---
created_at: '2017-04-15T04:57:43.000Z'
title: 'Measures of Complexity: A non-exhaustive list (2001) [pdf]'
url: http://web.mit.edu/esd.83/www/notebook/Complexity.PDF
author: breck
points: 92
story_text: 
comment_text: 
num_comments: 7
story_id: 
story_title: 
story_url: 
parent_id: 
created_at_i: 1492232263
_tags:
- story
- author_breck
- story_14119465
objectID: '14119465'

---
[Source](http://web.mit.edu/esd.83/www/notebook/Complexity.PDF "Permalink to ")

%PDF-1.2 % 10 0 obj << /Length 11 0 R >> stream BT 222.75 705.75 TD 0 0 0 rg /F0 15.75 Tf 0.164 Tc 0.0235 Tw (Measures of Complexity) Tj 11.25 -24 TD 0.0872 Tc -0.2747 Tw (a non--exhaustive list) Tj 42 -21.75 TD /F0 12.75 Tf 0.0308 Tc 0.5317 Tw (Seth Lloyd) Tj -123 -20.25 TD /F1 12 Tf -0.042 Tc 0.417 Tw (d'Arbeloff Laboratory for Information Systems and Technology) Tj 58.5 -20.25 TD -0.0162 Tc 0.0162 Tw (Department of Mechanical Engineering) Tj 2.25 -19.5 TD -0.0362 Tc 0.5362 Tw (Massachusetts Institute of Technology) Tj 54 -20.25 TD 0.0373 Tc 0 Tw (slloyd@mit.edu) Tj -177.75 -19.5 TD -0.0884 Tc 0.4634 Tw ( The world has grown more complex recently, and the number of ways of measuring) Tj 0 -13.5 TD -0.066 Tc 0.4698 Tw (complexity has grown even faster. This multiplication of measures has been taken by) Tj 0 -14.25 TD -0.0559 Tc 0.3372 Tw (some to indicate confusion in the field of complex systems. In fact, the many measures) Tj 0 -13.5 TD -0.0551 Tc 0.4013 Tw (of complexity represent variations on a few underlying themes. Here is an (incomplete)) Tj 0 -14.25 TD -0.0688 Tc 0.4855 Tw (list of measures of complexity grouped into the corresponding themes.) Tj 0 -19.5 TD -0.0461 Tc 0.2768 Tw (An historical analog to the problem of measuring complexity is the problem of describing) Tj 0 -13.5 TD -0.0157 Tc 0.0907 Tw (electromagnetism before Maxwell's equations. In the case of electromagnetism,) Tj 0 -14.25 TD -0.0272 Tc 0.2147 Tw (quantities such as electric and magnetic forces that arose in different experimental) Tj 0 -13.5 TD -0.0364 Tc 0.241 Tw (contexts were originally regarded as fundamentally different. Eventually it became) Tj 0 -14.25 TD -0.0808 Tc 0.4557 Tw (clear that electricity and magnetism were in fact closely related aspects of the same) Tj 0 -13.5 TD -0.0317 Tc 0.2567 Tw (fundamental quantity, the electromagnetic field. Similarly, contemporary researchers in) Tj T* -0.0437 Tc 0.377 Tw (architecture, biology, computer science, dynamical systems, engineering, finance, game) Tj 0 -14.25 TD -0.0759 Tc 0.3973 Tw (theory, etc., have defined different measures of complexity for each field. Because) Tj 0 -13.5 TD -0.0338 Tc 0.2213 Tw (these researchers were asking the same questions about the complexity of their different) Tj 0 -14.25 TD -0.0973 Tc 0.4973 Tw (subjects of research, however, the answers that they