N o f a ge nt s , as a f u n c t i o n o f the mitial n u m b e r o f o p i n i o n s in a time a d ap ti v e D e f f u a n t - l i k e mo de l . Th e c o n s e n s u s t h r e s h o l d c ha ng e s ov e r time

d e p e n d i n g on the a g en t e xper ienc e: it i n c re as e s (d ec rea ses ) o f a f a c t o r I d e p e n d i n g on w h e t h e r the a g en t is s u cc e ss f ul (or u n s u c c e s s f u l ) in fi nd i ng a

l i k e - m i n d e d n e i g h b o u r . Th e re sults o b t a i n e d for T = 1 0 , 50, 333, 500 are c o m p a r e d to t h o s e o b t a i n e d for the n o n a d ap ti v e case. T h e a ver age s are p e r f o r m e d o ve r the di re ct n e i g h b o u r h o o d . T h e n u m b e r o f a g en ts is equal to N —14. E a c h agent has m — \ dire ct c o n n e c t i o n s . T h e s i m u l a t i o n s have b e e n c ar ri ed o u t o ve r 1000 sampl es. T h e e r r o r bars are o f the same size as the p o i n t - m a r k e r s .

F i gu re 6.7: N u m b e r o f su rvi vi ng o p i n i o n s d iv id e d by the n u m b e r N o f a g en ts , as a f u n c t i o n o f the initial n u m b e r o f o p i n i o n s in a time a dapt i ve D e f f u a n t - l i k e mo de l. Th e c o n s e n s u s t h r e s h o l d c ha ng e s o ve r time

d e p e n d i n g on the a gen t e xper ien ce : it i n c re as e s (d ec rea ses ) o f a f a c t or X d e p e n d i n g on w h e t h e r the agent is s u cc e ss f ul (or u n s u c c e s s f u l ) in fi nd i ng a

l i k e - m i n d e d n e i g h b o u r . T h e r es ults o b t a i n e d for T = 1 0 , 50, 333, 2 5 0 0 are c o m p a r e d to t h o s e o b t a i n e d for the n o n a d ap t i v e case. T h e a ver age s are p e r f o r m e d o ve r the di re ct n e i g h b o u r h o o d . T h e n u m b e r o f a gent s is equal to A/=104. E a c h a g en t has w = 4 di re c t c o n n e c t i o n s . T h e s i m u l a t i o n s have b e e n c ar r ie d o u t o ve r 1000 sampl es. T h e e r r o r bars are o f the same size as the p o i n t - m a r k e r s .

D e f f u a n t - l i k e m o d e l . T h e c o n s e n s u s t h r e s h o l d c h a n g e s o v e r t i me d e p e n d i n g o n t he a g e n t e x p e r i e n c e : it i n c r e a s e s ( d e c r e a s e s ) o f a f a c t o r

I d e p e n d i n g o n w h e t h e r t he a g e n t is s u c c e s s f u l (or u n s u c c e s s f u l ) in f i n d i n g a l i k e - m i n d e d n e i g h b o u r . T h e r e s u l t s o b t a i n e d f or x = 1 0 , 50, 333, 2 5 0 0 are c o m p a r e d t o t h o s e o b t a i n e d f or t he n o n a d a p t i v e case. T h e a v e r a g e s are p e r f o r m e d o v e r t he di r e c t n e i g h b o u r h o o d . T h e n u m b e r o f a g e n t s is e qual t o N = 1 0 0 4 . E a c h a g e n t has m —\ d i r e c t c o n n e c t i o n s . T h e s i m u l a t i o n s h a ve b e e n c a r r i e d o u t o v e r 1000 s a mp l e s . T h e e r r o r b a r s are o f t he s a me size as t he p o i n t - m a r k e r s .

F i g u r e 6.9: N u m b e r o f s u r v i v i n g o p i n i o n s d i v i d e d by t he n u m b e r o f a g e n t s , as a f u n c t i o n o f t he i ni t i al n u m b e r o f o p i n i o n s in a t i me a d a p t i v e D e f f u a n t - l i k e m o d e l . T h e c o n s e n s u s t h r e s h o l d c h a n g e s o v e r t i me

d e p e n d i n g o n t he a g e n t e x p e r i e n c e : it i n c r e a s e s ( d e c r e a s e s ) o f a f a c t o r x d e p e n d i n g o n w h e t h e r t he a g e n t is s u c c e s s f u l (or u n s u c c e s s f u l ) in f i n d i n g a

l i k e - m i n d e d n e i g h b o u r . T h e r e s u l t s o b t a i n e d f or x = 10, 50, 333, 2 5 0 0 are c o m p a r e d t o t h o s e o b t a i n e d f or t he n o n a d a p t i v e case. T h e a v e r a g e s are p e r f o r m e d o v e r t he d i r e c t n e i g h b o u r h o o d . T h e n u m b e r o f a g e n t s is e qual t o N = 1 0 0 0 4 . E a c h a g e n t h a s m —A di r e c t c o n n e c t i o n s . T h e s i m u l a t i o n s have b e e n c a r r i e d o u t o v e r 1000 s a mp l e s . T h e e r r o r ba r s are o f t he s a me size as t he p o i n t - m a r k e r s .

