Romain Boulet

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.

DISJOINT UNIONS OF COMPLETE GRAPHS CHARACTERIZED BY THEIR LAPLACIAN SPECTRUM

A disjoint union of complete graphs is in general not determined by its Laplacian spectrum. It is shown in this paper that if one only considers the family of graphs without isolated vertex, then a disjoint union of complete graphs is determined by its Laplacian spectrum within this family. Moreover, it is shown that the disjoint union of two complete graphs with a and b vertices, a b > 5 3 and b> 1 is determined by its Laplacian spectrum. A counter-example is given when a b = 5 3 .

Network analysis of the French environmental code

We perform a detailed analysis of the network constituted by the citations in a legal code, we search for hidden structures and properties. The graph associated to the Environmental code has a small-world structure and it is partitioned in several hidden communities of articles that only partially coincide with the organization of the code as given by its table of content. Several articles are also connected with a low number of articles but are intermediate between large communities. The structure of the Environmental Code is contrasting with the reference network of all the French Legal Codes that presents a rich-club of ten codes very central to the whole French legal system, but no small-world property. This comparison shows that the structural properties of the reference network associated to a legal system strongly depends on the scale and granularity of the analysis, as is the case for many complex systems.

Simplicial simple-homotopy of flag complexes in terms of graphs

A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs have the same simplicial simple-homotopy type. This result is closely related to similar results established by Barmak and Minian [J.A. Barmak, E.G. Minian, Simple homotopy types and finite spaces, Adv. Math. 218 (1) (2008) 87-104. doi:10.1016/j.aim.2007.11.019] in the framework of posets and we give the relation between the two approaches. We conclude with a question about the relation between the s-homotopy and the graph homotopy defined in [B. Chen, S.-T. Yau, Y.-N. Yeh, Graph homotopy and Graham homotopy, Selected papers in honor of Helge Tverberg, Discrete Math. 241 (1-3) (2001) 153-170. doi:10.1016/S0012-365X(01)00115-7.]

A Complex-System Approach: Legal Knowledge, Ontology, Information and Networks

In this paper we first briefly summarize the process used for building ontology from a legal corpus given in natural language. Current ontology-building supposes a particular structure and a finite number of relation types. The corresponding architecture is mainly driven by tree-like structures that capture a part of the full complexity that is effectively at work in any legal system. We propose to endow a legal ontology with further functionalities related to its mapping in a given corpus. We define posterior probability functions related to the frequency of occurrence of any term or concept, and information functions that measure the mutual information shared by terms in the corpus, whatever might be the a priori links represented between them in the ontology. We then show how these probabilistic tools can be also associated with a scale-dependent view on the network structure of a legal corpus (from the larger scale of the network of all codes or laws of a legal system, to the much finer scale of articles). New perspectives mixing semantic web and some properties of complex systems are described.

Network approach to the French system of legal codes part II: the role of the weights in a network

Unlike usual real graphs which have a low number of edges, we study here a dense network constructed from legal citations. This study is achieved on the simple graph and on the multiple graph associated to this legal network, this allows exploring the behavior of the network structural properties and communities by considering the weighted graph and see which additional information are provided by the weights. We propose new measures to assess the role of the weights in the network structure and to appreciate the weights repartition. Then we compare the communities obtained on the simple graph and on the weighted graph. We also extend to weighted networks the amphitheater-like representation (exposed in a previous work) of this legal network. Finally we evaluate the robustness of our measures and methods thus taking into account potential errors which may occur by getting data or building the network. Our methodology may open new perspectives in the analysis of weighted networks.

Graphs for Ontology, Law and Policy

Since the post-war decades, Public Policies are required increasingly as a preferred tool to promote collective action. Today these public policies are developed through sophisticated participatory schemes involving a variety of actors, public or private. Indeed in the current context of globalization though the State and its administration are key actors (Henry, 2004), their influence fades gradually into a more diffuse institutional environment (Ostrom, 2005) involving a multitude of other actors (Hassenteufel, 2008). For example at the two ends of the spectrum of governance, we find on one hand the increasing involvement of supranational entities, on the other hand the involvement of nongovernmental organizations.

s-homotopy for finite graphs

Abstract We introduce the notion of “s-dismantlability” which will give in the category of finite graphs an analogue of formal deformations defining the simple-homotopy type in the category of finite simplicial complexes. More precisely, s-dismantlability allows us to define an equivalence relation whose equivalence classes are called “s-homotopy types” and we get a correspondence between s-homotopy types in the category of graphs and simple-homotopy types in the category of simplicial complexes by the way of classical functors between these two categories (theorem 3.6). Next, we relate these results to similar results obtained by Barmak and Minian (2006) within the framework of posets (theorem 4.2).