Computational Legal Studies™

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 the 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 begins in work such as How Long is the Coastline of the Law? Thoughts on the Fractal Nature of Legal Systems by David G. Post & Michael B. Eisen and continues in more recent work 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.

dmartink Uncategorized classic models, evolution of law, Law as a Complex System

The Forest Fire Model is a commonly invoked example of non-linear system–where a very small perturbation can generate significant differences in observed outcomes. Consider the above Netlogo–to Run the Model: (1) Adjust the Density Slider to set the concentration within the Forest.  (2) Hit the Setup Button (3) Hit the Go Button  …. Rinse and Repeat at different levels of Density.

Above is the output for a run of the model at several levels of Density {48%, 56%, 62%}.  Notice the differences in the Percent Burned {1.6%, 5.2%, 86.5%}.

This is obviously a theoretical model but it has potential application to a wide class of substantive questions including regulatory failure.  In addition, the Forest Fire Model is important because it has been invoked in the critique of the popular book The Tipping Point. Specifically, in discussing the book network scientist Duncan Watts notes ”It sort of sounds cool … But it’s wonderfully persuasive only for as long as you don’t think about it.” Watts notes “…trends are more like forest fires: There are thousands a year, but only a few become roaring monsters. That’s because in those rare situations, the landscape was ripe: sparse rain, dry woods, badly equipped fire departments. If these conditions exist, any old match will do…. and nobody… will go around talking about the exceptional properties of the spark that started the fire.” (Quotes from Jan 2008 Is the Tipping Point Toast? Fast Company Magazine).

dmartink Uncategorized agent based models, classic models, evolution of law

As was stated in Part 1 of this thread, it is by no means a given that the statistical artifact displayed above would appear. Namely, such large scale patterns need not assume this flavor as many social and physical systems feature substantially different properties.

For purpose of generating an empirically grounded theory of American Common Law development … explaining these artifacts would seem to critical. Fortunately, with respect to the above pattern, there exist a definable set of generative processes plausibly responsible for producing what is displayed. While certainly not the only generative process responsible for a power law, the preferential attachment model, first outlined in the physics literature by Barabási & Albert, is among the likely candidates.

Confronting much of the extant literature, query as to whether a closed form equilibria based analytical apparatus (punctuated or otherwise) is up to the task of describing the relevant dynamics? If anything, the distributions displayed above provide first-order evidence of a system which is likely to feature dynamics of a non-linear flavor. Indeed, while significant work still remains, the weight of available evidence indicates Law is a Complex Adaptive System. As such, we believe it would be appropriate to leverage the methods typically reserved for the study of complexity.  For purposes of generating positive legal theory, we believe agent based models, dynamic network analysis and other methods of computational social science offer great potential. We encourage scholars to consider learning more about these approaches.

dmartink Uncategorized computational legal studies, evolution of law, judicial citation network, Judicial Peer Effects, Law as a Complex System

This is the third installment of posts related to our paper Law as a Seamless Web? Comparison of Various Network Representations of the United States Supreme Court Corpus (1791-2005) previous posts can be found (here) and (here). As previewed in the earlier posts, we believe comparing the Union, the Intersect and the Compliment of the SCOTUS semantic and citation networks is at the heart of an empirical evaluation of Law as a Seamless Web …. from the paper….

“Though law is almost certainly a web, questions regarding its interconnectedness remain. Building upon themes of Maitland, Professor Solum has properly raised questions as to whether or not the web of law is “seamless”. By leveraging the tools of computer science and applied graph theory, we believe that an empirical evaluation of this question is at last possible.  In that vein, consider Figure 9, which offers several possible topological locations that might be populated by components of the graphs discussed herein. We believe future research should consider the relevant information contained in the union, intersection, and complement of our citation and semantic networks.

While we leave a detailed substantive interpretation for subsequent work, it is worth broadly considering the information defined in Figure 9.  For example, the intersect (∩) displayed in Figure 9 defines the set of cases that feature both semantic similarity and a direct citation linkage. In general, these are likely communities of well-defined topical domains.  Of greater interest to an empirical evaluation of the law as a seamless web, is likely the magnitude and composition of the Citation Only and Semantic Only subsets.  Subject to future empirical investigation, we believe the Citation Only components of the graph may represent the exact type of concept exportation to and from particular semantic domains that would indeed make the law a seamless web.”

dmartink Uncategorized computational legal studies, computational linguistics, evolution of law, network analysis, supreme court

In our paper Law as a Seamless Web, we offer a first-order method to generate case-to-case and opinionunit-to-opinionunit semantic networks. As constructed in the figure above, nodes represent cases decided between 1791-1865 while edges are drawn when two cases possess a certain threshold of semantic similarity. Except for the definition of edges, the process of constructing the semantic graph is identical to that of the citation graph we offered in the prior post. While computer science/computational linguistics offers a variety of possible semantic similarity measures, we choose to employ a commonly used measure. Here a description from the paper:

“Semantic similarity measures are the focus of significant work in computational linguistics. Given the scope of the dataset, we have chosen a first-order method for calculating similarity.  After lemmatizing the text of the case with WordNet, we store the nouns with the top N frequencies for each case or opinion unit. We define the similarity between two cases or opinion units A and B as the percentage of words that are shared between the top words of A and top words of B.

An edge exists between A and B in the set of edges  if  σ (A,B) exceeds some threshold.  This threshold is the minimum similarity necessary for the graph to represent the presence of a semantic connection.”  

