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

Explore the relation between static network structure and the dynamic flow of information in landscapes. Investigate the impact of different generation rules on network structures and emergent landscapes. Analyze attractor landscapes, ruggedness, and attractor homogeneity in Boolean networks. Discover how ordering influences network behaviors and landscape characteristics. Gain insights into thought attractors and cognitive processes in complex networks.

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

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  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. A B C D Components of an NK Boolean Network • static network: binary nodes and wiring • attractor landscapes The flow of state vectors over time falls into attractors, which we may think of as basins in a landscape

  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. 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

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

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

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

  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.

  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

  18. Attractor Homogeneity as Function of Generation Rule Mean number of attractor cycle lengths (L’s)

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

  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.

  27. References • Barabási A. L. & Réka A. (1999). Emergence of Scaling in Random Networks. Science 286, 509 - 512 • Barabási, A.L. (2003) Linked. Cambridge: Plume. Bateson, G. (2002) Mind and Nature. Cresskill: Hampton Press, Inc. (originally published by Dutton, 1979). • Butner, J., Pasupathi, M., Vallejos, V. (2008). When the facts just don’t add up: The fractal nature of conversational stories. Social Cognition, 26, 670-699. • Erdős, P. & Rényi, A. (1959) On Random Graphs I. Publ. Math. Vol ?, 290–297. Erdős, P. & Rényi, A. (1960). On the evolution of random graphs. Publications of the Matkemafical Insfifufe of the Hungarian Academy of Sciences, 5. • Kauffman, S. A. (1969). Metabolic stability and epigenesis in randomly connected nets. Journal of Theoretical Biology, 22, 437. • Kauffman, S. A. (1971). Gene regulation networks. Current Topics in Developmental Biology, • 6, 145. • Kauffman, S. A. (1993). The origins of order: self-organization and selection in evolution. • Oxford: Oxford University Press. • Malloy, T. E., Bostic St Clair, C. & Grinder, J. (2005). Steps to an ecology of emergence. Cybernetics & Human Knowing, 12, 102-119. • Malloy, T.E., Butner, J., & Jensen, G. C. (2008). The emergence of dynamic form through phase relations in dynamic systems. Nonlinear Dynamics, Psychology, and Life Sciences, 12, 371-395. • Malloy, T.E., Jensen, G.C. (2008). Dynamic constancy as a basis for perceptual hierarchies. Nonlinear Dynamics, Psychology, and Life Sciences, 12, 191-203. • Malloy, T. E., Jensen, G. C., & Song, T. (2005) Mapping knowledge to Boolean dynamic systems in Bateson’s epistemology. Nonlinear Dynamics, Psychology, and Life Sciences, 9, 37- 60. • Mitchell, M. (2009). Complexity. Oxford: Oxford University Press. • Turing,A.M.(1952).The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237, 37-72. • Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of 'small world' networks. Nature 393, 440-42.

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