Design Degrees of Freedom and Mechanisms for Complexity

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Design Degrees of Freedom and Mechanisms for
Complexity

• David Reynolds
• J. M. Carlson

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Question to Answer

• The relationship between design and complexity
• Two extreme theoretical points
• Differentiated by their stance regarding the role
  of design.
• Theory of Self-organized criticality (SOC), edge
  of chaos (EOC).
• Complexity emerges in systems that are
  otherwise internally homogeneous and simple.
• Large-scale structure arises naturally and at no
  apparent cost through collective fluctuations in
  systems with generic interactions between
  individual agents.
• Structure is associated with bifurcation points
  and critical phase transitions.
• Highly optimized tolerance (HOT)
• Complexity is associated with intricately
  designed or highly evolved systems.
• Role of robustness to uncertainties in the
  environment as a driving force towards increasing
  complexity in biological evolution and
  engineering design.
• Robustness design is the primary mechanism for
  complexity.
• This paper is based on HOT

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Question to Answer

• To determine how the characteristics of designed
  systems change as the resolution of the design is
  varied
• A measure for design, design degrees of freedom,
  (DDOF)
• Varying the number of DDOFs to interpolate
  between systems with minimal design, and those
  that are highly designed
• Using the model, percolation forest fire model

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Percolation forest fire (PFF) model

• Two-dimensional N-by-N lattice
• Each site is either occupied by a tree or is
  vacant
• Each contiguous set of nearest neighbor occupied
  sites defines a connected cluster (forest)
• The forest is subject to external perturbations,
  represented by sparks
• A spark hits a vacant site on the lattice nothing
  happens a spark hits an occupied site it burns
  all the trees in the connected cluster associated
  with the site.
• Distribution of the spark, P(i, j). With
  probability P(i,j), the spark hits the site
  (i,j).
• In random percolation, the state of the system is
  fully characterized by the density ?.
• Individual sites are independently occupied with
  probability ?
• Properties of the system are determined by
  ensemble averages in which all configurations at
  density ? are taken to be equally likely

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Percolation forest fire model Contd

• Yield Y to be the number of trees remaining after
  a single spark,
• ? the density before the spark
• ?l? average loss due to the fire, computed over
  the distribution of sparks P(i, j) as well as the
  configurations in the ensemble
• Objective optimize yield as a function of the
  tunable parameters, given a distribution of
  sparks P(i, j)
• DDOFs in PFF (number of tunable design
  parameters)
• 1 random percolation the density ?
• N2 specifically choose whether each site
  individually is occupied or vacant.
• DDOFs ? ? as N ? ?
• Intractable as 2(N2) candidate lattices

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• For finite DDOFs, the authors adopt an local
  incremental algorithm for increasing the density.
• local optimization in configuration space
• Sites are occupied one at a time, always choosing
  the next site to occupy in order to maximize
  yield for the incremental change in density
• Plot a yield curve, Y(? ), which has a maximum at
  some ? ?max.
• Beyond this value there is a sharp drop in yield.
• In the thermodynamic limit, ?max approaches unity

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• Between, 1 and N2 , we consider intermediate
  numbers of DDOFs.
• Interpolate between 1 and infinite DDOFs
• The approach
• subdividing the N-by-N lattice into equal square
  cells
• M-by-M lattice, with each cell containing n2
  (N/M)2 sites
• Individual cells are characterized by a density
  ?IJ where (I, J) defines the cell coordinate on
  the M-by-M design lattice.

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NUMERICAL RESULTS

• M 1 single DDOF, optimization of the yield
  leads to criticality ?c
• N finite, M small as N gets large, the
  subregion densities converge to either unit
  density or a density that approaches the critical
  density ?c. ?c - ?
• N??, M small.
• brute force calculation of the globally optimal
  configuration
• design lattice breaking up into compact domains
  of unit density, separated by uncrossable
  barriers of density, ?c - ?
• N ??, M large
• cellular patterns similar to HOT patterns
• Spark distribution, exponential

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Criticality- the optimal solution for a single
DDOF

• M 1.
• The state of the system is characterized by the
  density ?
• The choice of P(x,y) is is irrelevant for large
  N.
• The model exhibits a continuous phase transition
  at density
• ?c ? 0.592
• In the limit N?? ,
• for ? lt ?c, no infinite cluster
• for ? lt ?c , an infinite cluster exists
  somewhere on the lattice with probability one
• for ? ?c , the probability of an infinite
  cluster lies between zero and unity and depends
  on the shape of the lattice.
• For a square-shaped, ½
• .