came up with for how to measure) Tj 0 -13.5 TD -0.0249 Tc 0.0249 Tw (complexity bear a considerable similarity to ) Tj 213 0 TD -0.0366 Tc 0.3366 Tw (eachother. Three questions that researchers) Tj -213 -13.5 TD -0.0651 Tc 0.3378 Tw (frequently ask to quantify the complexity of the thing (house, bacterium, problem,) Tj 0 -14.25 TD -0.0624 Tc 0.3124 Tw (process, investment scheme) under study are) Tj 0 -19.5 TD -0.1553 Tc 0.6553 Tw (1. How hard is it to describe?) Tj 0 -20.25 TD -0.1653 Tc 0.6653 Tw (2. How hard is it to create?) Tj 0 -19.5 TD -0.0587 Tc 0.3087 Tw (3. What is its degree of organization?) Tj T* -0.0679 Tc 0.4679 Tw (Here is a list of some measures of complexity grouped according to the question that they) Tj 0 -14.25 TD -0.0304 Tc 0.1668 Tw (try to answer. Measures within a group are typically closely related quantities.) Tj 0 -39.75 TD /F0 12 Tf 0.1003 Tc -0.3503 Tw (1. Difficulty of description) Tj 135 0 TD /F1 12 Tf -0.1262 Tc 0.4262 Tw (. Typically measured in bits.) Tj -135 -19.5 TD -0.055 Tc 0 Tw (Information;) Tj 0 -19.5 TD -0.1875 Tc (Entropy;) Tj 0 -20.25 TD -0.041 Tc 0.341 Tw (Algorithmic Complexity or Algorithmic Information Content;) Tj 0 -19.5 TD -0.0734 Tc 0.8234 Tw (Minimum Description Length;) Tj 0 -20.25 TD -0.0367 Tc 0.0367 Tw (Fisher Information;) Tj ET endstream endobj 11 0 obj 3758 endobj 4 0 obj << /Type /Page /Parent 5 0 R /Resources << /Font << /F0 6 0 R /F1 8 0 R >> /ProcSet 2 0 R >> /Contents 10 0 R >> endobj 13 0 obj << /Length 14 0 R >> stream BT 90 709.5 TD 0 0 0 rg /F1 12 Tf -0.186 Tc 0.936 Tw (Renyi Entropy;) Tj 0 -20.25 TD -0.0369 Tc 0.0369 Tw (Code Length (prefix-free, Huffman, Shannon-) Tj 222 0 TD -0.0255 Tc 0.4005 Tw (Fano, error-correcting, Hamming);) Tj -222 -19.5 TD -0.024 Tc 0.024 Tw (Chernoff Information;) Tj 0 -20.25 TD -0.0504 Tc 0 Tw (Dimension;) Tj 0 -19.5 TD -0.1534 Tc -0.5966 Tw (Fractal ) Tj 36 0 TD -0.1254 Tc 0 Tw (Dimension;) Tj -36 -19.5 TD -0.0503 Tc (Lempel--) Tj 44.25 0 TD -0.096 Tc 0.096 Tw (Ziv Complexity.) Tj -44.25 -39.75 TD /F0 12 Tf 0.0957 Tc -0.5957 Tw (2. Difficulty of creation) Tj 119.25 0 TD /F1 12 Tf -0.073 Tc 0.3542 Tw (. Typically measured in time, energy, dollars, etc.) Tj -119.25 -20.25 TD -0.0646 Tc -0.6854 Tw (Computational ) Tj 73.5 0 TD -0.0693 Tc 0 Tw (Complexity;) Tj -73.5 -19.5 TD -0.0512 Tc 0.0512 Tw (Time Computational Complexity;) Tj 0 -19.5 TD -0.0606 Tc 0.0606 Tw (Space Computational Complexity;) Tj 0 -20.25 TD -0.0571 Tc 0.0571 Tw (Information--Based Complexity;) Tj 0 -19.5 TD -0.1403 Tc 0.1403 Tw (Logical Depth;) Tj 0 -20.25 TD -0.0913 Tc 0.0913 Tw (Thermodynamic Depth;) Tj 0 -19.5 TD -0.1188 Tc 0 Tw (Cost;) Tj T* -0.1293 Tc (Crypticity.) Tj 0 -20.25 TD /F0 12 Tf 0.1135 Tc -0.4885 Tw (3. Degree of organization. ) Tj 135.75 0 TD /F1 12 Tf -0.053 Tc 0.2576 Tw (This may be divided up