L I S T O F T A B L E S

T a b l e l . P a y o f f o f d i f f e r e n t s t r a t e g i e s i n t h e P r i s o n e r ’ s d i l e m m a . T h e o u t c o m e f or a s i n g l e p l a y e r d e p e n d s o n t h e a c t i o n s o f b o t h p l a y e r s . I f t h e y b o t h c o o p e r a t e t h e y b o t h ge t 4 p o i n t s e a c h ; i f o n e c o o p e r a t e a n d t h e o t h e r s q u e a l s , t h e f i r s t g e t 0 p o i n t s a n d t h e s e c o n d 5. I f t h e y b o t h s q u e a l t h e y o n l y g e t 1 p o i n t e a c h .

A b b r e v i a t i o n s

p d f = p r o b a b i l i t y d e n s i t y f u n c t i o n .

A C F = A u t o C o v a r i a n c e F u n c t i o n .

C T R W = C o n t i n u o u s T i m e R a n d o m Wa l k .

S T D P = S u r v i v i n g T i m e P r o b a b i l i t y D i s t r i b u t i o n .

M - L = M i t t a g - L e f f l e r .

G L V = G e n e r a l i z e d L o t k a - V o l t e r r a .

G L V P P = G e n e r a l i z e d L o t k a - V o l t e r r a w i t h P e e r P r e s s u r e .

P P = P e e r P r e s s u r e

R E M = R e l a t i v e E x p e c t a t i o n M o d e l .

C H A P T E R 1

G E N E R A L I N T R O D U C T I O N

1.1 C o m p l e x i t y

T o d e s c r i b e a n d u n d e r s t a n d n a t u r e it is f r e q u e n t l y n e c e s s a r y t o a p p e a l to s t a t i s t i c a l or p r o b a b i l i s t i c c o n c e p t s . In p h y s i c s , t h a t is t h e c a s e w h e n w e d e a l w i t h l a r g e n u m b e r s o f p a r t i c l e s or w i t h s y s t e m s t h a t ar e c o m p o s e d o f a l a r g e n u m b e r o f s u b s y s t e m s . In g e n e r a l o n e t r i e s t o r e l a t e t he b e h a v i o u r o f t h e c o m p o s i t e s y s t e m t o t h e p r o p e r t i e s o f t h e p a r t i c l e s or s u b s y s t e m s a n d t h e i r m u t u a l i n t e r a c t i o n s .

T h e r e ar e s y s t e m s c o m p o s e d o f g r o u p s o f r e l a t e d u n i t s ( s u b s y s t e m s ) , f or w h i c h t h e d e g r e e a n d n a t u r e o f t h e r e l a t i o n s h i p s is i m p e r f e c t l y k n o w n . T h e i r o v e r a l l e m e r g e n t b e h a v i o r is d i f f i c u l t t o p r e d i c t , e v e n w h e n t he s u b s y s t e m b e h a v i o r is r e a d i l y p r e d i c t a b l e . T h e t i m e - s c a l e s o f v a r i o u s s u b s y s t e m s m a y b e v e r y d i f f e r e n t . B e h a v i o r in t h e l o n g - t e r m a n d s h o r t ­ t e r m m a y be m a r k e d l y d i f f e r e n t a n d s m a l l c h a n g e s i n i n p u t s or p a r a m e t e r s m a y p r o d u c e l a r g e c h a n g e s i n b e h a v i o r . S y s t e m s w i t h t h e s e f e a t u r e s a r e d e f i n e d as C o m p l e x . E x a m p l e s o f C o m p l e x S y s t e m s ar e : t h e a t m o s p h e r e , t u r b u l e n t f l u i d s , b i o l o g i c a l s t r u c t u r e s a n d c o m m u n i t i e s , w h i c h i n c l u d e h u m a n s o c i e t i e s .

1.2 S t u d y i n g c o i t i mu n i t i e s

T h e i n t e r e s t o f p h y s i c i s t s in h u m a n - b a s e d c o m m u n i t i e s h a s i n d e e d r o o t s t h a t d a t e b a c k t o 19 3 6 w h e n t h e p h y s i c i s t M a j o r a n a w r o t e a p i o n e e r i n g

CHAPTER 1. GENERAL INTRODUCTION

i n c o m e s i n e c o n o m y a n d t h e d i s t r i b u t i o n o f c o t t o n p r i c e s : b o t h s e e m e d to

f l u c t u a t e r a n d o m l y a n d c o u l d n o t be f i t t e d to t h e n o r m a l d i s t r i b u t i o n

c u r v e u s e d as a s t a n d a r d m o d e l f o r p l o t t i n g v a r i a t i o n . M a n d e l b r o t [2]

b e g a n t o a n a l y z e a c e n t u r y ' s w o r t h o f c o t t o n p r i c e d a t a a n d , i n s t e a d o f

j u s t c o n c e n t r a t i n g o n t he l a r g e l o n g t e r m c h a n g e s , he i n c l u d e d t h e s m a l l

s c a l e f l u c t u a t i o n s t oo. T h i s g a v e M a n d e l b r o t a d e t e r m i n a t i o n t o e x p l o r e

t h e p h e n o m e n o n o f s c a l i n g .

