>[!warning] >This content has not been peer reviewed. # Networks, graphs, and connectivity in complex systems ## What it is (established result) Many complex systems can be modelled as **networks**: entities as nodes and interactions as edges. Classical results from graph theory and network science show that: - As the **edge probability** or average degree increases in random graphs, there is a **percolation threshold** above which a **giant component** emerges that contains a finite fraction of all nodes \([1,2]\). - Near this threshold, universal **critical exponents** and fractal structures (backbones, dangling ends) appear, largely independent of microscopic details \([2,3]\). - Many real-world systems (infrastructure, biology, social networks) can be understood in terms of connectivity, robustness, and flow on such graphs. These results provide a general language for connectivity, transport, and critical phenomena in discrete systems. --- ## How RRT / RST uses it RST treats the substrate as a **network of relations**: - Nodes and edges are the basic degrees of freedom and connections in the **Format**; connectivity controls where identity can persist and propagate. - The emergence of a giant component corresponds to the formation of a **signal-carrying backbone**, where identity can be maintained across large regions with high fidelity. - The **backbone dimension** \(d_B \approx 1.22\) at critical percolation links to the core RST sharpness parameter \(n \approx 1.25\), tying network geometry to the exponent in the Resource Triangle. Thus, network science supplies the **structural substrate language** for RST: connectivity, percolation, and backbones describe how and where the substrate supports long-range, high-fidelity relations. --- ## Links | Role | Link | |:---|:---| | Foundation mapping | [[foundation/Networks and connectivity/Networks and connectivity (RST)]] | | RST script | [[foundation/Networks and connectivity/Networks and connectivity - Code]] | | Substrate properties | [[Concrete properties of the substrate]] | --- ## References [1] P. Erdős and A. Rényi, "On the evolution of random graphs," *Publ. Math. Inst. Hung. Acad. Sci.* **5**, 17–61 (1960). [2] M. E. J. Newman, *Networks: An Introduction*, Oxford University Press, 2010. [3] D. Stauffer and A. Aharony, *Introduction to Percolation Theory*, 2nd ed., Taylor & Francis, 1994.