networks and hypernetworks 3model of bouchaud and mØzard (bm) ∑ ∑ ≠ ≠ = + − j i ji i j i...

NetworksNetworks andandhypernetworkshypernetworks 33

NetworksNetworks inin economyeconomy

Rui Vilela MendesRui Vilela Mendeshttp://label2.ist.utl.pt/http://label2.ist.utl.pt/vilelavilela//

Business ties in US biotechBusiness ties in US biotech--industryindustry

Nodes: companies: investmentpharmaresearch labspublicbiotechnology

Links: financialR&D collaborations

http://ecclectic.ss.uci.edu/~drwhite/Movie

black: opinion leadersred: influenced green: uninfluencedgrey: undecided

Viral marketing

http://www.orgnet.com

Hubs:

‘broadcast’ weakly infectious viruses,

ideas

Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8 Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8

500 randomly chosen users 500 most active users

Empirical wealth distributionsTwo typical forms:

Large wealth: Pareto�s law (power-law distribution):

α--1wP(w) ∝

Small wealth: Gibrat�s law (log-normal distribution):

=0

222 w

wlog2

1-exp2w1P(w)

σπσ

Cumulative distribution: ∫∞ ≡=> x

ww/x)dx'P(x'(x)P tot

Personal Income Distribution

Log-normal distribution with power-law tails (mixed form)

U.S.A. 1935-36 ($) Japan 1998 (M¥)

Gross Domestic Product Distribution (GDP)

All countries; 1998, 1999, 2000, 2001 (G$): log-normal and power-law (mixed)

Empirical forms of �wealth� distributions:

The most general form of P(w) is �mixed�:

Combination of a power-law and a log-normal distribution

Theoretical models that can reproduce the mixed form

Purely multiplicative stochastic process:

wi (t) = wealth of agent i at time t

ηi (t) = Gaussian process (mean m and variance 2σ2 )

(t)(t)wη(t)w iii =&

Independent agents models

Log-normal distribution

...(t)]ηlog[11)]-(tηlog[11)]-(tlog[w

(t)]ηlog[1(t)]log[w1)](tlog[w

(t)(t)]wη[1(t)(t)wη(t)w1)(tw

iii

ii

i

iiiiii

==++++=

=++==+

+=+=+

Multiplicative stochastic process with a lower boundary:

Multiplicative-additive stochastic process:

miniiii w(t)w(t)(t)wη(t)w >=&

Power-law distribution

0<+= (t)η log(t)ξ(t)(t)wη(t)w iiiii&

Independent agents models

Wealth evolution with N agents:

wi (t) = wealth of agent i at time t

ηi (t) = Gaussian process (mean m and variance 2σ2 )

Jij = fraction of wealth flowing from j to i

Model of Bouchaud and Mézard (BM)

∑∑≠≠

−+=ij

ijiij

jijiii (t)wJ(t)wJ(t)(t)wη(t)w&

Interactive multiplicative stochastic process:wealth evolution is determined by the interactions among economic agents

Independent agentsJij =0 ∀∀∀∀ i,j

(t)(t)wη(t)w iii =&

Mean field

(t)Jw(t)wJ(t)(t)wη(t)w iiii -+=&

Jij =J/N ∀∀∀∀ i,j

α=1+J/σ2totw/wx

xP(x) --1

≡∝ α

● Start with a set of N isolated vertices;

● For each pair of vertices draw a link with uniform probability p.

p=0 p=0.1

p=0.5 p=1

BM model on random graphs

p=N-1.5 p=N-1

p=N-0.5 p=N0=1

The wealth distribution P(w) changes suddenlyfrom log-normal (p<pc) to power-law (p>pc)

BM model on random graphs

Simulation parameters: N=3000 T=10000

J=0.05 �η�=1 �η2�-�η�2 = 0.1

TheThe networksnetworks ofof thethe corporatecorporate eliteeliteTheThe Elite 16 Elite 16 inin Canada (2004)Canada (2004)

!! ffff

TheThe networksnetworks ofof thethe corporatecorporate eliteelite

TheThe networksnetworks ofof thethe corporatecorporate eliteelite!! IndividualsIndividuals, , atat thethe core core ofof thethe networknetwork, , controlcontrol

thethe diffusiondiffusion ofof informationinformation inin thethe networknetwork!! CorporateCorporate governancegovernance practicespractices spreadspread throughthrough

sharedshared directorsdirectors!! FirmsFirms are more are more likelylikely to to adoptadopt anan acquisitionacquisition

strategystrategy ifif theythey shareshare a director a director withwith a a companycompany thatthat hashas anan acquisitionacquisition strategystrategy

!! AntiAnti--takeovertakeover strategiesstrategies diffusediffuse alongalong director director networksnetworks

TheThe global global corporatecorporate eliteelite

!! NetworkNetwork ofof overlappingoverlapping membershipmembership amongamongdirectorsdirectors ofof thethe world’sworld’s (500) (500) leadingleadingcorporationscorporations andand transnationaltransnational policypolicy boardsboards

!! 500 500 leadingleading corporationscorporations!! 7 global 7 global policypolicy groupsgroups!! 4 4 transnationaltransnational businessbusiness councilscouncils

((W. K. W. K. CarrollCarroll andand J. P. J. P. SapinskySapinsky, , InternationalInternationalSociologySociology 25 (2010) 50125 (2010) 501--538538))

TheThe global global corporatecorporate eliteelite!! Global Global policypolicy groupsgroups


!!

