Mapping inter-city trade networks to maximum entropy models using electronic invoice data

We analyze the network of transactions among cities in Ceará state, Brazil, using a database from municipal electronic invoices containing 3.7 billion records from 2016 to 2019. These transactions form an asymmetrical adjacency matrix, representing a directed graph with connections weighted by transaction volumes. Considering Ceará’s vast area, its unequal distribution of wealth, and spatially heterogeneous population density, we identify city communities based on trade intensity, assessing their economic “cohesiveness”. Utilizing the Infomap algorithm, we first determine the network’s optimal community structure in terms of its associated flow dynamics, revealing five distinct modules, whose two-dimensional geographical projections are all simply-connected domains, i.e., consisting of single pieces without holes. We then proceed with the analysis from the perspective of traded products by mapping out bipartite structures between municipalities and products for both buying and selling contexts. Employing the revealed comparative advantage (RCA) approach, we propose a non-monetary, binary activity index to distinguish the trade strength of a city for a class of goods or services as evidenced by trade flows. Finally, through the pairwise Maximum Entropy Model, we can associate to the largest communities their corresponding binary Ising-like Hamiltonian models. For a given community, the local fields and couplings computed are those that best reproduce the average product activities of all cities as well as the statistical correlations between the product activities of all pairs of cities. Remarkably, our findings show that, despite not being explicitly used in the inference method, the observed correlations among triplets of cities are consistently replicated by the derived Ising-like models. Moreover, in an analogy with critical phenomena, our results reveal that each community operates at a “temperature” that is close to the corresponding “critical point”, suggesting a high degree of “economic cohesiveness” in its trade network of cities.