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A measurable definition of Emergence in quantitative systems

This article presents a measurable definition of emergence in quantitative systems using the concept of persistent mutual information (PMI). PMI captures the non-trivial behavior that persists over time in a system. It discusses the limitations and generalizations of this measurement approach.

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A measurable definition of Emergence in quantitative systems

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  1. A measurable definition of Emergence in quantitative systems Input ideas: Shannon: Information -> Entropy transmission -> Mutual Information Crutchfield: Complexity <-> Information MacKay: Emergence = system evolves to non-unique state Emergence measure: Persistent Mutual Information across time.

  2. B B B A A A Entropy & Mutual Information Shannon 1948

  3. time Entropy density (rate) Shannon ? Excess Entropy Crutchfield & Packard 1982 space Statistical Complexity Shalizi et al PRL 2004 MI-based Measures of Complexity A B Persistent Mutual Information - candidate measure of Emergence

  4. Measurement of Persistent MI • Measurement of I itself requires converting the data to a string of discrete symbols (e.g. bits) • above seems the safer order of limits, and computationally practical • The outer limit may need more careful definition

  5. Examples with PMI • Oscillation (persistent phase) • Spontaneous ordering (magnets) • Ergodicity breaking (spin glasses) – pattern is random but aspects become frozen in over time Cases without with PMI • Reproducible steady state • Chaotic dynamics

  6. 0 PMI = 0 log 2 log 4 log 2 log 8 log 4 log 3 Logistic map

  7. A definition of Emergence • System self-organises into a non-trivial behaviour; • there are different possible instances of that behaviour; • the choice is unpredictable but • it persists over time (or other extensive coordinate). • Quantified by PMI = entropy of choice • Shortcomings • Assumes system/experiment conceptually repeatable • Measuring MI requires deep sampling • Appropriate mathematical limits need careful construction • Generalisations • Admit PMI as function of timescale probed • Other extensive coordinates could play the role of time

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