icpsr - complex systems models in the social sciences - lecture 8 - professor daniel martin katz

Daniel Martin KatzMichigan State University - College of Law

Complex Systems Models in the Social Sciences

(Lecture 8)

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How Complex is a Particular System?

This is an Important Question Considered

in the Field

For an OverviewSee Chapter 7

Lots of Potential Candidate Measures

http://web.mit.edu/esd.83/www/notebook/Complexity.PDF

“1. How hard is it to describe?

2. How hard is it to create?

3. What is its degree of organization”

Questions Typically Considered:

Entropy

Algorithmic Complexity

Minimum Description Length

Fisher Information

Renyi Entropy

Chernoff Information

Dimension

Fractal Dimension

Lempel--Ziv Complexity

Code Length (prefix-free, Huffman, Shannon-Fano, error-correcting, Hamming)

1. Difficulty of Description(Typically measured in bits)

Computational Complexity

Time Computational Complexity

Space Computational Complexity

Information--Based Complexity

Logical Depth

Thermodynamic Depth

Crypticity

2. Difficulty of Creation(Typically measured in time, energy, dollars, etc.)

Metric EntropyFractal DimensionExcess EntropyStochastic ComplexitySophisticationEffective Measure ComplexityTrue Measure ComplexityTopological epsilon-machine sizeConditional InformationConditional Algorithmic Information ContentSchema lengthIdeal ComplexityHierarchical ComplexityTree-subgraph diversityHomogeneous ComplexityGrammatical Complexity

3. Degree of organization(a) Effective Complexity

Algorithmic Mutual Information

Channel Capacity

Correlation

Stored Information

Organization

3. Degree of organization(B) Mutual Information

http://vserver1.cscs.lsa.umich.edu/~crshalizi/notabene/complexity-measures.html

Measuring Complexity:An Applied Case

Measuring the Complexity

of Legal Systems

Optimal Precision in Legal RulesKaplow (1995) Tullock (1995)

The Complexity of the Law is Canonical Question

Applied Scholarship -- Tax, Environmental Law, Admin Law, etc.

Legal and Political TheoryJustifications for Law and the State

Aligns Incentives / Channels Behavior

Offers Focal Points / Coordination Mechanisms

Encourages Actors to Internalize Costs

Maintains Monopoly on Legitimated Violence

Protects Individual Rights and Liberties

Law as a Means of Solving Social Dilemmas

Complexity of Society & Complexity of the Law

Economic Exchange

What conditions must be met for law to Achieve these Ends?

Legal Rules Should Reflect The Nature of:

Social Interaction

Political Behavior

Applying the Tools of the ‘Big Data’ Era

A Perspective on the Scope of Law in a Modern Society

Using an Important Corpus of Written Law

This Paper is an Effort (Albeit Imperfect)to Measure the Complexity of the Law

That is Large and Cross Cutting

This is the United States Code

Compiled Version of Federal Statutory Law

This is the United States Code

Drawn from the U.S. Statutes at Large

Does not Include Fed Admin Regulations

The 50 titles in US Code

How Complex is the Code and its Components?

How has its Composition Changed Over Time?

Some Potential Questions

Can we understand if those changes Scale to changes in the complexity of broader Society?

How Large is the United States Code?

How Large is the United States Code?

How Large is the United States Code?

How Large is the United States Code?

How Large is the United States Code?

~Title 29 - Labor

How Large is the United States Code?

How Large is the United States Code?

This is the United States Code

The Case for a Computational Approach

The United States Code is Large ...

Computational Methods are Arguably Required

Need Tools and Methods that Scale to the Size and Scope of this Body of Information

Computational Approach to the Measurement

of Complexity

(2) Generate a Measurement Strategy for that Object

(1) Provide a Mathematical Representation for an Object

A Mathematical Approach to the Study of the United States Code

389 Physica A 4195 (2010 Forthcoming)

Additional Treatment Available Here

The US Code as a Mathematical Object

Hierarchical Structure

Title 26

Subtitle A

Chapter 1

Subchapter F

Part I

Section 501

Subsection (c)

Paragraph (3)

Citation Network

Example: Tax Evasion

Title 26 - Tax

Might Cite

Title 18 - Crimes &

Criminal Procedure

The US Code as a Mathematical Object

http://computationallegalstudies.com/2009/09/14/the-structure-of-the-united-

states-code/

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Hierarchical Structure

CitationNetwork

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Hierarchical Structure

CitationNetwork

Linguistic Content

United States CodeFeatures

13 Million tokens

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Measuring the Complexity of the United States Code

Knowledge Acquisition Framework

Knowledge Acquisition

Subfield at Intersection of Computer Science and Psychology

Interested in protocols used bysubjects as they acquire, store and analyze information

Interested in Mirroring the “Protocols”Used by a Hypothetical End User

Costs of Executing the Protocols Are Driven by the Complexity of the Object

We Consider an Individual Engaging in a Knowledge Acquisition Process

Measuring the Complexity of the United States Code

Protocol for United States Code Knowledge Acquisition

(1) Select an initial element of the Code corresponding to a concept of interest

(2) Beginning from this initial element, recursively assimilate the content of all sub-elements

(3) When a citation is encountered, apply this protocol recursively to the cited element

The Complexity of Knowledge Acquisition

3 Factors Influence Complexity ofExecuting the Protocol

Structure Language Dependence

Toward a Composite Measure

Multidimensional object--> Composite Measure Full Code and Individual Titles

Weighted RanksImperfect but a good start

A Positive Feature of weighted ranks (1) Change Weights(2) Add Additional Dimensions

Two Forms of Weighted Ranks

Unnormalized Measure

Considers the Complexity of reviewing a full title

Normalized Measure

Considers the Average or Emblematic Provision within the Title

Thus, we control for title size

The Unnormalized Measure

Structure Language Dependence

Unnormalized Measure

The Unnormalized Measure:Structure

Structure

Vertices(Nodes)

Structure

The Unnormalized Measure:Structure

Vertices(Nodes)

Structure

The Unnormalized Measure:Structure

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Vertices(Nodes)

Net Flow

Dependence

The Unnormalized Measure:Net Flow

Net Flow

Dependence

The Unnormalized Measure:Net Flow

The Unnormalized Measure: Language

Language

Entropy

The Unnormalized Measure: Language

Language

Entropy Uniform Random

The Unnormalized Measure: Language

Language

Entropy

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The Unnormalized Measure: Language

Language

Entropy Title 42 - Public Health & Welfare

LeprosySocial Security National Flood Insurance US Synthetic Fuels Corp Intl Child Abduction Remedies

Entropy ≈ Diversity

The Unnormalized Measure

Structure Language Dependence

Unnormalized Measure

Vertices EntropyNet Flow

The Unnormalized Measure

The Normalized Measure

Structure Language Dependence

Normalized Measure

Mean Depth Entropy

Net FlowPer

Section

Tokens Per

Section

Language

The Unnormalized Measure

Offered a framework designed to Measure the Complexity of the Law

Designed to open a dialogue regarding Legal complexity and its measurement

Demonstrated measurement principles that are potentially applicable to other classes of legal documents

Summary

Highlighted what computation might be able to offer to empirical legal studies

SupportingMaterials Available Online!

Thanks for Support