into two quantities: a) Difficulty of) Tj -135.75 -13.5 TD -0.0227 Tc 0.1165 Tw (describing organizational structure, whether corporate, chemical, cellular, etc.; b)) Tj 0 -14.25 TD -0.0434 Tc 0.2041 Tw (Amount of information shared between the parts of a system as the result of this) Tj 0 -13.5 TD -0.0163 Tc 0.0163 Tw (organizational structure.) Tj 0 -19.5 TD /F2 12 Tf 0.0646 Tc -0.0646 Tw (a) Effective Complexity) Tj 0 -20.25 TD /F1 12 Tf -0.0865 Tc 0.2911 Tw (Metric Entropy; Fractal Dimension; Excess Entropy;) Tj 0 -19.5 TD -0.0283 Tc 0.0283 Tw (Stochastic Complexity;) Tj 0 -20.25 TD -0.1008 Tc 0 Tw (Sophistication;) Tj 0 -19.5 TD -0.0207 Tc 0.0207 Tw (Effective Measure Complexity;) Tj T* -0.0788 Tc 0.0788 Tw (True Measure Complexity;) Tj 0 -20.25 TD -0.0343 Tc 0.0343 Tw (Topological epsilon-machine size;) Tj 0 -19.5 TD -0.0387 Tc 0.0387 Tw (Conditional ) Tj 59.25 0 TD 0.0075 Tc 0 Tw (Information;) Tj -59.25 -20.25 TD -0.0246 Tc 0.0246 Tw (Conditional Algorithmic Information Content;) Tj 0 -19.5 TD -0.082 Tc 0.082 Tw (Schema ) Tj 41.25 0 TD -0.2623 Tc 0 Tw (length;) Tj -41.25 -19.5 TD -0.0938 Tc 0.0938 Tw (Ideal Complexity;) Tj 0 -20.25 TD -0.0681 Tc 0.0681 Tw (Hierarchical Complexity;) Tj 0 -19.5 TD -0.246 Tc 0.246 Tw (Tree ) Tj 24.75 0 TD -0.0877 Tc 0.0877 Tw (subgraph diversity;) Tj ET endstream endobj 14 0 obj 2770 endobj 12 0 obj << /Type /Page /Parent 5 0 R /Resources << /Font << /F0 6 0 R /F1 8 0 R /F2 15 0 R >> /ProcSet 2 0 R >> /Contents 13 0 R >> endobj 18 0 obj << /Length 19 0 R >> stream BT 90 709.5 TD 0 0 0 rg /F1 12 Tf -0.0976 Tc 0.0976 Tw (Homogeneous ) Tj 71.25 0 TD 0.0671 Tc 0 Tw (Complexity;) Tj -71.25 -20.25 TD -0.0717 Tc 0.0717 Tw (Grammatical Complexity.) Tj 0 -19.5 TD /F2 12 Tf -0.0123 Tc 0.0123 Tw (b) Mutual Information:) Tj 0 -20.25 TD /F1 12 Tf -0.0861 Tc 0.4611 Tw (Algorithmic Mutual Information;) Tj 0 -19.5 TD -0.0308 Tc 0.0308 Tw (Channel Capacity;) Tj T* -0.1455 Tc 0 Tw (Correlation;) Tj 0 -20.25 TD -0.097 Tc 0.097 Tw (Stored ) Tj 33.75 0 TD 0.0075 Tc 0 Tw (Information;) Tj -33.75 -19.5 TD -0.0886 Tc (Organization.) Tj 0 -39.75 TD -0.0619 Tc 0.2962 Tw ( In addition to the above measures, there are a number of related concepts that are not) Tj 0 -13.5 TD -0.0301 Tc 0.1455 Tw (quantitative measures of complexity per se, but that are closely related. Such concepts) Tj 0 -14.25 TD -0.1183 Tc 0 Tw (include) Tj 0 -19.5 TD -0.0473 Tc 0.0473 Tw (Long--Range Order;) Tj 0 -20.25 TD -0.0954 Tc 0 Tw (Self--Organization;) Tj 0 -19.5 TD -0.0365 Tc 0.0365 Tw (Complex Adaptive Systems;) Tj T* -0.0755 Tc 0.0755 Tw (Edge of Chaos.) Tj 0 -20.25 TD -0.135 Tc 0.6037 Tw (Please feel free to send me additions to this list, whether or not they fall in the) Tj 0 -13.5 TD -0.0863 Tc 0.8363 Tw (classification scheme.) 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