In s p i t e o f t h i s , it w a s o n l y f r o m t h e 9 0 ’ s t h a t p h y s i c s c o m m u n i t y

b e c a m e p r o g r e s s i v e l y i n t e r e s t e d in m o d e l i n g f i n a n c i a l m a r k e t s , u r b a n

d e v e l o p m e n t , w e a l t h d i s t r i b u t i o n s a n d t he d i f f u s i o n o f o p i n i o n s a n d

d i s e a s e s . D r a m a t i c e v e n t s s u c h as t h e N e w Y o r k s t o c k m a r k e t c r a s h a n d

t h e f a s t s p r e a d o f H I V in t h e 8 0 ’ s b r o u g h t i n t o q u e s t i o n s t h e e x i s t i n g

q u a n t i t a t i v e m o d e l s o f h u m a n i n t e r a c t i o n a n d , w i t h i n t h e l a s t d e c a d e ,

t r i g g e r e d a s e r i e s o f n e w i n t e r d i s c i p l i n a r y s t u d i e s . F o r i n s t a n c e :

• M a n t e g n a a n d S t a n l e y [3] w e r e t h e f i r s t to p u b l i s h an

e x t e n s i v e s t u d y o f t h e s c a l i n g p r o p e r t i e s o f d a t a t a k e n f r o m

t h e s t o c k m a r k e t . T h e y w e r e a b l e t o s h o w t h e e x i s t e n c e o f

s i g n i f i c a n t d i f f e r e n c e s b e t w e e n t h e s t a t i s t i c a l d i s t r i b u t i o n s

a n d t h e e c o n o m e t r i c m o d e l s ( l i k e A R C H , G A R C H , e t c ) u s e d

in f i n a n c e a n d w h a t is o b s e r v e d f or r e a l d a t a .

• B a r a b a s i a n d A l b e r t [ 4 ] [ 5 ] i n v e s t i g a t e t h e s c a l e - f r e e

p r o p e r t i e s o f r e a l w o r l d n e t w o r k s a n d p r o p o s e d a m o d e l b a s e d

on a ‘p r e f e r e n t i a l a t t a c h m e n t ’ o f n o d e s m e c h a n i s m . T h i s w a s

a m i l e s t o n e a c h i e v e m e n t , s i n c e t h e s p r e a d i n g o f o p i n i o n s a n d

d i s e a s e s in a s o c i e t y is h i g h l y d e p e n d e n t on t h e t o p o l o g i c a l

p r o p e r t i e s o f t h e s y s t e m .

• F a r m e r [6] p r o p o s e d a m a r k e t m o d e l b a s e d o n t r a d e r e c o l o g y

a n d , w i t h L i l l o , s t u d i e d t he p r o p e r t i e s o f t h e i m p a c t - p r i c e

f u n c t i o n [7].

CHAPTER 1. GENERAL INTRODUCTION

• B o u c h a u d [8] i n v e s t i g a t e d t h e s t a t i s t i c a l f o u n d a t i o n s o f e x i s t i n g m a r k e t m o d e l s a n d p r o p o s e d s o m e n e w i n s i g h t f u l m o d e l s . He a l s o s u g g e s t e d an e x p l a n a t i o n b a s e d o n g r a n d - c a n o n i c a l m i n o r i t y g a m e s f or t h e l o n g - t i m e - m e m o r y e f f e c t s o b s e r v e d in m a r k e t v o l u m e a n d v o l a t i l i t y t i m e s e r i e s [9].

• S o l o m o n p r o p o s e d a n a g e n t - b a s e d g e n e r a l i z a t i o n o f L o t k a - V o l t e r r a m o d e l s , c a l l e d G e n e r a l i z e d L o t k a V o l t e r r a ( G L V ) m o d e l , a b l e t o a c c o u n t f or t h e p o w e r l a w d i s t r i b u t i o n s f o u n d in m a n y s e l f - o r g a n i z e d s o c i a l s y s t e m s ( e . g . w e a l t h d i s t r i b u t i o n , c i t y p o p u l a t i o n , e t c . ) . T h e G L V m o d e l s o f f e r a ‘m e s o s c o p i c ’ d e s c r i p t i o n o f s y s t e m s . T h e i r a c t u a l c o n n e c t i o n w i t h m i c r o s c o p i c i n t e r a c t i o n s is n o t o b v i o u s , t h o u g h [ 10] .

• G o p i k r i s h n a n a n d P l e r o u i n v e s t i g a t e d t h e s t a t i s t i c a l p r o p e r t i e s o f m a r k e t v o l u m e s a n d v o l a t i l i t y , i d e n t i f y i n g s o m e q u a s i - u n i v e r s a l f e a t u r e s o f t h e s e q u a n t i t i e s [ 11] [ 12] .

• C h a l l e t s t u d i e d t h e M i n o r i t y G a m e p r o b l e m , i n s p i r e d by t h e , so c a l l e d , ‘El F a r o l ’ b a r p r o b l e m [ 13] .

• A u s l o o s i n v e s t i g a t e d t h e s t a t i s t i c a l p r o p e r t i e s o f e x c h a n g e r a t e t i m e s e r i e s [ 1 4 ] , t h e s t o c k m a r k e t c r a s h p r e c u r s o r s [ 15] a n d s o u g h t a m o r e s c i e n t i f i c a p p r o a c h t o t h e t e c h n i c a l a n a l y s i s m e t h o d o l o g y u s e d b y f i n a n c i a l p r a c t i t i o n e r s [ 16] .

