The Right Coast

January 04, 2005
The Web of Law
By Tom Smith

I finally have a draft of the paper I've been obsessed with since last August. You can download the comment draft in PDF format here. (I love the idea so much, I've given it its own blog.)

The abstract is below. If you are interested in law and in the WWW, I am almost certain you'll find the paper quite interesting. I have gone to some lengths to make some pretty abstruse stuff accessible. (The paper will be available on SSRN shortly; this is a preview for the elite readers of TRC. I plan to send it out to law reviews in the Spring, assuming I can find one that will print the beautiful color pictures, and all the cool graphs.)


Scientists and mathematicians in recent years have become intensely interested in the structure of networks. Networks turn out to be crucial to understanding everything from physics and biology, to economics and sociology. This article proposes that the science of networks has important contributions to make to the study of law as well. Legal scholars have yet to study, or even recognize as such, one of the largest, most accessible, and best documented human-created networks in existence. This is the centuries-old network of case law and other legal authorities into which lawyers, judges, and legal scholars routinely delve in order to discover what the law is on any given topic. The network of American case law closely resembles the Web in structure. It has the peculiar mathematical and statistical properties that networks have. It can be studied using techniques that are now being used to describe many other networks, some found in nature, and others created by human action. Studying the legal network can shed light on how the legal system evolves, and many other questions. To initiate what I hope will become a fruitful new type of legal scholarship, I present in this article the preliminary results of a significant citation study of nearly four million American legal precedents, which was undertaken at my request by the LexisNexis corporation using their well-known Shepard’s citation service. This study demonstrates that the American case law network has the overall structure that network theory predicts it would.

The article has three parts. First, I introduce some basic concepts of network science, including such important ideas as nodes, links, random graphs, evolving networks, scale-free networks, small worlds, the “rich get richer” dynamic, node fitness, and clusters. Oddly enough, the mathematical tools that have proven most useful for studying networks (or at least scale-free networks) come from statistical mechanics, a branch of physics. Having introduced network theory in Part I, and having presented evidence that American case law is a scale-free network in Part II, I argue for the significance of this discovery in Part III. I hope that by the time they reach Part III, readers will already be realizing the potential richness of applying network theory to legal systems. In Part III, I describe some insights that appear from this application and suggest areas for future research.

The most famous hypothesis about the structure of law is that it is a “seamless web.” This old phrase, however, is just a metaphor we have used to grope for a reality we have not been in a position to express more precisely. Network science changes that. The Web of Law can be considered as a mathematical object whose topology can be analyzed using the tools pioneered by physicists and others who wanted to explore the structure of the Web and other real networks. The Web of Law has a structure very similar to that of other real networks, such as the Web and the network of scientific papers. The Web of Law is in substantial part a scale-free network, organized with hub cases that have many citations and the vast majority of cases, which have very few. The distribution of citation frequency approximates a power-law distribution, as is common with real scale-free networks, with truncations at either extreme of its distribution, which is also common.

Many promising hypotheses can be generated by considering the law as a scale-free network. State and federal systems can be examined empirically to measure how well integrated each is with itself, and with each other, and how this is changing over time. Legal authorities can be measured to determine whether their authority is emerging or declining. Institutional bodies, such as courts, can be examined in the same way. Clusters of cases, which will reveal the semantic topology of law, can be mapped to determine whether traditional legal categories are accurate or require reform. These methods can be used to develop computer programs to improve the efficiency of searching electronic legal databases. The topology of American law can be compared to that of other legal systems to determine whether legal systems share universal architectural features, and in what respects different systems are unique. Changing dynamics of the citation frequency and the fitness of particular cases can be studied over historical periods to test historiographical hypotheses. So, for example, Farber’s hypothesis that changes in constitutional interpretation occur suddenly, and many others, may be tested rigorously. The dynamics of authority in law generally can be studied much more rigorously. The mere fact that law is a scale free, not a random network, suggests a high degree of intellectual coherence, contrary to what some critics have suggested. The shape of the degree distribution graph of the Web of Law, in its similarity to the scientific citation network, also suggests that cases age, in the sense of losing the ability to attract citations, over time, just as scientific papers do. Yet Supreme Court cases seem to age more slowly. How nodes age profoundly affects overall network structure and therefore affects the shape of the Web of Law. Network theory hints at complex, but analyzable, interactions between the legal doctrines of precedent, and the systems of common law and multiple sovereignties.

Because law grows and because it has doctrines of authority, it creates a network of a certain shape, which spontaneously organizes itself. This is the product of laws that govern networks of computers as inexorably as they govern networks of cases, laws arising from the underlying mathematics of networks. Legal databases, which are huge, precisely documented, and readily accessible, present a perfect opportunity for the application of network science. This research would produce new knowledge of general jurisprudence that has simply been impossible until now, when we have the necessary advances in network science, the fast computers, and the existence of a complete record of the legal network in electronic form, waiting to be explored.