# Higher-Order System #definition A **higher-order system** is a system characterized by interactions, relationships, or dynamics that transcend simple pairwise or first-order connections, encompassing more complex structures, dependencies, or hierarchies among its components. In the context of system science, mathematics, and network theory, a higher-order system may refer to one where the behavior, evolution, or organization arises from interactions involving three or more elements simultaneously, or from nested subsystems forming higher levels of organization. Such systems are described using frameworks that capture their multidimensional or multilevel structure, such as hypergraphs, hypermatrices, or higher-order differential equations. Higher-order systems are crucial for modeling collective phenomena, emergent behaviors, and complex dependencies not reducible to lower-order interactions[^1][^2][^3][^4][^5]. **Definition in the Context of Wardley Mapping:** In Wardley Mapping, “higher-order system” describes an emergent structure or value-creating capability that is enabled by the commoditization and ubiquity of underlying components in the value chain. The identification and anticipation of higher-order systems are central to strategic advantage, innovation, and navigating cycles of change in business ecosystems[^6][^7][^8]. --- **References** [^1]: Higher-order systems. In *Handbook of Geometric Constraint Systems Principles* (Springer, 2022). [Read PDF](https://link.springer.com/content/pdf/10.1007/978-3-030-91374-8.pdf) [^2]: Benson, A. R. (2023). What are higher-order networks?. *SIAM Review, 65*(2), 251-272. [Read abstract](https://epubs.siam.org/doi/abs/10.1137/21M1414024) [^3]: Qi, L., & Ye, Y. (2023). Hypermatrix algebra and irreducible arity in higher-order systems: Concepts and perspectives. *Mathematics in Computer Science, 17*(1), 109-134. [Read abstract](https://www.worldscientific.com/doi/abs/10.1142/S0219525923500078) [^4]: Gundlach, C., & Martín-García, J. M. (2015). Hyperbolicity of high-order systems of evolution equations. *International Journal of Modern Physics D, 24*(12), 1550001. [Read abstract](https://www.worldscientific.com/doi/abs/10.1142/S0219891615500010) [^5]: Iacopini, I., Petri, G., & Barrat, A. (2024). Contagion dynamics on higher-order networks. *Nature Reviews Physics, 6*, 255–267. [Read article](https://www.nature.com/articles/s42254-024-00733-0) [^6]: Simon Wardley, *Wardley Maps - Coach agile* (2020), [PDF](https://coach-agile.com/wp-content/uploads/2022/09/Wardley-Maps-Simon-Wardley-compresse.pdf) [^7]: Simon Wardley, “Open source as weapon”, *Bits or Pieces* (2015), [blog](https://blog.gardeviance.org/2015/12/open-source-as-weapon.html) [^8]: Simon Wardley, “I don't like Hayek vs Keynes”, *Bits or Pieces* (2012), [blog](https://blog.gardeviance.org/2012/08/i-dont-like-hayek-vs-keynes.html)