CHLr\PTER 1. GENERAL INTRODUCTION

a s t a t e i n w h i c h it d o e s not . N e v e r t h e l e s s , q u i t e u n r e a l i s t i c a l l y , in t h e s e s i m u l a t i o n s a g e n t s a r e t h o u g h t t o be w i t h o u t m e m o r y a n d r a n d o m c h a n g e s o f o p i n i o n a r e n o t a l l o w e d . S t a u f f e r a l s o s i m u l a t e d a s e c o n d c o n s e n s u s m o d e l , p r e v i o u s l y i n t r o d u c e d by D e f f u a n t [ 2 0 ] , on a s c a l e - f r e e n e t w o r k [ 21] a n d p r o p o s e d a n e w v e r s i o n o f it b a s e d on d i s c r e t e o p i n i o n v a l u e s [ 22] , as o p p o s e d to t h e c o n t i n u o u s o p i n i o n v a l u e s ( w i t h i n a c e r t a i n r a n g e ) u s e d in t h e o r i g i n a l m o d e l . In t h e s e f i r s t v e r s i o n s o f t h e D e f f u a n t m o d e l a g e n t s ar e n o t a l l o w e d to a d a p t t o t h e i r l o c a l e n v i r o n m e n t a n d to l e a r n f r o m t h e pa s t .

• M a i n a r d i a n d S c a l a s p r o p o s e d an a p p r o a c h t o t h e m o d e l i n g o f w a i t i n g t i m e d i s t r i b u t i o n s in f i n a n c i a l m a r k e t s b a s e d o n c o n t i n u o u s t i m e r a n d o m w a l k a n d f r a c t i o n a l B r o w n i a n m o t i o n [ 23] . T h e y s u c c e s s f u l l y t e s t e d t h e i r m o d e l u s i n g s o m e f i n a n c i a l a s s e t s ( L I F F E b o n d f u t u r e ) , b u t t h e g e n e r a l v a l i d i t y o f t h e i r a p p r o a c h h a s n o t b e e n p r o v e d yet .

• R i c h m o n d s t u d i e d t h e G e n e r a l i z e d L a n g e v i n E q u a t i o n s , L o t k a - V o l t e r r a a n d P e e r P r e s s u r e M o d e l s a n d s h o w e d h o w t h i s m o d e l s a r e r e l a t e d a n d h o w t h e y m a y a c c o u n t f o r s o m e i m p o r t a n t s t a t i s t i c a l f e a t u r e s f o u n d in s o c i o - e c o n o m i c s y s t e m s [ 10] [ 56] .

O t h e r s i g n i f i c a n t c o n t r i b u t i o n s c a m e f r o m : G a l a m [ 2 4 ] [ 2 5 ] , H e l b i n g a n d H o l y s t [ 26] , M a r s i l i [ 27] , S c h w e i t z e r [ 28] , l or i [ 29] a n d m a n y o t h e r s .

CHAPTER 1. GENERAL INTOODUCTION

T h e i r w o r k p a v e d t h e w a y t o a n e w p r o m i s i n g s c i e n t i f i c f i e l d , t o d a y k n o w n as s o c i o - e c o n o - p h y s i c s .

1.3 A g e n t M o d e l i n g

A g e n t - b a s e d m o d e l s ar e w i d e l y u s e d to d e s c r i b e s y s t e m s in w h i c h m a n y u n i t s i n t e r a c t in a w a y t h a t d e p e n d s on s o m e i n t e r n a l d e g r e e s o f f r e e d o m . O n e m a y n a i v e l y t h i n k o f a n a g e n t as a p a r t i c l e t h a t m a y be e i t h e r a t t r a c t e d or r e p e l l e d b y a g i v e n e x t e r n a l f i e l d . T h e a t t r a c t i o n o r r e p u l s i o n d e p e n d s o n t h e v a l u e t a k e n b y a n i n t e r n a l v a r i a b l e ( w h i c h i t s e l f o b e y s a d y n a m i c s p o s s i b l y d e t e r m i n e d b y t h e e x t e r n a l c o n d i t i o n s ) .

A r e l e v a n t e x a m p l e o f a g e n t m o d e l is g i v e n b y t h e M i n o r i t y G a m e m o d e l ( s e e a p p e n d i x ) , w i d e l y s t u d i e d w i t h i n t h e p h y s i c s c o m m u n i t y in r e c e n t y e a r s .

