Over the last decades, an interdisciplinary community of scientists, especially physicists, has carried out both empirical and theoretical investigations aimed at understanding economic and financial systems within the framework of complex systems. This approach, which defines the so-called field of Econophysics, differs from the one traditionally adopted in mainstream economics in various respects. First of all, emphasis is put on induction from empirical evidence to theoretical models, rather than deduction from strict and often ureasonable mathematical assumptions about the expected behaviour of individuals. Second, the preferred objects of analysis are large socio-economic systems with many uderlying units, which often interact with each other forming intricate networks with a complex topology. In this talk, I will briefly introduce various examples of complex networks encountered in Econophysics, and then focus in particular on the World Trade Web formed by the import-export relationships among all world countries. I will conclude with a discussion of the implications that recent results about the properties of the World Trade Web have for international macroeconomics

1. Interdisciplinary challenges:
Complex Networks and Econophysics
Diego Garlaschelli
Assistant Professor
Lorentz Institute for Theoretical Physics
Leiden Institute of Physics
garlaschelli@lorentz.leidenuniv.nl

2. Econophysics and networks
Vertices = assets traded in the stock market
Links = equal-time correlations
From synchronous
correlations
to distances:
↑
Minimum Spanning Tree

3. Vertices = assets traded in the stock market
Links = delayed correlations
Delayed correlations

Directed network of influence

Econophysics and networks

4. Italian Stock Market (2002)
Vertices = companies
Links = shareholdings (who owns whom)
Econophysics and networks

5. Vertices = boards of directors
Links = interlock (shared directors)
Econophysics and networks

6. Austrian Banks (2000-2003)
Vertices = banks
Links = liability
Econophysics and networks

7. The World Trade Web
Vertices = world countries
Links = trade relationships
Econophysics and networks

8. Product space of world
economy
Vertices = commodities
Links = similar trade patterns by countries
Econophysics and networks

9. Network description: added value?
Example (WTW):
1) Does the network approach add nontrivial information to traditional international-
economics analyses that explain trade in terms of country-specific variables only?
2) In network jargon: are higher-order topological properties (indirect interactions,
n-steps-away) explained by local ones (direct interactions, 1-step-away)?
Fix degrees
(1st order)
Study effect on degree correlations
(2nd order)
Study effect on clustering properties
(3rd order)

10. Rewiring the WTW: 4 choices of local constraints
degree
in-degree &
out-degree
strength
in-strength &
out-strength
We employ a maximum-entropy method:
Squartini, Garlaschelli – arxiv:1103.0701
Squartini, Fagiolo, Garlaschelli – arxiv:1103.1243
Squartini, Fagiolo, Garlaschelli – arxiv:1103.1249

11. (average number of trade partners of country i’s trade partners)
- Both the observed disassortativity and the temporal evolution of ANND
are fully explained by the degree sequence (red=real, blue=randomized).
- Focusing on local properties captures higher-order patterns.
Binary undirected WTW (constraint = degrees)
Degree-degree correlations:

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

13. Arms Coffee & Tea
Plastics Optical inst.
Nuclear react. Top 14
Arms Coffee & Tea
Plastics Optical inst.
Nuclear react. Top 14
Binary undirected WTW – disaggregated commodities
Degree correlations vs Degree: Clustering vs Degree:
- 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!

14. Same as in the binary undirected case: perfect accordance (red=real, blue=randomized).
Binary directed WTW (constraint = in & out degrees)
Average Nearest Neighbour Degree (ANND, 4 types):

15. Same as in the binary undirected case: perfect accordance (red=real, blue=randomized).
Binary directed WTW (constraint = in & out degrees)
Clustering Coefficients (4 types, see Fagiolo PRE 2007):

16. Binary directed WTW – disaggregated commodities
Tot.ANND vs Tot.Degree: Tot.Clustering vs Tot.Degree:
- 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!
Arms Coffee & Tea
Plastics Optical inst.
Nuclear react. Top 14
Arms Coffee & Tea
Plastics Optical inst.
Nuclear react. Top 14

17. (average bidirectional trade intensity of country i’s trade partners)
Scatter plot: ANNS vs strength (2002); ANNS mean & 95% conf. int. (1992-2002)
- Both the observed disassortativity and the temporal evolution of ANNS
are not explained by the strength sequence (red=real, blue=randomized).
- Focusing on local properties only (strength sequence = total trade of countries)
does not capture higher-order patterns.
- As the formula shows, deviations are in the topology (real sparser than random)
Weighted undirected WTW (constraint = strengths)
Strength-strength correlations (ANNS):

18. (relative intensity of interconnections among country i’s trade partners)
Scatter plot: clustering vs strength (2002); Clust. mean & 95% conf. int. (1992-2002)
- 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 from the null model are more evident when disaggregating (see next).
Weighted undirected WTW (constraint = strengths)
Weighted Clustering Coefficient:

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

20. Scatter plots: ANNS vs strength (2002); ANNS mean & 95% conf. int. (1992-2002)
Same as in the weighted undirected case: no accordance (red=real, blue=randomized).
Weighted directed WTW (constraint = in & out strengths)
Strength correlations (ANNS, 4 types):

