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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.