Emergence: the relation between static network structure and the dynamic qualities of landscapes

Jeremy Yamashiro, University of UtahJonathan Butner, University of UtahChase Dickerson, University of UtahThomas 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

•attractor landscapes

A B

C D

The flow of state vectors over time falls into

attractors, which we may think of as basins in a

landscape

IN OUTA 2 2B 2 1C 2 2D 2 3

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

• Simultaneous• Each Randomly connects to

2 other nodes

• Ordered• Each Randomly connects to 2

other nodes• But only connect to those before

it

Generation Rule determines how nodes are connected to each

other

Types of NetworkVery different network structures emerge

from different Generation Rules

Ordered RandomSimultaneous Random

Distribution of Output Links

Simultaneous Random

Ordered Random

N=1000, K=3, 100% self referencing

Network BehaviorBoolean landscapes: attractors/basins

Nodes 1-100{

State Vector at Time T to T+NL=12

Network BehaviorBoolean landscapes: attractors/basins

Nodes 1-100{

L=4State 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 OR

-1.0314+/-0.0194

-1.4865+/- 0.2485

Analysis of Attractor

Landscapes

•Generation Rule

•Log-log slope

As a function of:

Landscapes as a function of

Generation Rule

•ruggedness

•attractor homogeneity

We generated 50 SR and 50 OR at N=100, K=3 for landscape analysis.

Ruggedness as Function of

Generation RuleMean number of basins

SR OR

859.34 Standard Deviation = 249.11

970.48 Standard Deviation = 94.27

Attractor Homogeneity as Function of

Generation RuleMean number of attractor cycle lengths (L’s)

SR SO

24.54Standard Deviation = 21.16

2.02 Standard Deviation

= 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 SlopeLog-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

DiscussionMapping:

Boolean Network => Neural Net

Attractor Thought=>

Stream of Cognition/Thought

Attractor Landscapes=>

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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