random boolean networks. attractors a boolean network has 2 ^n possible states. sooner or later it...
Post on 05-Jan-2016
212 Views
Preview:
TRANSCRIPT
Random Boolean Networks
Random Boolean Networks
Random Boolean Networks
Attractors
A Boolean network has 2^N possible states. Sooner or later it will reach a previously visited state, and thus, since
the dynamics are deterministic, fall into an attractor. If the attractor has only a single state it is called a point
attractor, and if the attractor consists of more than one state it is called a cycle attractor. The set of states that
lead to an attractor is called the basin of the attractor. States with no incoming connections are called garden-of-
Eden states and the dynamics of the network flow from these states towards attractors. The time it takes to reach
an attractor is called transient time. (Gershenson 2004)
Random Boolean networks (RBNs) are known as NK networks or Kauffman
networks (Dubrova 2005). An RBN is a system of N binary-state nodes
(representing genes) with K inputs to each node representing regulatory
mechanisms. The two states (on/off) represent respectively, the status of a
gene being active or inactive. The variable K is typically held constant, but it
can also be varied across all genes, making it a set of integers instead of a
single integer. In the simplest case each gene is assigned, at random, K
regulatory inputs from among the N genes, and one of the possibleBoolean
functions of K inputs. This gives a random sample of the possible
ensembles of the NKnetworks. The state of a network at any point in time is
given by the current states of all N genes. Thus the state space of any such
network is 2^N.
Genetic Algorithms
Chapter 9, Complexity: A Guided Tour
Evolution by Natural Selection
Charles Darwin
Evolution by Natural Selection
Charles Darwin
• Organisms inherit traits from parents
Evolution by Natural Selection
Charles Darwin
• Organisms inherit traits from parents
• Traits are inherited with some variation, via mutation and sexual recombination
Evolution by Natural Selection
Charles Darwin
• Organisms inherit traits from parents
• Traits are inherited with some variation, via mutation and sexual recombination
• Due to competition for limited resources, the organisms best adapted to the environment tend to produce the most offspring.
Evolution by Natural Selection
Charles Darwin
• Organisms inherit traits from parents
• Traits are inherited with some variation, via mutation and sexual recombination
• Due to competition for limited resources, the organisms best adapted to the environment tend to produce the most offspring.
• This way traits producing adapted individuals spread in the population
Evolution by Natural Selection
• Organisms inherit traits from parents
• Traits are inherited with some variation, via mutation and sexual recombination
• Due to competition for limited resources, the organisms best adapted to the environment tend to produce the most offspring.
• This way traits producing adapted individuals spread in the population
in computers
Computer programs
Charles Darwin
Evolution by Natural Selection
• Organisms inherit traits from parents
• Traits are inherited with some variation, via mutation and sexual recombination
• Due to competition for limited resources, the organisms best adapted to the environment tend to produce the most offspring.
• This way traits producing adapted individuals spread in the population
in computers
Computer programs
Charles Darwin
John Holland
Evolution by Natural Selection
• Organisms inherit traits from parents
• Traits are inherited with some variation, via mutation and sexual recombination
• Due to competition for limited resources, the organisms best adapted to the environment tend to produce the most offspring.
• This way traits producing adapted individuals spread in the population
in computers
Computer (e.g., programs)
Charles Darwin
John Holland
Genetic Algorithms (GAs)
Some real-world uses of genetic algorithms
Some real-world uses of genetic algorithms
• Used by GE to automate parts of aircraft design
Some real-world uses of genetic algorithms
• Used by GE to automate parts of aircraft design
• Used by pharmaceutical companies to discover new drugs
Some real-world uses of genetic algorithms
• Used by GE to automate parts of aircraft design
• Used by pharmaceutical companies to discover new drugs
• Used by the London Stock Exchange to automatically detect fraudulent trades
Some real-world uses of genetic algorithms
• Used by GE to automate parts of aircraft design
• Used by pharmaceutical companies to discover new drugs
• Used by the London Stock Exchange to automatically detect fraudulent trades
• Used to generate realistic computer animation in the movies Lord of the Rings: The Return of the King and Troy
Some real-world uses of genetic algorithms
• Used by GE to automate parts of aircraft design
• Used by pharmaceutical companies to discover new drugs
• Used by the London Stock Exchange to automatically detect fraudulent trades
• Used to generate realistic computer animation in the movies Lord of the Rings: The Return of the King and Troy
• Used to model and understand evolution in nature!
