Jan Tinbergen s legacy for economic networks: from gravity to quantum statistics

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1 Jan Tinbergen s legacy for economic networks: from gravity to quantum statistics Diego Garlaschelli Assistant Professor Lorentz Institute for Theoretical Physics Leiden Institute of Physics garlaschelli@lorentz.leidenuniv.nl

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3 Jan Tinbergen - 1 s t Nobel Memorial Prize in Economics, Leiden, 1929: PhD Thesis Minimumproblemen in de natuurkunde en economie, supervisor P. Ehrenfest The Gravity Model of international trade: Simplest case: and 1, = 0 (as in Newton s law) Main limitations, in the light of a modern network approach: - Fitted only on non-zero flows; - Predicts fully connected network; - Bad performance if used to estimate link probabilities [Duenas & Fagiolo, J. of Economic Interaction and Coordination (2013)]

4 The International Trade Network (ITN) or World Trade Web (WTW)

5 Networks with local constraints degree C = { } k i in-degree & out-degree C =, { in out k } i k i strength C = { } s i in-strength & out-strength C =, { in out s } i s i We employ our analytical maximum-entropy method: Squartini, Garlaschelli New J. Physics 13, (2011) Squartini, Fagiolo, Garlaschelli Phys. Rev. E 84, (2011) Squartini, Fagiolo, Garlaschelli Phys. Rev. E 84, (2011)

6 Binary networks: fixed degree sequence C = { } k i Equiprobable configurations: P( )=P( )=P( ) (must hold for all vertices)

7 Binary networks: fixed degree sequence C = { k } = i a ij j i where FERMI- DIRAC! [Park & Newman, Phys. Rev. E 70, (2004)]

8 H { }) = Quantum analogy ( n a ε ana H( A) = α = ijaij a i< j i α k i i n a occupation number a ij n x a a ε a x energy α = α + α a i j = Fermi-Dirac a 1 ij = + xa 1+ xi x j α x i = e ε a / kt i = e fugacity ij i x x j Note: quantum statistics is just combinatorics! (the quantum lies outside it) a ij =0,1 is the only required analogy!

9 Binary WTW (constraint = degrees) Degree-degree correlations: (average number of trade partners of country i s trade partners) - Disassortativity and temporal evolution fully explained by the degrees (red=real, blue=randomized). - The degree sequence predicts higher-order patterns.

10 Binary WTW (constraint = degrees) Clustering Coefficient: (fraction of country i s trade partners that are mutually connected) - Correlation profile and temporal evolution fully explained by the degrees (red=real, blue=randomized). - The degree sequence should be reproduced by models!

11 Binary WTW (disaggregated commodities) Degree correlations vs Degree: Clustering vs Degree: Arms Coffee & Tea Arms Coffee & Tea Plastics Optical inst. Plastics Optical inst. Nuclear react. Top 14 Nuclear react. Top 14 - Note: from a) to f), the volume of trade and level of aggregation increases. - The results obtained in the aggregated case are robust to disaggregation. - Commodity-specific degree sequences are still maximally informative!

12 Weighted networks: fixed strength sequence C = { } s i Equiprobable configurations: P( )=P( )=P( ) =P( )=P( ) (must hold for all vertices)

13 Weighted networks: fixed strength sequence C = { s } = i w ij j i if w*=+ then BOSE- EINSTEIN! [Garlaschelli & Loffredo, Phys. Rev. Lett. 102, (2009)]

14 Weighted WTW (constraint = strengths) Strength-strength correlations (ANNS): (average bidirectional trade intensity of country i s trade partners) Scatter plot: ANNS vs strength (2002); ANNS mean & 95% conf. int. ( ) - Disassortativity and temporal evolution are not explained by the strengths (red=real, blue=randomized). - Local properties only (strength sequence = total trade of countries) do not capture higher-order patterns. - Deviations due to the topology (real sparser than random, see formula)

15 Weighted WTW (constraint = strengths) Weighted Clustering Coefficient: (relative intensity of interconnections among country i s trade partners) Scatter plot: clustering vs strength (2002); Clust. mean & 95% conf. int. ( ) - Partial accordance, improving over time (red=real, blue=randomized). - However deviations in the topology (denominator) and in the weigths (numerator) are both large, but largely compensate each other. - Deviations more evident when disaggregating (see next).

16 Weighted WTW (disaggregated commodities) Strength correlations vs Strength: W.Clustering vs Strength: Arms Coffee & Tea Arms Coffee & Tea Plastics Optical inst. Plastics Optical inst. Nucl.react. Top 14 Nucl.react. Top 14 - Note: from a) to f), the volume of trade and level of aggregation increases. - Sparser and less aggregated commodities are more scattered. - Local properties less informative as sparseness and resolution increase.

17 Consequences for macroeconomics The WTW as a binary network: 1) country-specific properties (number of partners) fully capture the WTW; 2) theories and models of trade should aim at explaining the degree sequence, which is maximally informative and encodes almost entirely the observed topology. The WTW as a weighted network: 1) country-specific properties (total trade) do not capture the WTW; 2) again, the topology deserves more attention in models of trade, as it is responsible the empirical deviations from the expected behavior.

18 Doubling the constraints C =, { } k i s i P( ) P( ) P( ) P( )=P( 2 )=P( )

19 The generalized Bose-Fermi distribution: Now [Garlaschelli & Loffredo, Phys. Rev. Lett. 102, (2009)] [Mastrandrea, Squartini, Fagiolo, Garlaschelli, in preparation]

20 Binary WTW (constraint = strengths + degrees) Average nearest-neighbour degree:

21 Binary WTW (constraint = strengths + degrees) Clustering Coefficient:

22 Weighted WTW (constraint = strengths + degrees) Average nearest neighbour strength:

23 Weighted WTW (constraint = strengths + degrees) Weighted Clustering Coefficient:

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25 Looking back at previous results from a new perspective The binary topology of the World Trade Web can be completely reproduced using only the knowledge of the GDP of each country ( fitness model ) GDP data connection probability expected topology WTW data test the model Directionality and reciprocity can also be taken into account and reproduced. Garlaschelli, Loffredo Physical Review Letters 93, (2004) Garlaschelli, Loffredo Physica A 355, 138 (2005) Garlaschelli, Di Matteo, Aste, Caldarelli, Loffredo European Physical Journal B 57,159 (2007) Garlaschelli, Loffredo Physical Review E 78, (R) (2008)

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28 Consequences/speculations Extensive and intensive margins of economic (and not only) networks 1) weighted properties are NOT necessarily more informative than topological ones (typically the opposite is true); 2) intensive margins (degree) are as fundamental as extensive ones (strengths) and irreducible to the latter. Network reconstruction (equilibrium vs out-of-equilibrium) 1) at equilibrium, node-specific (countries or banks) information is enough in order to reconstruct the network (but must include topology); 2) out of equilibrium, node-specific information is NOT enough, and actually deviations from equilibrium might be the most important piece of information (early-warning signals).