Many of you are aware of my obsession (here) (here) (here) with fractals, power law distributions, etc. and their role in understanding a variety of phenomena.  In this spirit, I recently came across this video on MIT World from the late Benoît B. Mandelbrot.  It is well worth the watch – enjoy!

Do legal systems have physical properties? Considered in the aggregate, do the distinctions upon distinctions developed by common law judges self-organize in a manner that can be said to have definable physical property (at least at a broad level of abstraction)? The answer might lie in fractal geometry.

Fractal geometry was developed in a set of classic papers by mathematician Benoît Mandelbrot. The original paper in the field How Long is the Coastline of Britain describes the coastline measurement problem.  In short form, the length of the coast line is a function of the size of measurement one employs.  As shown below, as the unit of measurement decreases the length of the coastline increases.  The ideas expressed in this and subsequent papers have been applied to a wide class of substantive questions. In particular, the application to economic systems has been particularly illuminating. Given recent economic events, we agree with views of the Everyday Economist arguing the applied economic theory built upon his work should earn Mandelbrot a share of the Nobel Prize.

A more abstract fractal is the simple version of the Sierpinski triangle displayed at the top of this post. Here, there exists self similarity at all levels. Specifically, at each iteration of the model, the triangles at the tip of each of the lines replicate into self similar versions of the original triangle. If you click on the visual above, you can run the applet (provided you have java installed on your computer). {Side note: those of you NKS Wolfram fans out there will know the Sierpinski triangle can be generated using cellular automata Rule 90.}

For those who are interested in another demonstration consider the Koch Snowflake — a fractal which also offers a view of the relevant properties.  The Koch Snowflake is a curve with infinite length (i.e. there is no convergence even though it is located in a bounded region around the original triangle).  Click here to view an online demo of the Koch Snowflake.

So, you might be wondering … what is the law analog to fractals? As a first-order description of one important dynamic of the common law, we believe significant progress can be made by considering the conditions under which legal systems behave in a manner similar to fractals. For those interested, a number of important papers have discussed the fractal nature of legal systems.  While discussing legal argumentation, the original idea is outlined in two important early papers The Crystalline Structure of Legal Thought and  The Promise of Legal Semiotics both by Jack Balkin.  The empirical case began more than ten years ago in the important paper How Long is the Coastline of the Law? Thoughts on the Fractal Nature of Legal Systems by David G. Post & Michael B. Eisen. It continues in more recent scholarship such as The Web of the Law by Thomas Smith.

In our view, the utility of this research is not to adjudicate the common law to be a fractal. Indeed, there exist mechanisms which likely prevent legal systems from actually behaving as unbounded fractal.  The purpose of the discussion is determine whether describing law as a fractal is a reasonable first-order description of at least one dynamic within this complex adaptive system. While full adjudication of these questions is still an area of active research, we highlight these ideas for their important potential contribution to positive legal theory.

One thing we want to flag is the important relationship between the power law distributions we discussed in these prior posts (here and here) and the original work of  Benoît Mandelbrot. The mapping of the power law like properties displayed by the common law and its constitutive institutions is part of the larger empirical case for the fractal nature of legal systems. Building upon the prior work, in two recent papers, which are available on SSRN here and here, we mapped this property of self organization among two sets of legal elites — judges and law professors.

As a new semester is here at Michigan CSCS, I have made several revisions to the content of our global reading list for the Computational Legal Studies Working Group. The content of this interdisciplinary reading list features work from economics, physics, sociology, biology, computer science, political science, public policy, theoretical and empirical legal studies and applied math. I wanted to highlight this reading list for anyone who is interesting in learning more about the state of the literature in this interdisciplinary space.  Also, for those interested in learning model implementation, please consult my my slides from the 2010 ICPSR Course Introduction to Computing for Complex Systems. Feel free to email me if you have any questions.

Understanding the sources of complexity in legal systems is a matter long considered by legal commentators. In tackling the question, scholars have applied various approaches including descriptive, theoretical and, in some cases, empirical analysis. The list is long but would certainly include work such as Long & Swingen (1987), Schuck (1992), White (1992), Kaplow (1995), Epstein (1997), Kades (1997), Wright (2000) and Holz (2007). Notwithstanding the significant contributions made by these and other scholars, we argue that an extensive empirical inquiry into the complexity of the law still remains to be undertaken.