Robby the Robot
Example: Evolving Strategies for Robby the Robot
Example: Evolving Strategies for Robby the Robot
Input: Contents of N, S, E, W,
C(Current)
Example: Evolving Strategies for Robby the Robot
Input: Contents of N, S, E, W,
C(Current)
Possible actions:Move N
Move S
Move E
Move W
Move random
Stay put
Try to pick up can
Example: Evolving Strategies for Robby the Robot
Input: Contents of N, S, E, W,
C(Current)
Possible actions:Move N
Move S
Move E
Move W
Move random
Stay put
Try to pick up can
Rewards/Penalties (points): Picks up can: 10
Tries to pick up can on empty site: -1
Crashes into wall: -5
Example Strategy
Encoding a Strategy
Code: MoveNorth = 0MoveSouth = 1MoveEast = 2MoveWest = 3StayPut = 4PickUpCan = 5MoveRandom = 6
Encoding a Strategy
Code: MoveNorth = 0MoveSouth = 1MoveEast = 2MoveWest = 3StayPut = 4PickUpCan = 5MoveRandom = 6
Encoding a Strategy
0
Code: MoveNorth = 0MoveSouth = 1MoveEast = 2MoveWest = 3StayPut = 4PickUpCan = 5MoveRandom = 6
Encoding a Strategy
0 2
Code: MoveNorth = 0MoveSouth = 1MoveEast = 2MoveWest = 3StayPut = 4PickUpCan = 5MoveRandom = 6
Encoding a Strategy
0 2 6
Code: MoveNorth = 0MoveSouth = 1MoveEast = 2MoveWest = 3StayPut = 4PickUpCan = 5MoveRandom = 6
Encoding a Strategy
0 2 6 5 . . . 3 . . . 4
Code: MoveNorth = 0MoveSouth = 1MoveEast = 2MoveWest = 3StayPut = 4PickUpCan = 5MoveRandom = 6
Encoding a Strategy
0 2 6 5 . . . 3 . . . 4
243 values
Code: MoveNorth = 0MoveSouth = 1MoveEast = 2MoveWest = 3StayPut = 4PickUpCan = 5MoveRandom = 6
Genetic algorithm for evolving strategies for Robby
Genetic algorithm for evolving strategies for Robby
1. Generate 200 random strategies (i.e., programs for controlling Robby)
Random Initial Population
Genetic algorithm for evolving strategies for Robby
1. Generate 200 random strategies (i.e., programs for controlling Robby)
2. For each strategy, calculate fitness (average reward minus penalties earned on random environments)
Genetic algorithm for evolving strategies for Robby
1. Generate 200 random strategies (i.e., programs for controlling Robby)
2. For each strategy, calculate fitness (average reward minus penalties earned on random environments)
3. The strategies pair up and create offspring via “sexual recombination” with random mutations ― the fitter the parents, the more offspring they create.
Parent 1:
Parent 2:
Parent 1:
Parent 2:
Parent 1:
Parent 2:
Child:
Genetic algorithm for evolving strategies for Robby
1. Generate 200 random strategies (i.e., programs for controlling Robby)
2. For each strategy, calculate fitness (average reward minus penalties earned on random environments)
3. The strategies pair up and create offspring via “sexual recombination” with random mutations ― the fitter the parents, the more offspring they create.
4. Keep going back to step 2 until a good-enough strategy is found!
My hand-designed strategy:
My hand-designed strategy:
“If there is a can in the current site, pick it up.”
My hand-designed strategy:
“If there is a can in the current site, pick it up.”