CHAPTER 1. GENERAL INTRODUCTION

1.4 T h e p h y s i c a l p r o p e r t i e s o f a g e n t b a s e d s y s t e m s

1. 4.1 T h e t o p i c o f thi s t h e s i s

T h i s t h e s i s d e a l s w i t h s y s t e m s m a d e u p o f a g e n t s by u s i n g s t a t i s t i c a l p h y s i c s t o o l s ( s u c h as s c a l i n g , c o r r e l a t i o n s , p r o b a b i l i t y d i s t r i b u t i o n s , s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s a n d M o n t e C a r l o s i m u l a t i o n s ) . W e a i m at f i n d i n g a b a l a n c e b e t w e e n t h e l e v e l o f d e t a i l s n e e d e d f o r c a p t u r i n g t h e i n t e r n a l d e g r e e o f f r e e d o m a n d d e s c r i b i n g a g e n t h e t e r o g e n e i t y a n d a d a p t a b i l i t y , a n d t h e s i m p l i c i t y n e e d e d f or t h e m o d e l t o be h a n d l e d u s i n g d i f f e r e n t i a l e q u a t i o n s , c e l l u l a r a u t o m a t a a n d s i m i l a r t o o l s . S u c h an a p p r o a c h a l l o w s t h e s t u d y o f l a r g e s e l f - o r g a n i z e d s o c i a l s y s t e m s t h a t c a n n o t b e m a s t e r e d o t h e r w i s e .

In C h a p t e r 2 w e p r o v i d e a d e s c r i p t i o n o f s o m e o u t s t a n d i n g q u a n t i t a t i v e f e a t u r e s o f s o c i o - e c o n o m i c s y s t e m s . A m o n g t h e m : t h e a l m o s t ‘ u n i v e r s a l ’ c h a r a c t e r s o f w e a l t h d i s t r i b u t i o n s , t h e f at t a i l e d d i s t r i b u t i o n a n d t h e l o n g m e m o r y o f v o l u m e s a n d v o l a t i l i t y f l u c t u a t i o n s in f i n a n c i a l m a r k e t s , t h e s c a l e - f r e e n a t u r e o f m a n y r e a l s o c i a l n e t w o r k s , t h e b r a i n - t w i s t e r r e p r e s e n t e d by s o c i a l d i l e m m a s .

In C h a p t e r 3 w e s t u d y t h e w a i t i n g t i m e d i s t r i b u t i o n s o f t w o d a t a s e t s t a k e n f r o m t h e X I X c e n t u r y I r i s h s t o c k m a r k e t a n d f r o m t h e l a t e X X c e n t u r y w o r l d c u r r e n c y m a r k e t a n d w e t e s t a m o d e l p r o p o s e d b y S c a l a s R a b e r t o a n d M a i n a r d i [ 23] a n d b a s e d o n C o n t i n u o u s T i m e R a n d o m W a l k a n d f r a c t i o n a l B r o w n i a n m o t i o n r e s u l t s .

In C h a p t e r 4 w e s h o w h o w s t o c h a s t i c d y n a m i c a l e q u a t i o n s s u c h as G e n e r a l i z e d L o t k a - V o l t e r r a e q u a t i o n s m a y be u s e d i n a g e n t m o d e l l i n g . W e a l s o f i n d a l i n k b e t w e e n t h e s t a n d a r d G e n e r a l i z e d L o t k a - V o l t e r r a a n d a s i m p l e p e e r - p r e s s u r e - b a s e d m o d e l . T h i s p r o v i d e s a l s o a b e t t e r

CHAPTER 1. GENERAL INTRODUCTION

u n d e r s t a n d i n g o f t h e u n d e r l y i n g m i c r o s c o p i c i n t e r a c t i o n s in t h e G e n e r a l i z e d L o t k a - V o l t e r r a m o d e l s .

F u r t h e r m o r e i n C h a p t e r 5 we d e a l w i t h t h e so c a l l e d ‘ C o n s e n s u s d y n a m i c s ’ . U n d e r c e r t a i n c o n d i t i o n s , t h e ‘s t a t e ’ o f a n a g e n t m a y s p r e a d t h r o u g h t h e s y s t e m a n d b e c o m e t h e ‘s t a t e ’ o f t h e s y s t e m . W e w i l l r e f e r to m o d e l s d e s c r i b i n g t h i s s p r e a d i n g p r o c e s s as ‘C o n s e n s u s M o d e l s ’ . In p a r t i c u l a r , w e c o n s i d e r t h e so c a l l e d , S z n a j d c o n s e n s u s m o d e l w i t h s y n c h r o n o u s u p d a t i n g . W e s h o w h o w i n d i v i d u a l a g e n t m e m o r y a n d r a n d o m f l u c t u a t i o n s m a y i n c r e a s e t h e d e g r e e o f c o n s e n s u s w i t h i n t h e s y s t e m ( i n a w a y t h a t , at f i r s t g l a n c e , m a y l o o k c o u n t e r - i n t u i t i v e ) . We d e s c r i b e h o w an i n t e r d e p e n d e n c e m e c h a n i s m b e t w e e n t h e p r o b a b i l i t y o f r a n d o m f l u c t u a t i o n s a n d t h e g l o b a l c o n s e n s u s f l u c t u a t i o n s m a y a c c o u n t f o r t h e o b s e r v e d m a r k e t v o l u m e d y n a m i c s .

CHAPTER 1. GENERAL INTRODUCTION

1 . 4 . 2 O r i g i n a l i t y o f t h i s t h e s i s .

In t hi s t h e s i s I wi l l s h o w that:

1. Th e w a i t i n g t i me d i s t r i b u t i o n for X I X C e n t u r y Ir i sh S t o c k M a r k e t c an be d e s c r i b e d by a M i t t a g - L e f f l e r f u n c t i o n , o b t a i n e d u n d e r the a s s u m p t i o n o f a p o w e r - l a w m e m o r y f u n c t i o n .

