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Modeling collective behavior in hierarchically-organized systems

Modeling collective behavior in hierarchically-organized systems

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Modeling collective behavior in hierarchically-organized systems

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  1. Modeling collective behavior in hierarchically-organized systems Jordan Snyder, UC Davis with Anatoly Zlotnik and Aric Hagberg

  2. “Complex Systems” -Herbert A. Simon, The Architecture of Complexity, 1962

  3. The Watchmaker Parable • Large systems will evolve faster if they are made of stable sub-systems • We should expect to see hierarchical organization Simon, H. A. (1991). The Architecture of Complexity. In Facets of Systems Science (Vol. 27, pp. 457–476). Boston, MA: Springer US. https://doi.org/10.1007/978-1-4899-0718-9_31 Wu, J. (2013). Hierarchy Theory: An Overview. In Linking Ecology and Ethics for a Changing World (Vol. 1, pp. 281–301). Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-94-007-7470-4_24

  4. Intrinsic meets Extrinsic • Within a module, units interact • Whole modules also interact with each other • At the module level, this looks like mutual coupling versus external forcing

  5. Order via mutual coupling: Kuramoto Model • Key features: • Heterogeneity: • Coupling drives phases together • Trade-off occurs at

  6. Order via External Forcing: Entrainment • Weakly driven phase oscillators: • In rotating reference frame: • Where • Entrainment: , but remain different • Possible Result: splay state

  7. External forcing meets coupling • How does external forcing affect ability of coupling to align phases? Plan of attack: • Pick and such that there is a splay state fixed point • Find the value of at which the splay state becomes unstable

  8. Math • Goal: entrainment with phases evenly spread from –π to π • Take and • Then • The splay state is a fixed point for all values of • The splay state is stable for and unstable for • With the same natural frequencies but no forcing, phase alignment develops at

  9. Results

  10. Conclusions • Interplay of internal coupling and external forcing is a building block of complex systems • Forcing and coupling can impose different types of order, yet interact in surprising ways