“Otherwise, if there is a can in one of the adjacent
sites, move to that site.”
My hand-designed strategy:
“If there is a can in the current site, pick it up.”
“Otherwise, if there is a can in one of the adjacent
sites, move to that site.”
“Otherwise, choose a random direction to move in.”
My hand-designed strategy:
“If there is a can in the current site, pick it up.”
“Otherwise, if there is a can in one of the adjacent
sites, move to that site.”
“Otherwise, choose a random direction to move in.”
Average fitness of this strategy: 346
(out of max possible ≈500)
My hand-designed strategy:
“If there is a can in the current site, pick it up.”
“Otherwise, if there is a can in one of the adjacent
sites, move to that site.”
“Otherwise, choose a random direction to move in.”
Average fitness of this strategy: 346
(out of max possible ≈500)
Average fitness of GA evolved strategy: 486
(out of max possible ≈500)
My hand-designed strategy:
“If there is a can in the current site, pick it up.”
“Otherwise, if there is a can in one of the adjacent
sites, move to that site.”
“Otherwise, choose a random direction to move in.”
Average fitness of this strategy: 346
(out of max possible ≈500)
Average fitness of GA evolved strategy: 486
(out of max possible ≈500)???
One Run of the Genetic AlgorithmB
est
fitne
ss in
pop
ulat
ion
Generation number
Generation 1
Best average score = −81
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 0 Score: 0
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 1 Score: 0
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 2 Score: −5
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 2 Score: −5
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 3 Score: −10
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 3 Score: −10
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 4 Score: −15
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 4 Score: −15
Generation 10
Best average score = 0
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 0 Score: 0
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 1 Score: 0
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 2 Score: 0
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 3 Score: 0
Generation 200
Fitness = 240
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 0 Score: 0
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 1 Score: 0
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 2 Score: 0
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 3 Score: 10
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 4 Score: 10
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 5 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 6 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 7 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 8 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 9 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 10 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 11 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 12 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 13 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 14 Score: 30
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 15 Score: 30
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 16 Score: 40
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 17 Score: 40
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 18 Score: 50
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 19 Score: 50
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 20 Score: 60
Generation 1000
Fitness = 492
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 0 Score: 0
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 1 Score: 0
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 2 Score: 10
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 3 Score: 10
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 4 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 5 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 6 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 7 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 8 Score: 20
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 9 Score: 30
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 10 Score: 30
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 11 Score: 40
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 12 Score: 40
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 13 Score: 50
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 14 Score: 50
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 15 Score: 60
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 16 Score: 60
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 17 Score: 70
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Time: 18 Score: 70
Why Did The GA’s Strategy Outperform Mine?
My Strategy
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
The GA’s Evolved Strategy
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9
Principles of Evolution Seen in Genetic Algorithms
Principles of Evolution Seen in Genetic Algorithms
• Natural selection works!
Principles of Evolution Seen in Genetic Algorithms
• Natural selection works!
• Evolution proceeds via periods of stasis “punctuated” by periods of rapid innovation
Principles of Evolution Seen in Genetic Algorithms
• Natural selection works!
• Evolution proceeds via periods of stasis “punctuated” by periods of rapid innovation
Bes
t fit
ness
in p
opul
atio
n
Generation number
Principles of Evolution Seen in Genetic Algorithms
• Natural selection works!
• Evolution proceeds via periods of stasis “punctuated” by periods of rapid innovation
• Exaptation is common
Principles of Evolution Seen in Genetic Algorithms
• Natural selection works!
• Evolution proceeds via periods of stasis “punctuated” by periods of rapid innovation
• Exaptation is common
• Co-evolution speeds up innovation
Principles of Evolution Seen in Genetic Algorithms
• Natural selection works!
• Evolution proceeds via periods of stasis “punctuated” by periods of rapid innovation
• Exaptation is common
• Co-evolution speeds up innovation
• Dynamics and results of evolution are unpredictable and hard to analyze
top related