Cluster Expansion Method for Evolving Weighted Networks Having Vector-like Nodes

The Cluster Variation Method known in statistical mechanics and condensed matter is revived for weighted bipartite networks. The decomposition of a Hamiltonian through a finite number of components, whence serving to define variable clusters, is recalled. As an illustration the network built from data representing correlations between (4) macro-economic features, i.e. the so called $vector$ $components$, of 15 EU countries, as (function) nodes, is discussed. We show that statistical physics principles, like the maximum entropy criterion points to clusters, here in a (4) variable phase space: Gross Domestic Product (GDP), Final Consumption Expenditure (FCE), Gross Capital Formation (GCF) and Net Exports (NEX). It is observed that the $maximum$ entropy corresponds to a cluster which does $not$ explicitly include the GDP but only the other (3) ''axes'', i.e. consumption, investment and trade components. On the other hand, the $minimal$ entropy clustering scheme is obtained from a coupling necessarily including GDP and FCE. The results confirm intuitive economic theory and practice expectations at least as regards geographical connexions. The technique can of course be applied to many other cases in the physics of socio-economy networks.

[1]  Alessandro Pelizzola,et al.  Cluster Variation Method in Statistical Physics and Probabilistic Graphical Models , 2005, ArXiv.

[2]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[3]  Padhraic Smyth,et al.  Belief networks, hidden Markov models, and Markov random fields: A unifying view , 1997, Pattern Recognit. Lett..

[4]  Stephen G. Brush,et al.  Improvement of the Cluster‐Variation Method , 1967 .

[5]  R. Kikuchi.  A Theory of Cooperative Phenomena , 1951 .

[6]  R. Mantegna.  Hierarchical structure in financial markets , 1998, cond-mat/9802256.

[7]  Marcel Ausloos,et al.  AN ATTEMPT TO OBSERVE ECONOMY GLOBALIZATION: THE CROSS CORRELATION DISTANCE EVOLUTION OF THE TOP 19 GDP'S , 2006 .

[8]  Albert-László Barabási,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW , 2004 .

[9]  Evidencing European regional convergence clubs with optimal grouping criteria , 2005 .

[10]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[11]  Salvador Barrios,et al.  Industry mobility and geographic concentration in the European union , 2004 .

[12]  M. Ausloos,et al.  Cluster structure of EU-15 countries derived from the correlation matrix analysis of macroeconomic index fluctuations , 2006 .

[13]  Ryoichi Kikuchi,et al.  A Theory of Cooperative Phenomena. III. Detailed Discussions of the Cluster Variation Method , 1953 .

[14]  M. Ausloos,et al.  Clusters or networks of economies? A macroeconomy study through Gross Domestic Product , 2007 .

[15]  David Bawden,et al.  Book Review: Evolution and Structure of the Internet: A Statistical Physics Approach. , 2006 .

[16]  Marcel Ausloos,et al.  Convergence and cluster structures in EU area according to fluctuations in macroeconomic indices , 2008 .