A simple system (Kauffman 1993)
Imagine a system of three (n=3) light bulbs (A,B,C), each of which
can be either off (0) or on (1) (s=2).
Each light bulb is connected to the other two (k=2)
There are 23 (sn) possible initial states, also known as the “design
space” at (t=0):
t=0
A
B
C
1
0
0
0
2
1
0
0
3
0
1
0
4
0
0
1
5
1
1
0
6
1
0
1
7
0
1
1
8
1
1
1
Next, establish a “rule” to determine whether a bulb is “on” or “off” in subsequent
iterations:
A bulb will turn on or stay on only if both of its partners are on (the Boolean and
function)
What happens?
t=0
•
•
A
B
C
and
and
and
1
0
0
0
2
1
0
3
0
4
t=1
A
B
C
and
and
and
1
0
0
0
0
2
0
0
1
0
3
0
0
0
1
4
5
1
1
0
6
1
0
7
0
8
1
t=2
A
B
C
and
and
and
1
0
0
0
0
2
0
0
0
0
0
3
0
0
0
0
0
0
4
0
0
0
5
0
0
1
5
0
0
0
1
6
0
1
0
6
0
0
0
1
1
7
1
0
0
7
0
0
0
1
1
8
1
1
1
8
1
1
1
Our eight initial states have resolved down to just two in only two iterations
If we subsequently turn one or two random bulb under conditions 1-7, the system
quickly reestablishes its equilibrium of all bulbs off
– We must turn on all three bulbs to escape this “basin of attraction”
•
However, if we turn off only one bulb under condition 8, within two iterations all
bulbs will be off!
Next, try the Boolean or: A bulb will turn on or stay on
if either of its partners is on.
t=0
A
B
C
or
or
or
1
0
0
0
2
1
0
3
0
4
t=1
A
B
C
or
or
or
1
0
0
0
0
2
0
1
1
0
3
1
0
0
1
4
5
1
1
0
6
1
0
7
0
8
1
t=2
A
B
C
or
or
or
1
0
0
0
1
2
1
1
1
0
1
3
1
1
1
1
1
0
4
1
1
1
5
1
1
1
5
1
1
1
1
6
1
1
1
6
1
1
1
1
1
7
1
1
1
7
1
1
1
1
1
8
1
1
1
8
1
1
1
The mirror effect of the Boolean and:
• Still only two equilibria
• “All on” becomes the strong attractor, “all off” is much
less stable
Now for some fun! Let bulb A have the or condition while B and C have the
and condition. What happens?
t=0
A
B
C
or
and
and
1
0
0
0
2
1
0
3
0
4
t=1
A
B
C
or
and
and
1
0
0
0
0
2
0
0
1
0
3
1
0
0
1
4
5
1
1
0
6
1
0
7
0
8
1
t=2
A
B
C
or
and
and
1
0
0
0
0
2
0
0
0
0
0
3
0
0
0
1
0
0
4
0
0
0
5
1
0
1
5
1
1
0
1
6
1
1
0
6
1
0
1
1
1
7
1
0
0
7
0
0
0
1
1
8
1
1
1
8
1
1
1
We now have three “basins of attraction”:
• Conditions 1-4 and 7 become “all off”
• Condition 8 becomes “all on”
• Conditions 5 and 6 are “A on, B and C alternating”
• The system is still quick to “adapt” to randomly changing the state of any
bulb, but will quickly fall into one of the three “attractors”
What does our simple system demonstrate?
The number of stable states is much smaller than the number of initial
states
• This “structure” is an “emergent” property not reducible to the individual bulbs or
the rules by which they “behave,” but rather results from their interactions
The system can “adapt” to changes
• There is a range of attractor “strength,” with some being quite resistant to change,
others unable to persist after any change, and still others somewhere in between
The system is sensitive to changes in initial conditions:
• Small differences in initial states can result in significantly different end states
• The system is described as “chaotic,” not because of any randomness at work in the
system but because of this sensitivity
• Imagine a chaotic system in which the differences in initial conditions are below the
level of measurement – seemingly identical initial states would resolve to very
different states, perhaps suggesting some “randomness” at play when there is
none!
Conway’s Game of Life
Extend Kauffman’s Boolean network example a bit. Think of a lattice grid
(i.e. a chessboard), in which each square is an “agent” surrounded by eight
neighbors.
• As with our light bulbs, each agent can be either “on” or “off”
• The decision rules are as follows:
•
•
•
If an agent is “off” it will turn “on” if it has exactly three neighbors “on”
If an agent is “on” it will stay “on” if it has either two or three neighbors “on”
Under all other conditions an agent will be “off”
Now, allow for a “landscape” of 100 x 180 agents.
• This is 218,000 (or ~105400) possible initial states (the design space)
• Number of seconds since the universe began: ~10 11
• Number of particles in the universe: ~10 82
Conway’s Game of Life
• Despite the Vast number of stable states, only a handful of
configurations are stable
•
Thus our models can meaningfully sample the design space!
•
The system is adaptive
•
The combination of rules is not arbitrary: most sets of rules either
generate games in which nothing interesting happens or the system
cycles through the design space
•
•
•
One question would be: what is the likelihood of randomly generating a set of rules
that displays complex adaptive behavior (A: very small for even simple systems)
Turning the question around: how many systems that don’t have rules generating
complexity and adaptation survive for long? A: very few!
So we find many complex adaptive systems not because they are relatively
common at the outset but they are the only ones that persist!
Shelling’s Segregation Model
The Standing Ovation Problem
One of a potentially very large number of problems in which “decentralized
dynamical systems consisting of spatially distributed agents who respond to
local information.”
• “Structure” (i.e. Standing Ovation) is an emergent property
• Agents are spatially related with limited information
In a simple version of the SOP, an agent will perceive the quality of a
performance, make an initial decision to stand based on the perceived
quality relative to an internal willingness to stand. After the first iteration,
agents will choose to change their initial behavior based on how many
neighbors are standing and the agent’s threshold for conformity (perhaps
relative to her initial preference).
The Standing Ovation Problem:
Modeling Issues
• How much external information does each agent
receive?
• How many neighbors does each agent have?
• Does the neighborhood size vary? Why?
• Does each neighbor have the same weight or does influence vary?
• Are some agents more influential to all agents who “see” them or
only to some (i.e. “friends”)?
• Why?
The Standing Ovation Problem:
Modeling Issues
• What is the relative balance between initial agent
preference (to stand or remain seated) and
neighborhood information?
• Is this balance the same for all agents or distributed in some other
way?
• What assumptions drive this decision?
• Is a decision to stand reversible? Why or why not?
• How many iterations should the model be run?
Comparing mathematical and
computational approaches to SOP
Both approaches:
• Often generate the “wrong” equilibrium (i.e. most
people end up standing even though most did not like
the play)
• Find the stronger the pressure to conform the more
often “wrong” equilibria occur
• The plot of people standing is an “s” curve, as diffusion
models predict
• People at the front of the audience can have a large
impact
Comparing mathematical and
computational approaches to SOP
However, mathematical models in some respects perform
poorly compared to computational models:
• They imply that all agents eventually agree, a rare
outcome in computational models
• They tend to ignore the sequencing of updating, which
is shown to be important
• The predicted “s” curve can be easily changed in
computational models to reduce the slope
• Mathematical models, by only counting the number of
standing agents, fail to replicate the spatial dynamics
generating the “s” curve