TheThe global global corporatecorporate eliteelite!! TransnationalTransnational businessbusiness councilscouncils


!! ffff

NetworksNetworks andand thethe productproduct spacespace!! Economies grow upgrading the products they produce and Economies grow upgrading the products they produce and

exportexport!! Technology, capital and skills needed to make newer products Technology, capital and skills needed to make newer products

are more easily adapted from some products than from othersare more easily adapted from some products than from others!! The network of relations between products, is called the The network of relations between products, is called the

“product space,”“product space,”!! Sophisticated products are located in a densely connected coreSophisticated products are located in a densely connected core!! Less sophisticated ones occupy a lessLess sophisticated ones occupy a less--connected periphery. connected periphery. !! Countries move through product space developing goods close Countries move through product space developing goods close

to those they currently produce. to those they currently produce. !! To reach the core most countries need to move through large To reach the core most countries need to move through large

distances, distances, !! Explains why poor countries have trouble developing Explains why poor countries have trouble developing

competitive exports and converge to the income level of rich competitive exports and converge to the income level of rich countriescountries

NetworksNetworks andand thethe productproduct spacespace!! ProductProduct codescodes, , sizesize andand proximityproximity

((C. A. C. A. HidalgoHidalgo, B. , B. KlingerKlinger, A. L. , A. L. BarabásiBarabási andand R. R. HausmannHausmann, , ScienceScience 317 (2007) 482317 (2007) 482--487487))

ModelsModels for for thethe formationformation ofof strategicstrategicnetworksnetworks

!! InIn anan economiceconomic contextcontext a a networknetwork linklink isis formedformed ifif andandonlyonly ifif bothboth agentsagents ((nodesnodes) ) findfind thatthat establishingestablishing thatthat linklinkisis beneficialbeneficial to to themthem

!! ThereforeTherefore modelsmodels requirerequire anan utilityutility functionfunction!! TheThe notionnotion ofof ““pairwisepairwise stabilitystability”” ofof a a networknetwork g g withwith

linkslinks betweenbetween agentsagents i i andand j j denoteddenoted ijij

!! DifferentDifferent fromfrom NashNash equilibriumequilibrium!! EfficientEfficient networknetwork whenwhen thethe total total utilityutility isis maximummaximum

TheThe connectionsconnections modelmodel

InnovationInnovation andand selfself--organizationorganization

InnovationInnovation andand selfself--organizationorganization!! A A multimulti--agentagent modelmodel!! EachEach agentagent isis characterizedcharacterized byby twotwo bit bit stringsstrings!! PP--stringstring: : WhatWhat thethe agentagent extractsextracts fromfrom thethe environmentenvironment

((thethe otherother agentsagents))!! NN--stringstring: : WhatWhat thethe otherother agentsagents maymay extractextract fromfrom himhim..!! TheThe modelmodel appliesapplies bothboth to to anan economyeconomy oror anan ecologicalecological

contextcontext!! FitnessFitness ofof eacheach agentagent

!! AtAt eacheach stepstep ofof thethe evolutionevolution eacheach agentagent matchesmatches hishis P P stringstring to to thethe N N stringsstrings ofof thethe otherother agentsagents..

!! ThenThen, , amongamong thosethose PP--stringsstrings withwith thethe higherhigher matchingmatchingwithwith a particular a particular NN--stringstring, , oneone isis chosenchosen atat randomrandom thatthatsuppliessupplies ((economyeconomy) ) oror preyspreys ((ecologyecology) ) thethe agentagent withwith thethecorrespondingcorresponding NN--stringstring. . TheThe q’sq’s inin thethe fitnessfitness are are thetheoverlapsoverlaps..

!! PP--innovationinnovation meansmeans to to changechange eacheach timetime oneone bit to bit to increaseincrease matchingmatching withwith thethe NN--stringsstrings

!! NN--innovationinnovation meansmeans to to changechange thethe bits to bits to decreasedecrease thethematchingmatching, , thereforetherefore reducingreducing whatwhat isis givengiven to to thethematchingmatching PP--stringstring..