2. G e n e r a l i z e d L o t k a - V o l t e r r a ( G L V ) a nd G e n e r a l i z e d Lo t k a - V o l t e r r a m o d e l s wi t h Pe e r P r e s s u r e ( G L V P P ) l ead to p o w e r l a w p r o b a b i l i t y d i s t r i b u t i o n s t hat fit the d i s t r i b u t i o n o f p e r s o n a l w e a l t h f o u n d in mo s t e c o n o mi e s .

3. In t he Sz n a j d C o n s e n u s mo d e l we o b s e r v e an a p p a r e n t p a r a d o x ; a l l o w i n g a g e n t s to c h a n g e t he i r st at e (or o p i n i o n ) t h r o u g h r a n d o m f l u c t u a t i o n s ma y r e s u l t in an i n c r e a s e o f the d e g r e e o f c o n s e n s u s w i t h i n t he s y s t e m. Th e S z n a j d mo d e l d y n a m i c s ma y be u s e d to r e p r o d u c e t he m a r k e t v o l u m e d i s t r i b u t i o n s o b s e r v e d in ma n y f i n a n c i a l ma r k e t s .

4. In an a d a p t i v e v e r s i o n o f the D e f f u a n t c o n s e n s u s mo d e l , we o b s e r v e t h a t t he r e l a t i v e n u m b e r o f s u r v i v i n g o p i n i o n s ( n u m b e r o f s u r v i v i n g o p i n i o n s S o v e r n u m b e r o f a g e n t s N) , as a f u n c t i o n o f the r e l a t i v e i ni t i al n u m b e r o f o p i n i o n s ( n u m b e r o f i ni t i al o p i n i o n s Q o v e r n u m b e r o f a g e n t s N) , p r e s e n t s a n o n - m o n o t o n i c b e h a v i o r . A m a x i m u m o c c u r s f or t he i ni t i al n u m b e r o f o p i n i o n s a p p r o x i m a t e l y e q u a l to t he n u m b e r o f a g e n t s N. A h i g h e r d e g r e e o f c o n s e n s u s is r e a c h e d not onl y for Q s ma l l e r t ha n N, but al s o for Q m u c h l a r g e r t h a n N.

C H A P T E R 2

P H E N O M E N O L O G I C A L D E S C R I P T I O N

OF S O C I O - E C O N O M I C S Y S T E M S

2.1 I n t r o d u c t i o n

S e l f - o r g a n i z e d c o m m u n i t i e s d i s p l a y a s t o n i s h i n g f e a t u r e s a r i s i n g f r o m t h e m u t u a l i n t e r a c t i o n o f i n d i v i d u a l s . M a j o r p r o g r e s s e s h a v e b e e n m a d e i n r e c e n t y e a r s i n s p o t t i n g t h e s t a t i s t i c a l r e g u l a r i t i e s o f t h e s e s y s t e m s , e s p e c i a l l y o f f i n a n c i a l m a r k e t s . F o r i n s t a n c e , t h e p r e s e n c e o f p o w e r l a w p r o b a b i l i t y d i s t r i b u t i o n s a n d m e m o r y a r e t y p i c a l ( a l m o s t u n i v e r s a l ) s t a t i s t i c a l f e a t u r e s a s s o c i a t e d t o t h e s e s y s t e m s .

T h e r o l e o f t o p o l o g y a n d i n f o r m a t i o n i n c o m p l e t e n e s s , at a m i c r o s c o p i c l e v e l , i s n o w b e t t e r u n d e r s t o o d b y t h e s c i e n t i f i c c o m m u n i t y , b u t i t s f u l l c o m p r e h e n s i o n i s s t i l l a n o p e n c h a l l e n g e .

2. 2 S t a t i s t i c a l q u a n t i t i e s .

L e t u s n o w o u t l i n e v e r y b r i e f l y s o m e q u a n t i t i e s o f i n t e r e s t i n t h e s t u d y o f S o c i o - E c o n o m i c S y s t e m s .

CHAPTER 2. PHENOM ENOLOGICAL DESCRIPTION OF SOCIO-ECONOMIC SYSTEMS

• P r o b a b i l i t y D e n s i t y F u n c t i o n ( p . d . f . ) T h e p r o b a b i l i t y d e n s i t y f u n c t i o n o f a c o n t i n u o u s r a n d o m v a r i a b l e is a f u n c t i o n w h i c h c a n be i n t e g r a t e d to o b t a i n t h e p r o b a b i l i t y t h a t t h e r a n d o m v a r i a b l e t a k e s a v a l u e i n a g i v e n i n t e r v a l .

M o r e f o r m a l l y , t h e p r o b a b i l i t y d e n s i t y f u n c t i o n , / ( x ) , o f a c o n t i n u o u s r a n d o m v a r i a b l e X is t h e d e r i v a t i v e o f t h e C u m u l a t i v e D i s t r i b u t i o n F u n c t i o n F( x ) .

