Introduction

This book is an overview of six years of joint research conducted within the Department of Applied Economics of ENSTA ParisTech. It introduces new tools for the analysis of economic exchange structures.

The first theme covered is that of the globalization of trade in industrial goods. In 2010, we published an article in the European Journal of Economic and Social Systems which introduced an original method to historically identify this phenomenon from international trade data for the period 1980-2004 [LEB 10]. These same tools were also used to quantify the scale of the phenomenon, identify contributing countries, in terms of industrial goods, whose exchange structure changed with time. In this method, trade in goods is represented in the form of graphs in which the nodes / poles correspond to countries and the links between these nodes to (physical or financial) flows that interconnect these countries. These flows can be divided according to the goods that are traded. Information on the direction of flow (who exports and who imports) and on their intensity (what amounts) is integrated in the graph. In the end, the structure of international trade, on a given date, looks like the graph in Figure I.1.

Figure I.1. Structure of international trade in industrial goods in 1980 (CEPII data, TradeProd database)

The graph is drawn such that the most "important" economies of the structure are positioned in its center. The term centrality here refers to a measure of the relative significance of a node. This significance is understood as a sum of influences that it exerts on the overall structure: capacity of transmitting to its partners the disruptions / economic shocks affecting it, or ability to control flows transiting through the structure.

In this book, we will present toolkits to interpret these graphs, both generally and also in a more restricted manner:

• - by "subgraphs", that is focusing the analysis on a restricted sample of countries (European or African for example);
• - by "partial graphs", that is for a particular product group (for example, low-medium- or high tech products).

These two "restrictions" can be combined, and we may occasionally analyze "partial subgraphs".

The book's added value is above all methodological. We present tools that help to navigate between the different scales of analysis while maintaining a solid basis for comparison. When we move from a complete graph to a subgraph, the links within the subgraph are relativized by inner and outer links. In other words, when studying trade relations between France and Germany in a "Europe" subgraph, the relationships that these countries have with their other economic partners play a role even if these partners are outside the European continent. The same applies for the partial graphs: relationships between these same two countries in high-tech products take into account those they develop for products with fewer new technologies. These tools are essentially derived from the economic dominance theory (EDT).

According to Lanther [LAN 74], economic dominance theory [LAN 74] initially applies to inter-industry trade flows as reported within the framework of the National Accounts. The indicators of centrality of nodes in exchange structures that Lantner presents, production multipliers and elasticities, reflect those traditionally handled within the framework of input-output analysis (which studies the interdependencies between productive sectors of an economy). The originality of the tool that the author develops in this context, the "influence graphs theory", is to articulate the mathematical graphs theory on the one hand and the fundamental elements of the input-output analysis on the other. Indeed, "the analysis of the effects of dominance in an exchange structure has been up to now subject to detrimental fragmentation", between matrix calculation, allowing for the understanding of global influences but not the process of disruptions, and the qualitative approach from unweighted graphs neglecting "unbalanced intensities of connections". The objective of the influence graphs theory is to "bridge the gap" between these two approaches "by revealing the conditions of general dependence and interdependence, related to the process of quantitative distribution of the influence" of poles in a given exchange structure [LAN 74].

This objective led the author to provide entirely new topological interpretations to inter-industry exchange structures. Roland Lantner showed that structural analysis is an intuitive way of calculating the determinant of matrices representative of directed and weighted exchange structures. This determinant will be subsequently considered as an indicator of the hierarchical distribution of influence through this structure. This led Lantner to formulate the following three theorems, which are informally1 presented here before attempting a synthetic interpretation:

• - Loops and circuits theorem: [LAN 74] shows that the value of the determinant associated with an exchange structure is a function of the value of "Hamiltonian partial graphs" (HPG) of the graph representing this structure. A Hamiltonian partial graph is a partial graph (i.e. initial graph without arcs interconnecting the poles) with the nodes having in and out "degrees" (number of connections) strictly equal to 1. The value of an HPG is in absolute value, the product of intensity coefficients that comprise it (see Chapter 3).
• - Amortization theorem: The value of the determinant is an increasing function of the general distribution of influence within an exchange structure. The looping effect generated by a non-Hamiltonian circuit (a "partial circularity"), disrupts the distribution of influence and reduces this value. The value of the determinant is therefore a decreasing function of partial circularities.
• - Partition theorem: This theorem defines the relationships between different sub-structures ("parts") of a given structure [LAN 00]. The determinant of the exchange structure is less than or equal to the product of the determinants of the parts. The difference measures "interdependence" between the parts. The general idea is to know the part of the general circulation of influence inside the structure (synthesized by the determinant) that is to be used in the circulation between the parts (which is, in our own word "external" to the parts) and that which is "internalized" in the parts. The aim is to try to determine if there exists a circulation base within the structure to, where appropriate, identify a hierarchy between the parts. In an extreme case where each pole constitutes a part, the difference between the product of the determinants of these parts (the product of the diagonal terms of the exchange structure) and the determinant of the structure measure the "general interdependence" of the structure. In the other extreme case where all the nodes are included in a single part, the difference (whose value is 0) indicates that all the interdependent relationships within the structure are internalized by the part.

We think the important point to note in these theorems, which we will refer to from time to time in the main development of the work, while re-explaining and illustrating them, is the following: the determinant of the matrix representing the graph allows for the separation of the results of influence / dependency, that is asymmetrical/ hierarchical relations between the poles, from the results of interdependence, that is symmetrical / circular relationships between these same poles. Exchange relationships are divided between these two structural phenomena: they reveal more or less dependence, and more or less interdependence. All things being equal, interdependence increases when the value of the determinant decreases. It is around this general result that the EDT toolkit will be constituted.

As [FRE 04] points out, the tools of mathematical graphs theory have been at the heart of the development of sociometric techniques (social network analysis) since the late 1940s, and more precisely since the pioneering intuitions of [BAV 48, LUC 49, SMI 50, LEA 51], to the more formal works of Frank Harary and of his colleagues from the University of Michigan [HAR 53, HAR 65]. What is required, according to Freeman, in what he called "l'école de la Sorbonne" (the school of Sorbonne), with [FLA 63] and [BER 58], is to establish "the earliest general synthesis showing explicitly that a wide range of problems could all be understood as special cases of a general structural model" [FRE 04]. The topological analyses of [PON 68, PON 72] and [LAN 72a, LAN 72b, LAN 74], continuing the reflections of François Perroux on the phenomena of power in economics (1973/1994 for a summary), are the most concrete manifestation of the breakthrough of this research tradition in the field of political economy.

The bridges between input-output analysis research traditions and social network analysis (SNA) have existed for a long time. The pioneering structural measures of global influence of [KAT 53, HUB 65, BON 72, COL 73, BUR 82] are thus, at least in part, derived from the application of matrix calculation concepts and techniques in SNA. These concepts and techniques are commonly...