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Criticality Contd

• At low densities, lattice sparse, at low
  densities there is on average zero macroscopic
  loss associated with a fire ignited by a single
  spark.
• At densities greater than or equal to the
  critical density, there is an infinite cluster
  and the probability that any given site is on the
  infinite cluster defines the percolation
  probability P?(?),
• At ?c, P?(?) 0, the infinite cluster (if
  exists) is fractal

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Explicit optimization on finite lattices with few
DDOFs

• M gt 1, but small.
• Determine the optimal yield configuration as a
  function of the M2 cell densities
• Fix M and increase N.
• Something interesting
• each of the densities ? IJ
• either converges rapidly to
• unity or more gradually
• towards ?c
• M 2, N 64.
• Yield Y(?11, ?12, ?21, ?22)

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• ?11 depends on the system size.

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Sample solutions for different distributions
ofsparks
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Global optimization for an infinite underlying
lattice andfew DDOFs

• N infinite, M small
• Cells density is either near critical or unit
• Calculate the yield.

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• Propagation
• Cells at ?c - ? experience no macroscopic loss
  in density in a fire, and fires do not propagate
  macroscopic distances across the cell.
• Fires do not propagate from left to right or from
  top to bottom across cells at density ?c - ? .
  The ?c - ? cells thus act effectively as fire
  breaks for vertical and horizontal propagation.
• Fires will propagate between adjacent edges of
  cells with density ?c - ?
• This implies a corner connection between cells
  at unity density is effectively the same as a
  shared edge.
• Cells at unit density experience total loss when
  a spark hits the cell
• cell, or when fires propagate into the cell from
  nearest (edge connected) or next nearest (corner
  connected) neighbor cells at unit density

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Local optimization for an infinite underlying
lattice andmany DDOFs
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Summary

• Degree of Design and Complexity?
• Higher Degree of Design -gt More complex system,
  specialized but fragile to rare events.

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Mutation, specialization, and hypersensitivity
inhighly optimized tolerance

• Tong Zhou, J. M. Carlson, and John Doyle

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• Using the Percolation forest fire (PFF) model to
  argue with self-organizes to a critical point
  (SOC) and edge of chaos (EOC)
• A key signature of a HOT system is that it is
  simultaneously
• robust, yet fragile simultaneously

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• Distribution of Spark exponential
• Genotype layout of the lattice
• Phenotype fn, ln
• ln event sizes
• fn the corresponding probabilities
• Two types to evolve.
• Tortoises Y? ? - ?l?,
• ?l? average loss
• Hares Y1 ? - ln
• ln stochastic loss
• evolve rapidly, over-specialize for common
  disturbances, win in the short term, but are
  vulnerable to extinction in rare events

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The experiment

• begin with 1,000 randomly generated lattices with
  N 16 divided equally between the tortoise and
  hare types.
• Each subsequent generation consists of two
  offspring from each
• parent lattice, each with a finite probability
  of mutation.
• Each resulting generation of offspring is then
  subject to natural selection based on fitness Y.
• An upper bound of 1,000 on the total population S
• Competition based on fitness occurs among all of
  the lattices for this limited space in the
  community.
• Any lattice with Y lt Yd is considered dead
  and automatically discarded, along with the
  lowest performers until the total size S of the
  community is S lt1,000.

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Observations of experiment

• Even at the relatively early stages of evolution
  the tortoises and hares begin to develop barrier
  patterns that are characteristic of their
  respective types.
• The tortoises sacrifice density to retain a
  higher concentration of barriers in regions that
  have less frequent sparks.
• If the tortoises win this initial competition,
  they persist forever at the maximum population of
  1,000.
• They are optimized for long-term
• If the hares survive, evolve more rapidly and to
  higher densities
• they do not sacrifice density to protect against
  rare events

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The second experiment

• Including niches
• the niches retain the top 50 of each type

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Observations of the Second Experiment

• For long periods when no rare events happen, the
  hares have higher fitness and thus dominate.
  Nearly all of the available 1,000 spaces belong
  to the hares, while the tortoise population is
  sustained at the minimum of 50.
• -The hares dominate!
• Rare event occurs,
• The hares quickly. Near extinction.
• Evolution cycle

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Interpretations

• Microevolutionary Features
• how intrinsic robust design tradeoffs interact
  with and constrain natural selection to generate
  highly ordered structure from randomness.
• creating habitats with different spark
  distributions P(i, j)
• Lattices in a habitat with skewed distributions
  become specialists and evolve like the hares,
• Lattices in a habitat with a spatially uniform
  distribution of sparks become generalists by
  developing a roughly uniform grid of barriers,
  which is optimal in their habitat.
• never extinction but suboptimal with skewed
  distributions
• Even though different genotypes, different
  layouts but essentially same phyenotypes (the
  pattern of the barriers)
• Macroevolutionary Consequences
• Initiation of large extinction events is
  typically associated with rare or anomalous
  external causes, such as meteoroid impacts and
  large-scale geological change, whereas more
  frequent, smaller events are typically associated
  with a mixture of competition between species, as
  well as more commonplace variability in the
  habitat and other environmental conditions