I f f ( x ) is a p r o b a b i l i t y d e n s i t y f u n c t i o n t h e n it m u s t o b e y t w o c o n d i t i o n s :

1. t h a t t h e t o t a l p r o b a b i l i t y f o r al l p o s s i b l e v a l u e s o f t he c o n t i n u o u s r a n d o m v a r i a b l e A" is 1:

2. t h a t t h e p r o b a b i l i t y d e n s i t y f u n c t i o n c a n n e v e r be n e g a t i v e ; f ( x ) > 0 f or all X.

• C e n t r a l L i m i t T h e o r e m ( C L T ) . T h e C e n t r a l L i m i t T h e o r e m s t a t e s t h a t w h e n e v e r a r a n d o m s a m p l e o f s i z e n is t a k e n f r o m a n y d i s t r i b u t i o n w i t h m e a n ^ a n d v a r i a n c e a , t h e n t h e s a m p l e m e a n m w i l l be a p p r o x i m a t e l y n o r m a l l y d i s t r i b u t e d w i t h m e a n |j, a n d v a r i a n c e a / « . T h e l a r g e r t h e v a l u e o f t h e s a m p l e s i z e n, t h e b e t t e r t h e a p p r o x i m a t i o n to t h e n o r m a l d i s t r i b u t i o n .

• A u t o - c o v a r i a n c e ( A C V F ) a n d A u t o c o r r e l a t i o n F u n c t i o n s . T h e A u t o c o v a r i a n c e is a m e a s u r e o f t h e l i n e a r d e p e n d e n c e b e t w e e n m e m b e r s o f a t i m e s e r i e s o f o b s e r v a t i o n s , s u c h as w e e k l y s h a r e p r i c e s or i n t e r e s t r a t e s , a n d t h e s a m e v a l u e s at a f i x e d t i m e i n t e r v a l l a t e r . A u t o c o r r e l a t i o n is o b t a i n e d b y d i v i d i n g t h e A u t o c o v a r i a n c e b y t h e V a r i a n c e o f t h e t i m e s e r i e s . A u t o c o r r e l a t i o n o c c u r s w h e n

(2 .1 )

C H A Pl’ER 2. PHENOM ENOLOGICAL DESCRIPTION OF SOCIO-ECONONflC SYSTEMS

r e s i d u a l e r r o r t e r m s f r o m o b s e r v a t i o n s o f t h e s a m e v a r i a b l e at d i f f e r e n t t i m e s a r e c o r r e l a t e d ( r e l a t e d ) .

• L e p t o k u r t o s i s . L e p t o k u r t i c is an a d j e c t i v e d e s c r i b i n g a d i s t r i b u t i o n w i t h h i g h k u r t o s i s . ' H i g h ' m e a n s t h e f o u r t h c e n t r a l m o m e n t ( k u r t o s i s ) is m o r e t h a n t h r e e t i m e s t h e s e c o n d c e n t r a l m o m e n t ; s u c h a d i s t r i b u t i o n h a s g r e a t e r k u r t o s i s t h a n a N o r m a l d i s t r i b u t i o n . T h i s t e r m is u s e d i n B o l l e r s l e v - H o d r i c k 19 9 2 t o c h a r a c t e r i z e s t o c k p r i c e r e t u r n s . L e p t o - m e a n s ' s l i m' i n G r e e k a n d r e f e r s t o t h e c e n t r a l p a r t o f t h e d i s t r i b u t i o n .

• W a i t i n g T i m e P r o b a b i l i t y D i s t r i b u t i o n ( W T P D ) . T h e W T D P is t h e p r o b a b i l i t y D i s t r i b u t i o n F u n c t i o n f o r t h e t i m e l a g b e t w e e n t w o c o n s e c u t i v e e v e n t s i n a t i m e s e r i e s . F o r i n s t a n c e , in C h a p t e r 3 we w i l l s t u d y t h e W T P D f o r t h e t i m e i n t e r v a l e l a p s i n g b e t w e e n t w o c o n s e c u t i v e t r a n s a c t i o n s in F i n a n c i a l M a r k e t s .

• S u r v i v a l T i m e P r o b a b i l i t y D i s t r i b u t i o n ( S J P D ) . T h e S T D P g i v e s t he p r o b a b i l i t y f or a c e r t a i n e v e n t to l a s t l o n g e r t h a n a g i v e n t i m e i n t e r v a l . In t h e F i n a n c i a l M a r k e t , t h e S T D P t e l l s us h o w w h a t is t h e p r o b a b i l i t y t h a t no t r a n s a c t i o n w i l l o c c u r d u r i n g a c e r t a i n t i m e i n t e r v a l .

• A u t o r e g r e s s i v e ( A R ) m o d e l . It d e n o t e s a s t o c h a s t i c p r o c e s s t h a t c a n be d e s c r i b e d by a w e i g h t e d s u m o f i t s p r e v i o u s v a l u e s a n d a w h i t e n o i s e e r r o r .

