Exploring Diversity in Evolutionary Systems Using Graph-based Genealogy

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Exploring Diversity in Evolutionary Systems Using
Graph-based Genealogy

• Chris Salzberg Antony Antony
• Hiroki Sayama

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Credits
This project is part of Thesis work leading to
the Master of Science degree in Computational
Science at the University of Amsterdam. Research
is supervised by Dr. Hiroki Sayama University of
Electro-Communications, Tokyo, Japan. Project
to be completed by the end of this year.
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Lecture Plan

• Context and statement of purpose
• Assumptions and definitions
• Graph-based genealogy
• Evolution as a flow in graph-space
• An example exploring evolutionary dynamics of
  the evoloop
• Summary and future goals

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Context

• Very little visualization research in Artificial
  Life / Evolutionary Systems.
• Research in visualization is mostly informal,
  context-specific.
• Few general methods of analysis exist for
  artificial evolutionary systems.

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Statement of Purpose

• Goal
• To understand complex evolutionary dynamics in
  systems of self-replicators.
• Method
• Devise an abstract framework for analysis.
• Visualize evolutionary dynamics of evolutionary
  systems within this framework.

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Assumptions

• Evolving system of self-replicators.
• Arbitrary but well-defined domain.
• Mutation-based evolution
• Birth is attributed to a single parent.
• Genetic crossover ruled out.
• Template reproduction, modular structure.

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Definitions
C
r

• Configuration space (C)
• Arbitrary well-defined domain
• Cellular automata space (SR Loop/Evoloop)
• Memory and CPU of a computer (Tierra)
• Continuous virtual liquid (JohnnyVon)

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Definitions
Q
q r-b-y-b

• Identifier space (Q)
• Sequence of digits from arbitrary alphabet
• Gene sequence
• Program code
• Unique and unambiguous.
• Controls template reproduction.

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Definitions
qk
qj
qi

• Species
• Self-replicators with the same identifier.
• Assign arbitrary but unique index to each.
• Species with index k we call qk.

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Definitions
qp r-b-r-y

• Birth
• Defined by parent species qp and child species
  qc.
• Mutations if qp ? qc .
• Death
• Defined by one species (qd).
• Birth and death specified in time and space.

qc r-b-y-y
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Genealogy as a Tree
qp y-r-b-y-y-r-r-r
C
T
qp y-r-b-y-r-r-b-r

• More possible species than individuals.
• Each individual is unique.
• Replication is never exact.
• Each species has only one instance and only one
  ancestor (parent species).

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Genealogy as a Graph
i
qj
qi
qk
j
k
C
G

• To each species qk in C we associate a node k in
  a directed graph G.
• Edges represent ancestral links.
• Multiple instances with the same identifier
  (species) mapped to the same node.

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Graph-space Transitions
t t1
t t2
t t3
i
i
j
j
k
k

• Define the traversal frequency F(k ? l, t)
• of link traversals from node k to node l at
  time t.
• Define the frequency of death D(k, t)
• of deaths at node k at time t.

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Derived Parameters

• Specify a window in time T (ti, tf) .
• Derive
• I(k, T) ? ? F(l ? k, t)
• O(k, T) ? ? F(k ? l, t)
• S(k, T) ? F(k ? k, t)

tf
tf
tf
tti
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Population Dynamics

• Population P(k, tf) of species qk at time t can
  be derived
• P(k, tf) P(k, ti) I(k, T) S(k, T) - D(k, T)
• Evolutionary dynamics described at the level of
  connectivity.

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Evolution as Flow

• Derive the production Prod(k, T)
• Prod(k, T) O(k, T) S(k, T) - I(k, T)
• Evolution in genealogy graph-space acts as a
  flow
• Prod(k, T) gt 0 source
• Prod(k, T) lt 0 sink

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Mapping to 2D
y
M
G
x

• Apply a mapping M qk ? (xk, yk)
• xk Mx(qk), yk My(qk).
• Size and colour points according to Prod(k, T)
• source, sink
• Convert traversal frequencies to line thickness.

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Visualization
T (0, 10)
T (10, 20)
T (20, 30)

• Animate series of plots to see evolution
  through graph-space.
• Critical evolutionary transitions are revealed.

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Example The Evoloop

• Configuration space
• 9-state cellular automata
• 5-cell neighbourhood
• Self-replicators
• Self-reproducing loops
• Mutation through extrinsic interaction (passive)
  and self-modification (active)

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Genealogy of the Evoloop
w
phenotype
l
genotype

• Evoloop composed of phenotype and genotype
• Phenotype inner and outer sheath of loop
• Genotype gene sequence within loop
• Define identifier as phenotype genotype
• q (phenotype, genotype)

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Replication of the Evoloop

• Exact self-replication qp qc
• Mutation qp ? qc

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Graph-based Genealogy

• Genealogy of the Evoloop is strongly graph-based
• Small species often have multiple ancestors.
• Evolutionary cycles exist.
• Species can be classified according to dynamics
  in free space
• Stable self-reproduce exactly (source).
• Transitional self-reproduce into a different
  species.
• Terminal do not reproduce (sink).

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Genealogy Tree ? Graph
Loop Size

• Tree-like for large species (many identifiers),
  graph-like for small species (few identifiers).

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Qualities of the Evoloop CA

• Simple and scalable
• Small rule set (95 59049 rules)
• No central operating system
• Purely deterministic (no stochastic operations in
  rule set).
• Adaptable.
• Realizes an emergent evolutionary process.

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Weakness of the Evoloop CA
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• On periodic domains, evolution favours
  smallest-sized loops
• Evolutionary dynamics are predictable.

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Evolution in Periodic Domains
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A Dynamic Environment

• New dissolver state spreads upon contact.
• Clears free space, partitions CA domain.
• Persists in time for a finite period of time,
  then dissolves away.

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Observations

• Emergence of large-sized loops, sustained
  diversity
• New phenomena
• Speciation
• Punctuated equlibrium
• Evolutionary bottlenecks

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Goals of Analysis

• Describe population dynamics at the level of
  evolutionary transitions.
• Understand the effect of environment on
  evolutionary dynamics.
• Quantify
• Speciation
• Diversity

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Mapping Genealogy to 2D
w
p
l
g

• Mapping functions separate genotype (g) and
  phenotype (p)
• Mx(q) Mx(p, g) Mx(g)
• My(q) My(p, g) My(p)

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Mapping Functions

• For genotype we use a hash function
• Mx(g) w1 W(g) w2 G(g) w3 A(g)
• For phenotype we use loop area
• My(p) My(l, w) l ? w

W(g) 8
G(g) 23
A(g) (0100001)2
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Genealogy in a Static Environment
phenotype
genotype

• Original evoloop system evolves to small
  phenotypes, confined graph-space.
• Small loops ? short gene sequence, small
  identifier space, no diversity

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Genealogy in a Dynamic Environment
phenotype
phenotype
phenotype
genotype
genotype
genotype

• Dynamic environment partitions graph-space,
  sustains large loops and diversity.
• Large-sized loops remain longer, complex
  evolutionary dynamics emerge

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Summary and Future goals

• Advantages of method
• General in scope.
• Reveals evolutionary trends at the level of
  genealogical transitions.
• Future goals
• Describe diversity/speciation with graph-based
  metrics.
• Expand applicability.