T. Mizokawa , J.-Y. Son, J. Quilty, D. Asakura, T.-T. Tran, K. Takubo
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...
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
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
!!
TheThe global global corporatecorporate eliteelite!! TransnationalTransnational businessbusiness councilscouncils
TheThe global global corporatecorporate eliteelite
TheThe global global corporatecorporate eliteelite
!! ffff
TheThe global global corporatecorporate eliteelite
TheThe global global corporatecorporate eliteelite
TheThe global global corporatecorporate eliteelite
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
TheThe connectionsconnections modelmodel
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
InnovationInnovation andand selfself--organizationorganization
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))
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
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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
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EigenvaluesEigenvalues comparedcompared withwith randomrandom permutationpermutation
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Geometria do Mercado FinanceiroMarket spaces
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Market spaces and crisis
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Distorsions andreduction of the
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Redes de MercadoStock market networks
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Redes de MercadoStock market networks
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1. hierarchical clustering2. minimal spanning tree3. LD
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1997 1998 2000 2001 2002 2004 2005 2006 20080
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