21. Scatter plots: clustering vs strength (2002); Clust. mean & 95% conf. int. (1992-2002)
Improved accordance with respect to the undirected case (red=real, blue=randomized).
Weighted directed WTW (constraint = in & out strengths)
Clustering Coefficients (4 types, see Fagiolo PRE 2007):

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

23. Interpretation
The WTW as a binary network:
1) Local constraints (direct interactions) fully explain higher-order properties (indirect
interactions);
2) This finding holds for both directed and undirected representations, and over time;
3) It also holds for disaggregated commodities with different volume and resolution;
4) Two consequences:
3a) country-specific properties (number of trade partners) fully capture the WTW;
3b) theories and models of trade should aim at explaining the degree sequence,
which is maximally informative and encodes all the observed topology.
The WTW as a weighted network:
1) Local constraints (direct interactions) do not completely explain higher-order
properties (indirect interactions);
2) The real WTW is sparser and more disassortative than explained by local trade;
3) This finding holds for both directed and undirected representations, and over time;
4) Deviations from the null model increase as sparser and less aggregated commodity
classes are considered;
5) Two consequences:
4a) country-specific properties (total trade of countries) do not capture the WTW;
4b) again, the topology deserves more attention in models of trade, as it is
responsible for most of the empirical deviations from the expected behavior.

24. The binary topology of the World Trade Web can be completely reproduced
using only the knowledge of the GDP of each country (“fitness model”)
• Garlaschelli, Loffredo Physical Review Letters 93, 188701 (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, 015101(R) (2008)
GDP data  connection probability
 expected topology
WTW data  test the model
Looking back at previous results from a new perspective
Directionality and reciprocity can also be taken into account and reproduced.

25. • Garlaschelli, Battiston, Castri, Servedio, Caldarelli, Physica A 350, 491 (2005).
• Caldarelli, Battiston, Garlaschelli, Catanzaro, Lecture Notes in Physics 650, 399 (2004).
• Battiston, Garlaschelli, Caldarelli, in Nonlinear Dynamics and Heterogeneous Interacting Agents (2005).
• Caldarelli, Battiston, Garlaschelli, in Practical Fruits of Econophysics (Springer, 2006).
(links are drawn from owned
companies to shareholders)
Italian Stock Market
(MIB) 2002
Effects of wealth on topology: Shareholding Networks

26.  

ij
iji
ij
jijiii (t)wJ(t)wJ(t)(t)wη(t)w
Effects of topology on wealth: Transaction networks
Real GDP distributions:
Bouchaud-Mézard model:
Empty graph: Complete graph:
Pajek
Mixed (log-normal + power-law)Core-periphery network:
• Garlaschelli, Loffredo, Physica A 338, 113 (2004)
• Garlaschelli, Loffredo, Journal of Physics A 41, 224018 (2008)

27. ‘Food webs’: Networks of interactions among biological species
Food Webs
• Garlaschelli, Caldarelli, Pietronero, Nature 423, 165 (2003)
• Garlaschelli, Caldarelli, Pietronero, Nature 435, E1 (2005)
• Garlaschelli, Sapere 6, year 69 (2003)
• Garlaschelli, European Physical Journal B 38(2), 277 (2004)
• Caldarelli, Garlaschelli, Pietronero, Lecture Notes in Physics 625, 148 (2003)
St. Martin St. Mark Grassland Silwood
Ythan 1 Little
Rock
Ythan 2
Similar allometric scaling laws
found in Mimimal Spanning Trees
obtained from real food webs
(common organising principle?):
C(A) Aη
η = 1.13  0.03

28. Analytical solution:
Interplay between topology and dynamics:
coupling Extremal Dynamics (Bak-Sneppen model) with the Fitness Model.
Naturally generates scale-free networks with clustering and correlations.
Self-Organized Adaptive Network Evolution
Dynamical process
Topological evolution
x1 x2
x3
x4
x5
x6x7
x8
x9
x10
• Garlaschelli, Capocci, Caldarelli, Nature Physics 3, 813-817 (2007)
• Caldarelli, Capocci, Garlaschelli, Eur. Jour. of Physics B 64, 585-591 (2008).

29. • Garlaschelli, Loffredo, Physical Review Letters 93, 268701 (2004)
• Garlaschelli, Loffredo, Physical Review E 73, 015101(R) (2006)
Reciprocity of Directed Networks
Reciprocity = tendency to form mutual connections.
Traditional measure of reciprocity in social network analysis:
Problem:
networks with different link densities have
different null values of r: rnull=L/N(N-1)
(so they cannot be compared with each other)
Our new measure of reciprocity (correlation coefficient):
This unbiased quantity allows cross-network comparisons
1
2
6
4
5
3

30. Comparing the
reciprocity of
real networks
WTW
WWW
Neural
Email
Metabolic
Food Webs
Words
Financial
Results:
-Real networks always
display a nontrivial degree
of reciprocity,
while models do not.
- Networks of the same kind
display similar values of the
reciprocity, so that
reciprocity classifies real
networks.
-For some networks, it is
possible to characterize the
reciprocity structure more
accurately.
Garlaschelli, Loffredo Phys. Rev. Lett. 93,268701(2004)