Interdisciplinary challenges:Complex Networks and Econophysics

Diego GarlaschelliAssistant Professor

Lorentz Institute for Theoretical PhysicsLeiden Institute of Physics

garlaschelli@lorentz.leidenuniv.nl

Econophysics and networksVertices = assets traded in the stock market

Links = equal-time correlations

From synchronouscorrelationsto distances:

↑Minimum Spanning Tree

Vertices = assets traded in the stock market

Links = delayed correlations

Delayed correlations

Directed network of influence

Econophysics and networks

Italian Stock Market (2002)

Vertices = companies

Links = shareholdings (who owns whom)

Econophysics and networks

Vertices = boards of directors

Links = interlock (shared directors)

Econophysics and networks

Austrian Banks (2000-2003)

Vertices = banks

Links = liability

Econophysics and networks

The World Trade Web

Vertices = world countries

Links = trade relationships

Econophysics and networks

Product space of world economy

Vertices = commodities

Links = similar trade patterns by countries

Econophysics and networks

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)

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:1103.0701Squartini, Fagiolo, Garlaschelli – arxiv:1103.1243Squartini, Fagiolo, Garlaschelli – arxiv:1103.1249

(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:

(fraction of country i’s trade partners that are mutually connected)

- Both the observed profile and the temporal evolution of clusteringare 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:

Arms Coffee & Tea

Plastics Optical inst.

Nuclear react. Top 14

Arms Coffee & Tea

Plastics Optical inst.

Nuclear react. Top 14

Binary undirected WTW – disaggregated commoditiesDegree 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!

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):

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):

Binary directed WTW – disaggregated commoditiesTot.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

(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):

(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:

Weighted undirected WTW – disaggregated commoditiesStrength 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

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):

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):

Arms Coffee & Tea

Plastics Optical inst.

Nucl.react. Top 14

Weighted directed WTW – disaggregated commoditiesTot.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

InterpretationThe 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.

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.

• 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 ownedcompanies to shareholders)

Italian Stock Market (MIB) 2002

Effects of wealth on topology: Shareholding Networks

ij

ijiij

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)

‘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 lawsfound in Mimimal Spanning Trees

obtained from real food webs(common organising principle?):

C(A) Aη η = 1.13 0.03

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 x2x3

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).

• Garlaschelli, Loffredo, Physical Review Letters 93, 268701 (2004)• Garlaschelli, Loffredo, Physical Review E 73, 015101(R) (2006)

Reciprocity of Directed NetworksReciprocity = 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

Comparing the reciprocity ofreal networks

WTWWWWNeural

Email

Metabolic

Food Webs

Words

Financial

Results:

-Real networks alwaysdisplay a nontrivial degree

of reciprocity, while models do not.

- Networks of the same kinddisplay similar values of the

reciprocity, so thatreciprocity classifies real

networks.

-For some networks, it ispossible to characterize the reciprocity structure more

accurately.

Garlaschelli, Loffredo Phys. Rev. Lett. 93,268701(2004)