We introduce a model for the retrieval of information hidden in legal texts. These are typically organised in a hierarchical (tree) structure, which a reader interested in a given provision needs to explore down to the “deepest” level (articles, clauses,...). We assess the structural complexity of legal trees by computing the mean first-passage time a random reader takes to retrieve information planted in the leaves. The reader is assumed to skim through the content of a legal text based on their interests/keywords, and be drawn towards the sought information based on keywords affinity, i.e. how well the Chapters/Section headers of the hierarchy seem to match the informational content of the leaves. Using randomly generated keyword patterns, we investigate the effect of two main features of the text – the horizontal and vertical coherence – on the searching time, and consider ways to validate our results using real legal texts. We obtain numerical and analytical results, the latter based on a mean-field approximation on the level of patterns, which lead to an explicit expression for the complexity of legal trees as a function of the structural parameters of the model.
In this work we propose a novel method to calculate mean first-passage times (MFPTs) for random walks on graphs, based on a dimensionality reduction technique for Markov State Models, known as local-equilibrium (LE). We show that for a broad class of graphs, which includes trees, LE coarse-graining preserves the MFPTs between certain nodes, upon making a suitable choice of the coarse-grained states (or clusters). We prove that this relation is exact for graphs that can be coarse-grained into a one-dimensional lattice where each cluster connects to the lattice only through a single node of the original graph. A side result of the proof generalises the well-known essential edge lemma (EEL), which is valid for reversible random walks, to irreversible walkers. Such a generalised EEL leads to explicit formulae for the MFPTs between certain nodes in this class of graphs. For graphs that do not fall in this class, the generalised EEL provides useful approximations if the graph allows a one-dimensional coarse-grained representation and the clusters are sparsely interconnected. We first demonstrate our method for the simple random walk on the $c$-ary tree, then we consider other graph structures and more general random walks, including irreversible random walks.
Background: legislation.gov.uk is a platform that enables users to explore and navigate the many sections of the UK’s legal corpus through its well-designed searching and browsing features. However, there is room for improvement as it lacks the ability to easily move between related sections or Acts and only presents a text-only rendering of provisions. With Graphie, our novel navigational tool (graphie.quantlaw.co.uk), we aim to address this limitation by presenting alternative visualizations of legal documents using both text and graphs. Methods: The building block of Graphie is Sofia, an offline data pipeline designed to support different data visualizations by parsing and modelling data provided by legislation.gov.uk in open access form. Results: Graphie provides a network representation of the hierarchical structure of an Act of Parliament, which is typically organized in a tree-like fashion according to the content and information contained in each sub-branch. Nodes in Graphie represent sections of an Act (or individual provisions), while links embody the hierarchical connections between them. The legal map provided by Graphie is easily navigable by hovering on nodes, which are also color-coded and numbered to provide easily accessible information about the underlying content. The full textual content of each node is also available on a dedicated hyperlinked canvas. Conclusions: While we focus on the Housing Act 2004 for illustrative purposes, our platform is scalable, versatile, and provides users with a unified toolbox to visualize and explore the UK legal corpus in a fast and user-friendly way.
We introduce a model for the retrieval of information hidden in legal texts. These are typically organised in a hierarchical (tree) structure, which a reader interested in a given provision needs to explore down to the "deepest" level (articles, clauses,...). We assess the structural complexity of legal trees by computing the mean first-passage time a random reader takes to retrieve information planted in the leaves. The reader is assumed to skim through the content of a legal text based on their interests/keywords, and be drawn towards the sought information based on keywords affinity, i.e. how well the Chapters/Section headers of the hierarchy seem to match the informational content of the leaves. Using randomly generated keyword patterns, we investigate the effect of two main features of the text -the horizontal and vertical coherence -on the searching time, and consider ways to validate our results using real legal texts. We obtain numerical and analytical results, the latter based on a mean-field approximation on the level of patterns, which lead to an explicit expression for the complexity of legal trees as a function of the structural parameters of the model.Policy implications of our results are briefly discussed.
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