A network approach to the French system of legal codes—part I: analysis of a dense network

Abstract

We explore one aspect of the structure of a codified legal system at the national level using a new type of representation to understand the strong or weak dependencies between the various fields of law. In Part I of this study, we analyze the graph associated with the network in which each French legal code is a vertex and an edge is produced between two vertices when a code cites another code at least one time. We show that this network distinguishes from many other real networks from a high density, giving it a particular structure that we call concentrated world and that differentiates a national legal system (as considered with a resolution at the code level) from small-world graphs identified in many social networks. Our analysis then shows that a few communities (groups of highly wired vertices) of codes covering large domains of regulation are structuring the whole system. Indeed we mainly find a central group of influent codes, a group of codes related to social issues and a group of codes dealing with territories and natural resources. The study of this codified legal system is also of interest in the field of the analysis of real networks. In particular we examine the impact of the high density on the structural characteristics of the graph and on the ways communities are searched for. Finally we provide an original visualization of this graph on an hemicyle-like plot, this representation being based on a statistical reduction of dissimilarity measures between vertices. In Part II (a following paper) we show how the consideration of the weights attributed to each edge in the network in proportion to the number of citations between two vertices (codes) allows deepening the analysis of the French legal system.

Appendix

Let us recall some aspects of a principal component analysis (PCA). If X is a data matrix with n rows and p columns (the ith row of X represents the coordinates in R ^p of the ith individual), a PCA gives a new coordinate system, that is a set of vectors u ₁, … ,u _p which is the new basis of R ^p. The individuals have new coordinates in this basis. The vector consisting of the jth coordinate of the individuals is called jth principal component and we denote it by c _j . This new coordinate system is such that maximizing the variance of the data projected on an r-dimensional subspace is done by projecting the data on the r first principal coordinates. It turns out that the vectors u _j are the orthonormal eigenvectors of the covariance matrix V = XX′ (X′ being the transpose of X). The principal coordinates c _j are given by $c_{j} = {\mathbf{X}}u_{j}$ with $u_{j} = \sqrt {\lambda_{j} } z_{j}$ where the z _j are the orthonormal eigenvectors of the scalar product matrix W = X′X. with matrix Λ of eigenvalues λ _j .

Proceeding in the same way with the distance matrix D we obtain the scalar product matrix W by the Torgerson formula

$$ w_{ij} = - \frac{1}{2}\left( {d_{i,j}^{2} - d_{i,.}^{2} - d_{.,j}^{2} - d_{.,.}^{2} } \right) $$

(5)

where

$$ d_{i,.}^{2} = - \frac{1}{n}\sum\limits_{j} {d_{i,j}^{2} } ,\quad d_{.,j}^{2} = - \frac{1}{n}\sum\limits_{i} {d_{i,j}^{2} } \quad {\text{and}}\quad d_{.,.}^{2} = - \frac{1}{n}\sum\limits_{i} {d_{i,.}^{2} } $$

(6)

Once we have the matrix W, a principal component analysis gives us the principal coordinates. Now, let X be a n × p data matrix and let a* ∈ (R ^p)* be a variable represented by the vector a ∈ R ^p (Riesz representation theorem). If we want to suppress the influence of the variable a* on the data matrix, we can consider the data projected on the orthogonal space of a (denoted $a^{ \bot }$). The new data matrix is Y = XP where P = I − aa′ is the matrix of the projection on $a^{ \bot }$ (with $\left\| a \right\| = 1$), we can then perform a principal component analysis on the new data matrix Y.

If we do not have X but the distance matrix D (and consequently the matrix W, the eigendecomposition of which is W = QΛQ ⁻¹) we can reconstruct a data matrix by setting X = Q√Λ (which is in fact the coordinates of individuals in the principal component system). Moreover we may not have directly the vector a ∈ R ^p representing the variable a* ∈ (R ^p)* but we have the value $\tilde{c}$ of the n individuals for the variable a*. The jth coordinate of a is the correlation coefficient between $\tilde{c}$ and the jth principal component c _j .

In practice, we obtained the angular coordinate θ(j) of the jth code in the hemicycle representation of the FLC network (see Sect. 4) in the following way: from the distance matrix D we get the scalar product matrix W, perform a first PCA and we recover X. Then we compute the vector a and perform a second PCA on XP (where P is the matrix of the projection on $a^{ \bot }$) and get a first principal component c ₁. Finally θ(j) is given (in rd) by:

$$ \theta (j) = \frac{{\pi \left[ {c_{1} (j) - \min_{k} \left( {c_{1} (k)} \right)} \right]}}{{\max_{k} \left( {c_{1} (k)} \right) - \min_{k} \left( {c_{1} (k)} \right)}} $$

(A3)