1. 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

2. How does static network structure relate to the dynamic flow of information across that structure? TILE IMAGE

3. Research Questions How do different Generating Rules structure different networks? How does network structure affect the qualities of its emergent landscapes?

4. Components of an NK Boolean Network static network: binary nodes and wiring attractor landscapes AB CD The flow of state vectors over time falls into attractors, which we may think of as basins in a landscape INOUT A22 B21 C22 D23

5. 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

6. 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

7. Types of Network Very different network structures emerge from different Generation Rules Ordered Random Simultaneous Random

8. Distribution of Output Links Simultaneous Random Ordered Random N=1000, K=3, 100% self referencing

9. Network Behavior Boolean landscapes: attractors/basins Nodes 1- 100 { State Vector at Time T to T+N L=12

10. Network Behavior Boolean landscapes: attractors/basins Nodes 1- 100 { L=4 State Vector at Time T to T+N

11. 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

12. 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

13. 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

14. Analysis of Network Structure.95 confidence interval for mean log-log regression slopes of nodes/output links distribution. SROR -1.0314+/- 0.0194 -1.4865+/- 0.2485

15. Analysis of Attractor Landscapes Generation Rule Log-log slope As a function of:

16. 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.

17. Ruggedness as Function of Generation Rule Mean number of basins SROR 859.34 Standard Deviation = 249.11 970.48 Standard Deviation = 94.27

18. Attractor Homogeneity as Function of Generation Rule Mean number of attractor cycle lengths (L’s) SRSO 24.54 Standard Deviation = 21.16 2.02 Standard Deviation = 0.77

19. 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

20. Landscapes as Function of Log-Log Slope Log-log slope fails to predict ruggedness (total number of attractors)

21. Landscapes as Function of Log-Log Slope Log-log slope predicts attractor homogeneity

22. 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

23. Discussion Mapping: Boolean Network => Neural Net AttractorThought=> Stream of Cognition/Thought Attractor Landscapes=>

24. 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

25. 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.

26. Acknowledgments This project was supported in part by a grant from the University of Utah’s Undergraduate Research Opportunities program.

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