While certainly just a slice of the broader legal universe, the United States Code represents a substantively important body of law familiar to both legal scholars and laypersons. In published form, the Code spans many volumes. Those volumes feature hundreds of thousands of provisions and tens of millions of words. The United States Code is obviously complicated, however, measuring its size and complexity has proven be non-trivial.

In our paper entitled, A Mathematical Approach to the Study of the United States Code we hope to contribute to the effort by formalizing the United States Code as a mathematical object with a hierarchical structure, a citation network and an associated text function that projects language onto specific vertices.

In the visualization above, Figure (a) is the full United States Code visualized to the section level. In other words, each ring is a layer of a hierarchical tree that halts at the section level. Of course, many sections feature a variety of nested sub-sections, etc. For example, the well known 26 U.S.C. 501(c)(3) is only shown above at the depth of Section 501.  If we added all of these layers there would simply be additional rings. For those interested in the visualization of specific Titles of the United States Code … we have previously created fully zoomable visualizations of Title 17 (Copyright), Title 11 (Bankruptcy),  Title 26 (Tax) [at section depth], Title 26 (Tax) [Capital Gains & Losses] as well as specific pieces of legislation such as the original Health Care Bill – HR 3962.

In the visualization above, Figure (b) combines this hierarchical structure together with a citation network.  We have previously visualized the United States Code citation network and have a working paper entitled Properties of the United States Code Citation Network. Figure (b) is thus a realization of the full United States Code through the section level.

With this representation in place, it is possible to measure the size of the Code using its various structural features such as vertices V and its edges E.  It is possible to measure the full Code at various time snapshots and consider whether the Code is growing or shrinking. Using a limited window of data, we observe growth not only in the size of the code but also its network of dependancies (i.e. its citation network).

Of course, growth in the size United States Code alone is not necessarily analogous to an increase in complexity.  Indeed, while we believe in general the size of the code tends to contribute to “complexity,” some additional measures are needed.  Thus, our paper features structural measurements such as number of sections, section sizes, etc.

In addition, we apply the well known Shannon Entropy measure (borrowed from Information Theory) to evaluate the “complexity” of the message passing / language contained therein.  Shannon Entropy has a long intellectual history and has been used as a measure of complexity by many scholars.  Here is the formula for Shannon entropy:

For those interested in reviewing the full paper, it is forthcoming in Physica A: Statistical Mechanics and its Applications. For those not familiar, Physica A is a journal published by Elsevier and is a popular outlet for Econophysics and Quantitative Finance. A current draft of the paper is available on the SSRN and the physics arXiv

We are currently working on a follow up paper that is longer, more detailed and designed for a general audience.  Even if you have little or no interest in the analysis of the United States Code, we hope principles such as entropy, structure, etc. will prove useful in the measurement of other classes of legal documents including contracts, treaties, administrative regulations, etc.

]]> http://computationallegalstudies.com/2010/08/02/measuring-the-complexity-of-the-law-the-united-states-code/feed/ 0 http://computationallegalstudies.com/2010/04/25/law-as-a-complex-adaptive-system-syllabus-updated-version-04-24-10/ http://computationallegalstudies.com/2010/04/25/law-as-a-complex-adaptive-system-syllabus-updated-version-04-24-10/#comments Sun, 25 Apr 2010 04:02:02 +0000 Daniel Martin Katz http://computationallegalstudies.com/?p=4088 Continue reading →]]>

For the past year, Mike and I have been running an undergraduate independent study course entitled Law as a Complex System? Well, it is the end of the academic year here at Michigan and we thought it would be a good idea to sit down and reflect upon the course content. We have made a few important changes to the syllabus and thought we would share the new version with anyone who might be interested.  We are really quite happy with where it now stands…

This weekend we participated in the Society for Evolutionary Analysis in Law (SEAL) annual meeting at William & Mary Law School. For those not already familiar, SEAL is devoted to the integration of the life sciences and social sciences into legal scholarship and teaching.  Relevant topics include but are not limited to evolutionary and behavioral biology, cognitive science, complex adaptive systems, economics, evolutionary psychology, primatology, etc.  SEAL boasts over 400 members from 30 countries — including the 2009 Economics Nobelist Elinor Ostrom. Anyway, this weekend witnessed a very interesting and exciting set of presentations …. we are looking forward to more great presentations at SEAL 12.

