Randomly Searching the Law: Mean First-Passage Times and Complexity of Legal Trees

Student thesis: Doctoral ThesisDoctor of Philosophy


This thesis proposes computational modelling to further the quantitative understanding of legal problems. We design a model for the behaviour of a law user retrieving information hidden in legal texts. The latter are typically organised in a hierarchical structure, which the reader needs to explore down to the ‘deepest’ level (Articles, Clauses, etc ), until they have identified an answer to their question. Following previous works on transport properties of networks, the mean first-passage time (MFPT) taken by a random reader to retrieve information planted in the leaves is nominated as a measure of structural complexity of legal trees. The reader is assumed to initially skim the contents of a text, identifying keywords based on their interests, and be drawn towards the sought information based on keyword affinity. That is, they estimate how well the Chapter and Section headers of the hierarchy seem to match the informational content they expect to see in their target-answer, and follow the more promising ones with higher probability. 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 derive a few plausible high-level consequences. We obtain numerical and analytical results, the latter by approximating the biases imposed on the reader by the keyword patterns with the average bias obtained by averaging over the distribution of keyword patterns. This method leads to an explicit expression for the complexity of legal trees as a function of the structural parameters of the model.

We present in a simple workflow how one can prepare a real Act of Parliament for calculating its complexity and how the model parameters, for the text ensemble, can be estimated to obtain the complexity of the ensemble, as well. This is demonstrated explicitly on the example of the Housing Act 2004. We briefly discuss potential steps to test the validity of the various assumptions and results of our model.

Our analytical results are powered by a novel method to calculate MFPTs for random walks on graphs. This is a general result based on a dimensionality reduction technique for Markov state models, known as local-equilibrium (LE).We can prove that for a broad class of graphs, LE coarse-graining preserves the MFPTs between certain nodes if the coarsegrained states (or clusters) are suitably chosen. The amenable class of graphs includes trees.

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. The proof produces a generalisation of a well-known lemma about MFPTs along bridges, or essential edges. This essential edge lemma (EEL) is valid for reversible random walks, whereas our generalisation also applies to irreversible walkers. It leads to explicit formulae for the MFPTs between certain nodes in the necklace-class of graphs. 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. This result is used to facilitate the analytical calculations for the reader model mentioned above.

For graphs that do not fall within the necklace class, we show that the generalised EEL provides useful approximations if the graph allows a one-dimensional coarse-grained representation and the clusters are either weakly interconnected or have a single dominating edge qualifying as a nearly essential edge. The error made in this approximation can be efficiently estimated by means of a perturbative expansion, given that the random walker is reversible. We provide references to other works on perturbations of Markov chains that may be useful in bounding the approximation error.
Date of Award1 Jul 2023
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorPierpaolo Vivo (Supervisor) & Alessia Annibale (Supervisor)

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