Networks and hypernetworks 3Model of Bouchaud and MØzard (BM) ∑ ∑ ≠ ≠ = + − j i ji i j i...
Transcript of Networks and hypernetworks 3Model of Bouchaud and MØzard (BM) ∑ ∑ ≠ ≠ = + − j i ji i j i...
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NetworksNetworks andandhypernetworkshypernetworks 33
NetworksNetworks inin economyeconomy
Rui Vilela MendesRui Vilela Mendeshttp://label2.ist.utl.pt/http://label2.ist.utl.pt/vilelavilela//
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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
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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
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black: opinion leadersred: influenced green: uninfluencedgrey: undecided
Viral marketing
http://www.orgnet.com
Hubs:
‘broadcast’ weakly infectious viruses,
ideas
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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
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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
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Personal Income Distribution
Log-normal distribution with power-law tails (mixed form)
U.S.A. 1935-36 ($) Japan 1998 (M¥)
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Gross Domestic Product Distribution (GDP)
All countries; 1998, 1999, 2000, 2001 (G$): log-normal and power-law (mixed)
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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
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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
==++++=
=++==+
+=+=+
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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
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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
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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
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● 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
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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
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TheThe networksnetworks ofof thethe corporatecorporate eliteeliteTheThe Elite 16 Elite 16 inin Canada (2004)Canada (2004)
!! ffff
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TheThe networksnetworks ofof thethe corporatecorporate eliteelite
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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
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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))
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TheThe global global corporatecorporate eliteelite!! Global Global policypolicy groupsgroups
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TheThe global global corporatecorporate eliteelite
!!
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TheThe global global corporatecorporate eliteelite!! TransnationalTransnational businessbusiness councilscouncils
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TheThe global global corporatecorporate eliteelite
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TheThe global global corporatecorporate eliteelite
!! ffff
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TheThe global global corporatecorporate eliteelite
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TheThe global global corporatecorporate eliteelite
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TheThe global global corporatecorporate eliteelite
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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
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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))
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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
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TheThe connectionsconnections modelmodel
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TheThe connectionsconnections modelmodel
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TheThe connectionsconnections modelmodel
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InnovationInnovation andand selfself--organizationorganization
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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
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!! 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
InnovationInnovation andand selfself--organizationorganization
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InnovationInnovation andand selfself--organizationorganization
!! 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))
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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
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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
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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
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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
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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
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-0.5
0
0.5
-0.5
0
0.50.5
0
0.5
Uti IT Fin Heal Cons Ind Ene
-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
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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
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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
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During the crisis theagents display highercorrelations
Redes de MercadoStock market networks
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
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Redes de MercadoStock market networks
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
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Redes de MercadoStock market networks
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
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SynchronizationSynchronization ((statesstates))Redes de MercadoStates
Sep2001
Uti IT Fin Heal Cons Ind Ene
Mar1998otherwises
Ljids
i
Dti
02
),(166
=
∃⇔= ≤
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Redes de MercadoTopologiaStates
Dec2007
Jan2008
Feb2008
Uti IT Fin Heal Cons Ind Ene
Aug2000
Sep2000