CHAPTER 2. PHENOM ENOLOGICAL DESCRIPTION OF SOCIO-ECONOMIC SYSTEMS

o n e . A n A R C H m o d e l is a d i s c r e t e t i m e s t o c h a s t i c p r o c e s s o f t h e f o r m;

= (

2

w h e r e t h e ar e i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d s t o c h a s t i c

v a r i a b l e s o v e r t i m e , w i t h z e r o m e a n a n d v a r i a n c e e q u a l t o 1. rj{t) is

a Standard Normal Variable ( i.e. a Gaussian distributed variable with zero mean and variance equal to 1). s{t) is a p o s i t i v e a n d t i m e - v a r y i n g q u a n t i t y ,

u s u a l l y o b t a i n e d as A u t o R e g r e s s i v e p r o c e s s f r o m t h e p a s t v a l u e s o f t h e t i m e s e r i e s v a r i a n c e

s ^ { t ) ~a Q + a^cr^{t - \ ) + ... + - n ) . ( 2 . 3 )

U s u a l l y s {t ) is f u r t h e r m o d e l e d to be an a u t o r e g r e s s i v e p r o c e s s . A c c o r d i n g t o A n d e r s e n a n d B o l l e r s l e v 1 9 9 5 / 6 / 7 , " A R C H m o d e l s a r e u s u a l l y e s t i m a t e d by m a x i m u m l i k e l i h o o d t e c h n i q u e s . " T h e y a l m o s t a l w a y s g i v e a l e p t o k u r t i c d i s t r b u t i o n o f a s s e t r e t u r n s e v e n i f o n e a s s u m e s t h a t e a c h p e r i o d ' s r e t u r n s a r e n o r m a l , b e c a u s e t h e v a r i a n c e is n o t t h e s a m e e a c h p e r i o d . E v e n A R C H m o d e l s , h o w e v e r , do n o t u s u a l l y g e n e r a t e e n o u g h k u r t o s i s i n e q u i t y r e t u r n s t o m a t c h U . S . s t o c k d a t a . A l s o , t h e i r s c a l i n g b e h a v i o u r d e v i a t e s f r o m w h a t o b s e r v e d in r e a l m a r k e t s [1]

• E r r o r a n d U n c e r t a i n t y

F o r p h e n o m e n o l o g i c a l d a t a e r r o r s a r i s e f r o m r e c o r d m i s t a k e s a n d r o u n d i n g up.

F o r M o n t e c a r l o s i m u l a t e d d a t a , e r r o r s h a v e a s t a t i s t i c a l o r i g i n . We p e r f o r m N s i m u l a t i o n s ( k e e p i n g al l t h e d e t e r m i n i s t i c p a r a m e t e r s u n c h a n g e d ) a n d w e t a k e t h e a r i t h m e t i c a v e r a g e as t h e m o s t l i k e l y v a l u e a n d (a q u a n t i t y p r o p o r t i o n a l t o ) t h e s t a n d a r d d e v i a t i o n , as

CHAPTER 2. PHENOM ENOLOGICAL DESCRIPTION O F SOCIO-ECONOMIC SYSTEMS

e r r o r ( t h i s w a y o f o p e r a t i n g is j u s t i f i e d b y t h e C e n t r a l L i m i t T h e o r e m ) .

T h e R e l a t i v e E r r o r ( o r u n c e r t a i n t y ) is t h e e s t i m a t e d a m o u n t ( or p e r c e n t a g e ) b y w h i c h a n o b s e r v e d or c a l c u l a t e d v a l u e m a y d i f f e r f r o m t h e t r u e v a l u e . A n e s t i m a t i o n o f u n c e r t a i n t y is g i v e n b y t h e r a t i o s t a n d a r d d e v i a t i o n / a v e r a g e v a l u e ( a s s u m i n g t h a t t h e a v e r a g e v a l u e is d i f f e r e n t f r o m z e r o )

• S t a t i s t i c a l A v e r a g e s .

I ' h e v a s t m a j o r i t y o f s o c i o - e c o n o m i c d at a are ‘s i n g l e r e a l i z a t i o n s ’. F o r i n s t a n c e , i f w e c o n s i d e r a c e r t a i n f i n a n c i a l i n d e x , w e m a y k n o w i ts t i m e e v o l u t i o n , b u t w e c a n n o t k n o w w h a t t h a t e v o l u t i o n w o u l d h a v e b e e n d u r i n g t h e s a m e p e r i o d o f t i m e , u n d e r d i f f e r e n t i n i t i a l c o n d i t i o n s . A s a c o n s e q u e n c e , u s u a l l y w e c a n o n l y p e r f o r m a v e r a g e s o v e r d at a t a k e n o v e r a c e r t a i n t i m e . W e w i l l u s e t h e s y m b o l < . . . > t o d e s i g n a t e t i m e a v e r a g e s .

2 . 3 D i s t r i b u t i o n o f w e a l t h

CHAPTER 2. PHENOM ENOLOGICAL DESCRIPTION O F SOCIO-ECONOMIC SYSTEMS

Understanding the origin and the ubiquity o f these distributions is still an open problem. Significant steps towards shedding light on it have been made in recent years, though.

100%

0)

g- 10%

a

c: 1 o>

s.

1

Zi

E

^

0 .1%

0

0 1

F i g u r e 2. 1: C u m u l a t i v e w e a l t h d i s t r i b u t i o n s i n t h e U n i t e d K i n g d o m in 19 9 6 ( c o u r t e s y o f D r a g u l e s c u & al . ; P h y s i c a A 2 9 9 2 1 3 - 2 2 1 ( 2 0 0 1 ) ) .

Pareto

10

100

1000

Total

not CtJ[)ital (wGallti),

kpounds