Together with Derek Stafford from the University of Michigan Department of Political Science, Hustle and Flow: A Social Network Analysis of the American Federal Judiciary represents our initial foray into Computational Legal Studies. The full paper contains a number of interesting visualizations where we draw various federal judges together on the basis of their shared law clerks (1995-2004). The screen print above is a zoom very center of the center of the network.  Yellow Nodes represent Supreme Court Justices, Green Nodes represent Circuit Court Justices, Blue Nodes represent District Court Justices.

There exist many high quality formal models of judicial decision making including those considering decisions rendered by judges in judicial hierarchy, whistle blowing, etc. One component which might meaningfully contribute to the extent literature is the rigorous consideration of the social and professional relationships between jurists and the impacts (if any) these relationships impose upon outcomes. Indeed, from a modeling standpoint, we believe the “judicial game” is a game on a graph–one where an individual strategic jurist must take stock of time evolving social topology upon which he or she is operating. Even among judges of equal institutional rank, we observe jurists with widely variant levels social authority (specifically social authority follows a power law distribution).

So what does all of this mean? Take whistle blowing — the power law distribution implies that if the average judge has a whistle, the “super-judges” we identify within the paper could be said to have an air horn. With the goal of enriching positive political theory / formal modeling of the courts, we believe the development of a positive theory of judicial social structure can enrich our understanding of the dynamics of prestige and influence. In addition, we believe, at least in part, “judicial peer effects” can help legal doctrine socially spread across the network. In that vein, here is a view of our operationalization of the social landscape … a wide shot of the broader network visualized using the Kamada-Kawai visualization algorithm:

Here is the current abstract for the paper: Scholars have long asserted that social structure is an important feature of a variety of societal institutions. As part of a larger effort to develop a fully integrated model of judicial decision making, we argue that social structure-operationalized as the professional and social connections between judicial actors-partially directs outcomes in the hierarchical federal judiciary. Since different social structures impose dissimilar consequences upon outputs, the precursor to evaluating the doctrinal consequences that a given social structure imposes is a descriptive effort to characterize its properties. Given the difficulty associated with obtaining appropriate data for federal judges, it is necessary to rely upon a proxy measure to paint a picture of the social landscape. In the aggregate, we believe the flow of law clerks reflects a reasonable proxy for social and professional linkages between jurists. Having collected available information for all federal judicial law clerks employed by an Article III judge during the “natural” Rehnquist Court (1995-2004), we use these roughly 19,000 clerk events to craft a series of network based visualizations.   Using network analysis, our visualizations and subsequent analytics provide insight into the path of peer effects in the federal judiciary. For example, we find the distribution of “degrees” is highly skewed implying the social structure is dictated by a small number of socially prominent actors. Using a variety of centrality measures, we identify these socially prominent jurists. Next, we draw from the extant complexity literature and offer a possible generative process responsible for producing such inequality in social authority. While the complete adjudication of a generative process is beyond the scope of this article, our results contribute to a growing literature documenting the highly-skewed distribution of authority across the common law and its constitutive institutions.

Several months ago, I put together this syllabus for use in a hypothetical seminar course entitled Law as a Complex System. I hoped to revise this reading list / syllabus and have now done so …. While this revised version might contain more content than would be practical for the typical 2-3 credit seminar, I do believe it is far more reasonable than Version 1.0. For anyone who is interested in learning more about the methodological tradition from which much of our scholarship is drawn … feel free to use this as a reading list. If you see any law related scholarship you believe should be included please feel free to email me.

As we mentioned in previous posts, Seadragon is a really cool product. Please note load times may vary depending upon your specific machine configuration as well as the strength of your internet connection. For those not familiar with how to operate it please see below. In our view, the Full Screen is best the way to go ….