THE COMPLEXITY OF MODELLING COMPLEXITY

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THE COMPLEXITY OF MODELLING COMPLEXITY
Fabio BoschettiCSIRO, Australia

• Complexity of the model
• Complexity of reconstructing the model
• Complexity of studying a complex system by
  modelling

The complexity of studying a complex system by
numerical modelling
Warning.. Work in progress Ideas still need
sharpening
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Outline

• What do Complex Systems do?
• Some information-theoretic place-holders
• How does this relate to our difficulty in
  understanding and modelling Complex Systems?
• Forward versus inverse problem
• Understanding a complex system is an inverse
  problem
• Complexity of the inverse problem

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a Complex System story

• many, but not too many, components interact in a
  non trivial fashion
• the system is open (receives energy/information/m
  atter from the environment)
• interactions ? symmetry breaking ? coordinated
  behaviour arises
• no central director/template ? the system
  self-organises
• coordination as patterns detectable by an
  external observer or structures convey new
  properties to the systems itself
• new behaviours emerge from the system
• coordination and emergence may arise from
  response to environment ? adaptation
• when adaptation occurs across generations at a
  population level we say that the system evolved
• now, at new scale, the system can be identified
  as a novel unit
• this becomes the building block ? new cycle at a
  new scale

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Info-theoretic framework

• A framework in which concepts like
• complexity,
• emergence,
• self-organisation,
• adaptation and evolution
• can be
• described,
• distinguished
• defined consistently

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Info-theoretic framework

• We chose an information-theoretic framework
• we borrow from work pioneered by the Santa Fe
  Institute
• it provides a well developed theory
• definitions can be formulated mathematically
• some computational tools are readily available

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Complexity

• Kolmogorovs algorithmic complexity size of the
  shortest program able to reconstruct a process
• Statistical Complexity minimum memory for
  optimal predictions

Complexity minimum number of rules needed to
capture the process minimum number of
statistically significant process states
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Complexity as difficulty to predict
The more complex the environment is, the more
rules an agent needs to use to predict/deal with
it. Higher complexity ? higher survival
probability ? selective advantage Self-organisati
on ? more complex behaviour ? selective
advantage Emergence ? simpler prediction ?
selective advantage Adaptation ? easeri
response ? selective
advantage Agents ? humans / scientists /
engineers
Key word Prediction Cause ? Effect
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complexity minimum number of rules needed to
capture the process minimum number of
statistically significant process states Does
this capture our intuition of the challenges
faced by a scientist in addressing a complex
problem?

• Typical CSS models like CAs (Game of life, SOC),
  ABMsGAs, Chaotic systems, are short ? they are
  simple
• A computer model carries out a mechanical
  process
• This does not account for the difficulty in
  reconstructingthe model

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Forward vs inverse modelling

• Forward modelling
• Take a model F of a physical process.
• Given a certain input i, F determines an output
  o ? o F i
• We can control the output o, by acting on i, ?
  operative definition of causality
• It represents our understanding of cause-effect
  relations.
• Since F respects our perception of the arrow of
  time (cause ? effect), it is usually called a
  forward model

• Examples
• Will a plane fly with this kind of wing?
• Will this building sustain a scale 5 earthquake?
• Does this disease display this symptom?

Prediction - What is the effect of this cause?
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Forward vs inverse modelling

• Real world questions
• Will a plane fly with this kind of wing?
• Will this building sustain a Richter scale 5
  earthquake?
• Does this disease display this symptom?

• What kind of wing does this plane need?
• How should I build in order to sustain a scale 5
  earthquake?
• What disease display this symptom?
• What is the cause of this effect?

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Forward vs inverse modelling
What is the cause of this effect?

• This implies a hypothetical model I which, given
  an effect o as input, gives a possible cause as
  an output ? i I o ).
• Because I reverses the arrow of time, it is
  usually called an inverse model (effect ? cause),

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Forward vs inverse modelling
Unfortunately, inverse models can be written
explicitly for only a very small set of forward
models. This is true not only for closed-form
models but also for purely numerical models.
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Forward vs inverse modelling
10² ? 100 10.5² ? 110.25 ? ? v115 10.75² ?
115.56 11² ? 121
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Forward vs inverse modelling
Model
Output
Forward Model
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Consider a typical Complex System Model CSM
global behaviour arising from a natural process/
local rules.
5. Judge whether the output matches the
expectations (possibly a result of the local
rules which he/she judges to be realistic or
interesting)
7. if something unexpected in the output sparks
new insights, change rules in CSM go back to 2
Modelling a complex problem inverse problem
1. Write CSM
2. Choose input i
3. Run CSM ? generate oCSM i (an image, an
animation, numerical data, etc)
4. analyse the output, often visually and
subjectively (sometimes numerically)
6. if the outcome is not satisfactory, change
input I go back to 2
CSM
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Forward vs inverse modelling

• We can summarise our discussion so far as
  follows
• information-theoretic measures are suitable to
  study model complexity
• However, they do not match our perception of what
  makes a natural processes complex to understand
• understanding a natural process is an inverse
  process
• this inverse process can not be carried out by an
  algorithm, rather it requires the modeller
  intervention
• Emphasis moves from
• the complexity of a model/process ? complexity of
  reconstructing the model

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What makes a modelling exercise complex?
the complexity of a modelling exercise is
represented by the number of levels at which
questions are asked and answered or number of
levels at which the modeller needs to
intervene within an inverse problem
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v115?
Complexity level 1
115
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Optimal scheduling
Complexity level 2
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Generate emergent patterns
Complexity level 3
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Ecosystem behaviour
Complexity level 4
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What makes a modelling exercise complex?
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De-complexification, understanding and cultural
development
v115 ?
Understanding decrease in complexity level?
Press a button
Time
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Summary

• There is a difference between the complexity of a
  process and the complexity we face in studying a
  complex process
• By using information theoretical framework we can
  define concepts like process complexity,
  self-organisation, emergence and adaptation in a
  consistent fashion Forward modelling
• Levels of analysis can be used to study the
  complexity of the human understanding of such
  process Inverse modelling
• Could this provide a setting for the study of
  cultural development and the way knowledge is
  transmitted ?
• Warning.. Work in progress . Ideas still need
  sharpening

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For more informationfabio.boschetti_at_csiro.au
http//www.per.marine.csiro.au/staff/Fabio.Boschet
ti/