Emergence: the relation between
static network structure and the
dynamic qualities of landscapes
Jeremy Yamashiro,
University
of
Utah
Jonathan Butner, University of Utah
Chase Dickerson, University of Utah
Thomas Malloy, University of Utah
How does static
network structure relate
to the dynamic flow of
information across that
structure?
TILE IMAGE
Research Questions
• How do different Generating Rules
structure different networks?
• How does network structure affect the
qualities of its emergent landscapes?
Components of an
NK Boolean Network
• static network: binary nodes and wiring
A
•
B
A
B
C
D
C
D
attractor landscapes
IN
2
2
2
2
OUT
2
1
2
3
The flow of state vectors
over time falls into
attractors, which we may
think of as basins in a
Generation rules
•
Simultaneous Random (SR)
All (N) nodes are inserted and form links
simultaneously; each node has equal probability
of taking one of its K inputs from any other node
•
Ordered Random (OR)
(N) nodes are inserted one at a time, and each
forms all K inputs before the next node is
inserted
Generation Rule
determines how nodes are connected to each other
•
Simultaneous
•
Each Randomly connects to 2
other nodes
•
Ordered
• Each Randomly connects to 2 other
nodes
• But only connect to those before it
Types of Network
Very different network structures emerge from
different Generation Rules
Simultaneous Random
Ordered Random
Distribution of Output
Links
Simultaneous Random
Ordered Random
N=1000, K=3, 100% self referencing
Network Behavior
Boolean landscapes:
attractors/basins
Nodes 1100
L=12
State Vector at Time T to T+N
Network Behavior
Boolean landscapes: attractors/basins
Nodes 1100
L=4
State Vector at Time T to T+N
Ruggedness
•
The relative number of attractors a
network can produce
•
Ruggedness is an indicator of the
behavioral complexity of a system;
greater ruggedness => greater
behavioral diversity, flexibility, adaptibility
Attractor
Homogeneity
•
Degree to which attractor lengths (L’s)
are limited or proliferate
•
Attractor homogeneity indicates
consistency between attractors; a
homogenous system can be more
coherent, despite high ruggedness, than
a heterogenous system
Analysis of Network
Structure
• Generated 1000 SR and 1000 OR
networks, N=100, K=3.
• log-log slope of regression line of
nodes/output links distribution of each
network
Analysis of Network
Structure
.95 confidence interval for mean log-log regression
slopes of nodes/output links distribution.
SR
-1.0314+/0.0194
OR
-1.4865+/0.2485
Analysis of Attractor
Landscapes
As a function of:
• Generation Rule
• Log-log slope
Landscapes as a
function of Generation
Rule
We generated 50 SR and 50 OR at N=100, K=3 for
landscape analysis.
• ruggedness
• attractor homogeneity
Ruggedness as
Function of
Generation
Rule
Mean number of basins
SR
OR
859.34
970.48
Standard Standard
Deviation = Deviation =
249.11
94.27
Attractor Homogeneity as
Function of Generation
Rule
Mean number of attractor
cycle lengths (L’s)
SR
SO
24.54
2.02
Standard Standard
Deviation = Deviation =
21.16
0.77
Landscapes as
Function of Log-Log
Slope
•We pooled the 2 samples of 50 networks of each
Generating Rule and pooled them into 1 sample of
100 networks
•regression analyses of log-log slope and
ruggedness, and log-log slope and attractor
homogeneity
Landscapes as
Function of Log-Log
Slope
Log-log slope fails to
predict ruggedness
(total number of
attractors)
Landscapes as
Function of Log-Log
Slope
Log-log slope
predicts attractor
homogeneity
Summary
• Ordered Random (OR) Generation Rule
produces networks with steeper (more
negative) log-log slope, within the
fractal range (Butner, Pasupathi,
Vallejos, 2008).
• OR networks produce more rugged
landscapes than SR networks.
• OR/fractal networks produce more
homogenous attractor landscapes
Discussion
Mapping:
Boolean Network => Neural Net
Attractor => Thought
Stream of
Attractor Landscapes =>
Cognition/Thought
Discussion
• Ruggedness and attractor homogeneity are a set of
freedoms and constraints on a system’s behavior
• Greater ruggedness means the diversity of thoughts
potential to a system is very high, even while
homogenous attractor landscapes may produce
greater coherence between different thoughts or sets
of thoughts.
• The landscapes at the intersection of high
ruggedness and high attractor homogeneity allow for
behavior that is simultaneously highly diverse
(creative?) and internally consistent
Discussion
Fractal-like networks produce attractor
landscapes of great consistency and
richness; fractal-like wiring of the neural net
may support more complex and coherent
cognitive processes.
Acknowledgments
This project was supported in part by a
grant from the University of Utah’s
Undergraduate Research Opportunities
program.
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