A Network Approach to the French System of Legal codes - Part I: Analysis of a Dense Network

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.

💡 Research Summary

The paper presents a novel network‑based analysis of the French legal system at the level of codified statutes. The authors construct an undirected, unweighted graph in which each of the 52 French legal codes (as defined by the 1989 codification policy and listed on the official Legifrance portal) is represented by a vertex, and an edge is placed between two vertices whenever one code cites the other at least once. By focusing solely on the existence of a citation (the number of citations is examined in a companion paper, Part II), the authors obtain a relatively small graph (V = 52) that is nevertheless extremely dense: it contains 531 edges, yielding an average degree of about 20.4 and a density of 0.40. This density is far higher than that typically observed in most empirical networks (social, technological, biological) where densities are usually below 0.1.

The authors argue that such a high‑density structure does not fit the classic “small‑world” paradigm (characterized by low density, high clustering, and short average path length). Instead, they introduce the notion of a “concentrated world” – a class of networks that are highly connected overall but still exhibit a discernible core‑periphery organization. Basic topological measures confirm this picture: the average shortest‑path length is about 1.9, the diameter is 3, and the clustering coefficient is 0.71, indicating that most vertices are only a step or two away from each other while still forming tightly knit triads.

Centrality analyses (degree, betweenness, closeness) reveal a small set of “hub” codes that dominate the citation landscape. The Civil Code, the Penal Code, and the Administrative Code emerge as the most central vertices, acting as hubs that link many other codes. Their prominence suggests that any legislative reform affecting these codes would have far‑reaching implications across the entire legal corpus.

Detecting communities in such a dense graph poses methodological challenges. Traditional modularity‑maximization algorithms (e.g., Louvain, Girvan‑Newman) tend to merge the whole network into a single community because the high edge density inflates the expected number of intra‑community edges under the null model. To overcome this, the authors adopt several strategies: (i) they adjust modularity to account for the overall density, (ii) they employ hierarchical clustering based on a dissimilarity matrix derived from shortest‑path distances, and (iii) they develop a bespoke visualization called the “hemicycle plot.” The hemicycle plot projects the dissimilarity matrix into two dimensions using multidimensional scaling, then arranges vertices along a semicircular arc. Vertices are colored and sized according to community membership and centrality, respectively, allowing a clear visual separation of groups despite the underlying density.

The community detection results identify three major clusters that correspond to substantive domains of regulation: (1) a central cluster of influential codes (the hub codes mentioned above), (2) a “social‑issues” cluster comprising codes related to labor, social security, health, and education, and (3) a “territories‑and‑resources” cluster containing environmental, land‑use, and local‑government codes. While inter‑cluster links are present, they are sparser than intra‑cluster connections, confirming a modular organization within an overall dense fabric.

The paper situates its contribution within two broader scholarly conversations. From a legal‑studies perspective, it offers a quantitative map of the French codified system that can inform policymakers about the structural importance of specific codes, potentially guiding codification priorities and impact assessments. From a network‑science standpoint, it highlights a class of empirical networks that have been largely overlooked: small‑scale but highly dense graphs. By formalizing the “concentrated world” concept, the authors invite further theoretical work on models that can generate such structures and on algorithms tailored to their analysis.

Limitations are acknowledged. The analysis ignores citation weights, which are examined in Part II; temporal dynamics (how the network evolves as new codes are added or citations change) are not addressed; and the study is confined to France, leaving open the question of whether other civil‑law jurisdictions exhibit similar dense patterns. Future research directions include weighted‑network extensions, longitudinal studies, and comparative cross‑national analyses.

In sum, the study demonstrates that the French legal code network, despite its modest size, is exceptionally dense and organized around a few central hubs and domain‑specific communities. By introducing the “concentrated world” classification and a novel visualization technique, the authors provide both substantive insights into French legal architecture and methodological contributions to the broader field of complex‑network analysis.