!! SupplySupply ((EconomyEconomy) ) oror PredationPredation ((EcologyEcology) ) networksnetworks



!! ConclusionsConclusions::!! WithWith PP--innovationinnovation alonealone: a : a winner(swinner(s))--taketake--allall

situationsituation!! WithWith NN--innovationinnovation alonealone: : diversifieddiversified supplierssuppliers, ,

lowlow costcost!! WithWith bothboth PP-- andand NN--innovationinnovation: similar to : similar to thethe

withoutwithout innovationinnovation casecase

((T. Araújo T. Araújo andand RVM, RVM, AdvancesAdvances inin ComplexComplexSystemsSystems 12 (2009) 23312 (2009) 233--253253))

The stock market: An undirectedweighted network

Nodes: CompaniesLinks: established by a metric dependending

on the fluctuation correlations

RVM, T. Araújo and F. Louçã, Physica A 323 (2003) 625-648T. Araújo and F. Louçã, Quantitave Finance 7 (2007) 63-74

MetricDistances defined from the returns correlation

)1(2 ijij cd −=

))(log())(log()( 1 kpkpkr tt −−= p: pricer: return

The market space1. Compute each stock coordinates from

the distances2. Define the center of mass as the origin3. Construct the inertia tensor4.4. IdentifyIdentify thethe relevantrelevant f f dimensionsdimensions byby

comparisoncomparison withwith a a randomrandompermutationpermutation ofof thethe datadata

Financial Financial marketmarket geometrygeometry

CC

IBMIBMIBM

GEGEGE

AAAAAA

BABABA

SSS

TTTKKK

The number of embedding dimensions

S&P500 and Dow Jones, daily data

" 35 companies, 10 years " 70 companies, 10 years" 249 companies, 33 years " 253 companies, 35 years" 253 companies, 22 years " 424 companies, 10 years

In all cases: No more than 6 dimensions !

Geometria do Mercado FinanceiroMarket spaces

dd

IBMIBM

GEGE

EigenvaluesEigenvalues comparedcompared withwith randomrandom permutationpermutation

λλλλλλλλ+(1+(1--λλλλλλλλ ’’))

λλλλ : actualλλλλ’ : random

Geometria do Mercado FinanceiroMarket spaces

0 10 20 300.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11998-2008 424 Stocks

λ +

(1- λ

*)0 10 20 30

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ +

(1- λ

*)

1988-2008 253 Stocks

Redes de Mercado

Shape modification at the crisis

Market spaces and crisis

“Spherical” formTypical of “surrogate data”

and “business-as-usual”periods

Distorsions andreduction of the

distances during crisis

-0.5

0

0.5

-0.5

0

0.5-0.5

0

0.5

Uti IT Fin Heal Cons Ind Ene

-0.5

0

0.5

-0.5

0

0.50.5

0

0.5


-0.5

0

0.5-0.5

0

0.5

-0.5

0

0.5

Uti IT Fin Heal Cons Ind En

Market shape: S&P500

Geometria do Mercado FinanceiroStructure index

ShapeShape distorsionsdistorsions

∑=

−=

6

11

)()('

i t

tt i

iSλλ

StructureStructure indexindex

After 1997 there are many periods withmarket distorsions

λλλλ : actualλλλλ�: random

17/04/71 07/10/76 30/03/82 20/09/87 12/03/93 02/09/98 23/02/04 15/08/090

10

20

30

40

50230 stocks

S

88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 080

5

10

15

20253 stocks A new regime after 1997

SS

>5

Redes de MercadoStock market networks

The market networks are weighted and fully connected

1. hierarchical clustering2. minimal spanning tree3. LD

6 smaller distance in R6 which insures network connectivity

4. Then

02

12

,,6

,,6

6

6

=⇒>

=⇒≤

jiDji

jiDji

bL

d

bL

d

Aug2000

28

Energy

73

Industrial

114

Consumer

44

Health

70

Financial

56

Technology

39

Utilities

Normal periods have few links

During the crisis theagents display highercorrelations


Jan2008

28

Energy

73

Industrial

114

Consumer

44

Health

70

Financial

56

Technology

39

Utilities

Crisis periods: increase of the number of links, mostly inside the sectors


Sep2001

28

Energy

73

Industrial

114

Consumer

44

Health

70

Financial

56

Technology

39

Utilities

In some crisis: general increase ofthe numberof links in allsectors


Strong versus weak links

1. hierarchical clustering2. minimal spanning tree3. LD

6

4. Strong-weak ratio

1997 1998 2000 2001 2002 2004 2005 2006 20080

0.5

1

1.5424 stocks From Jan.1998 to Mar.2008

R

1997 1998 2000 2001 2002 2004 2005 2006 2008

R>0

.5

∑

∑

>

≤=

66

66

),(

6

),(

6

),(

),(

Dt

Dt

Ljidt

Ljidt

tjid

jid

R

SynchronizationSynchronization ((statesstates))Redes de MercadoStates

Sep2001


Mar1998otherwises

Ljids

i

Dti

02

),(166

=

∃⇔= ≤

Redes de MercadoTopologiaStates

Dec2007

Jan2008

Feb2008


Aug2000

Sep2000

networks and hypernetworks 3model of bouchaud and mØzard (bm) ∑ ∑ ≠ ≠ = + − j i ji i j i...

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