# Properties of the United States Code Citation Network

## TL;DR

This paper analyzes the citation network of the United States Code, finding that while outdegree follows a power-law distribution, the indegree and degree per word are better modeled by a log-normal distribution, revealing insights into legal document structure.

## Contribution

The study introduces a model based on citation probability per word and demonstrates that degree distributions are log-normal, challenging the assumption of power-law distributions in legal citation networks.

## Key findings

- Outdegree distribution fits a power-law model.
- Indegree and degree per word follow a log-normal distribution.
- Power-law models are not generally applicable to the Code's citation network.

## Abstract

The United States Code (Code) is an important source of Federal law that is produced by the interactions of many heterogeneous actors in a complex, dynamic space. The Code can be represented as the union of a hierarchical network and a citation network over the vertices representing the language of the Code. In this paper, we investigate the properties of the Code's citation network by examining the directed degree distributions of the network. We find that the power-law model is a plausible fit for the outdegree distribution but not for the indegree distribution. In order to better understand this result, we construct a model with the assumption that the probability of citation is a per-word rate. We calculate the adjusted degree of each vertex under this model and study the directed adjusted degree distributions. These adjusted degree distributions indicate that both the adjusted indegree and outdegree distributions seems to follow a log-normal form, not a power-law form. Our findings indicate that the power-law is not generally applicable to degree distributions within the United States Code but that the distribution of degree per word is well-described by a log-normal model.