As this a technical paper, it is slanted toward demonstrating proof of methodological concept rather than covering significant substantive ground. With that said, we do offer a hint of our broader substantive goal of detecting the spread of legal concepts between various topical domains. Specifically, with respect to enriching positive political theory, we believe union, intersect and compliment of the semantic and citation networks are really important. More on this point is forthcoming in a subsequent post…

dmartink Uncategorized computational linguistics, evolution of law, judicial citation network, Law as a Complex System, network analysis

Several months ago, I put together this syllabus for use in a future seminar course Law as a Complex System. This contains far more content than would be practical for the typical 2 credit seminar. However, I have decided to repost this because it could also serve as a reading list for anyone who is interested in learning more about the methodological tradition from which must of our scholarship is drawn. If you see any law related scholarship you believe should be included please feel free to email me.

dmartink Uncategorized evolution of law, Law as a Complex System

We have recently posted Law as a Seamless Web? Comparison of Various Network Representations of the United States Supreme Court Corpus (1791-2005) to the SSRN. Given this is the first of several posts about the paper, I will speak broadly and leave details for a subsequent post. From the abstract “As research of judicial citation and semantic networks transitions from a strict focus on the structural characteristics of these networks to the evolutionary dynamics behind their growth, it becomes even more important to develop theoretically coherent and empirically grounded ideas about the nature of edges and nodes. In this paper, we move in this direction on several fronts …. Specifically, nodes represent whole cases or individual ‘opinion units’ within cases. Edges represent either citations or semantic connections.” The table below outlines several possible network representations for the USSC corpus. 

The goal of the paper is to do some technical and conceptual work. It is a small slice of broader project with James Fowler (UCSD) and James Spriggs (WashU). We recently presented findings from the primary project at the Networks in Political Science Conference. The main project is entitled The Development of Community Structure in the Supreme Court’s Network of Citations and we hope to have a version of this paper on the SSRN soon. In the meantime, we plan additional discussion of Law as a Seamless Web in the days to come.  

dmartink Uncategorized computational linguistics, computational social science, evolution of law, judicial citation, Law as a Complex System, network analysis, supreme court

Here is a cool visual for the Martin-Quinn Scores. For those of you not familiar, the Martin-Quinn paper and “MQ Scores” represented a significant breakthrough in the field of judicial politics. On that note, Stephen Jessee & Alexander Tahk have done a nice job both bringing their data up to date and extending their work.  For those interested, click on the visual above and check out all of the relevant links contained within this post.  

dmartink Uncategorized evolution of law, Judicial Decision Making, Public Law, supreme court, visualization

In my conversations with judicial politics scholars, many lament how many of our existing approaches tend to ignore opinion content.  For those interested in embedding opinion content into existing theories of judicial decision making … consider Yonatan Lupu & James Fowler’s paper recently posted to the SSRN.  

The authors present a strategic model of judicial bargaining over opinion content.  They note … “we find that the Court generates opinions that are better grounded in law when more justices write concurring opinions.”  To generate the specification for “grounding in law” the authors use Kleinberg’s Hubs and Authorities Algorithm calculated at the time the opinion was authored. The Strategic Content Paper is available here. 

The visual above is drawn from a related Fowler project located here.  Another very worthwhile paper authored by Fowler, Johnson, Spriggs, Jeon & Wahlbeck is located here.  

dmartink Uncategorized algorithms, evolution of law, Judicial Decision Making, Public Law, supreme court

This is the final installment of posts related to Reproduction of Hierarchy? A Social Network Analysis of the American Law Professoriate. Thanks for your emails.

Here is the plot we provide within the paper.  As a general proposition, we believe this represents an upper bound measure for the intellectual reach of an agenda offered by a given institution.  With respect to our version of the Reed Frost Epidemiological Model, we use the p parameter to model “idea infectiousness.”  When p = 1 every institution “contacted” by the idea is infected with the idea. When p = 0 no institution “contacted” by the idea is infected.  In this version, we use the programming language python to run the model 500 times per institution. The above plot represents an estimate of the “diffusion curve” for each of the 184 institutions in our model. Building off central limit type properties, this leaves a far better estimate of reach than is offered in the single model run from the previous Netlogo GUI.

A cursory review of the above plot demonstrates, we are far from the land of linearity.  Namely, a large number of institutions are able to reach much of the graph with very small changes in the value of p.

In the Structure of Scientific Revolutions, Kuhn quotes from Max Planck:  ”a new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.” Following Planck, we believe retirement is indeed be an important mechanism.  However, we also argue the nature of the p parameter is a relevant consideration.  In fact, unpacking various dimensions of p is the key to the broader model. Specifically, what are the properties of an idea that generate its infectiousness? Of course, we might like to believe infectiousness is related to a class of normatively attractive properties such as promoting efficiency or justice.  However, it is not clear that this follows.

We took no pass on the question of whether some institutions would be better or worse at producing ideas with greater or lesser values of p. The motivated question for this post considers whether, in general, the institutions which are top producers of law professors are (1) leaders in innovation, (2) subsequent ratifiers of a newly established paradigm or (3) defenders of the status quo. In a deep sense, we are asking how to reasonably model decision making by the heterogeneous agents located at such institutions.  Do institutions reward or punish intellectual risk-taking, search, etc.?

While this is an empirical question beyond the scope of this post, it worth asking because it partially informs the micro-dynamics plausibly responsible for generating the spread of new intellectual paradigms.

dmartink Uncategorized agent based models, evolution of law, law schools, python, social epidemiology, Sociology of Law, structure of science