>[!warning] >This content has not been peer reviewed. # The backbone dimension at the percolation threshold In **percolation theory**, a network of nodes and links has a **critical point** (e.g. a critical occupation probability) at which a giant connected component first appears. The **backbone** of that component is the set of links that carry current (or flow) between two distant points. The **fractal dimension** of that backbone — how its “mass” scales with linear size — is a **universal constant** for a given spatial dimension. In 3D, the backbone dimension is $d_B \approx 1.22 \pm 0.02$. --- ## What it is - **Critical percolation:** At the threshold, the cluster is fractal; its scaling exponents are universal (same for lattices, random graphs, etc.). - **Backbone:** Not all links in the cluster carry load; the backbone is the minimal subset that connects the endpoints. Its dimension $d_B$ is smaller than the cluster dimension. - **3D value:** $d_B \approx 1.22$ (established in percolation theory and simulations; see Stauffer & Aharony 1994; Deng & Blöte 2005). It does not depend on the microscopic details of the network. --- ## How RRT/RST uses it - **[[Backbone Dimension]]:** The substrate (A1) is a finite network; A5 (relational distance) makes it a resistance metric. For the universe to be observable, it must be “just barely” connected — at the **percolation threshold**. The dimension of the *signal-carrying* backbone is then $d_B \approx 1.22$. - **Transition sharpness $n_0$:** In RST, $n_0$ is the **relational scaling exponent** of the substrate’s signal backbone — i.e. $n_0 = d_B$. So **$n_0 \approx 1.22$** is derived from topology (A1 + A5 + connectivity + critical percolation). SPARC measures $n_0 \approx 1.25$; the ~2% difference is treated as frictional residue. So $n$ is not a fit parameter; it is the **fractal dimension of the format’s backbone**. --- ## Links | Concept | Note | |:---|:---| | Full derivation of $n_0 = d_B$ | **[[Backbone Dimension]]** | | Role of $n$ in fidelity and triangle | **[[Transition Sharpness]]** | | Cost scaling with distance | **[[Relational distance]]** | --- ## References - Stauffer, D. & Aharony, A. (1994). *Introduction to Percolation Theory*, 2nd ed. Taylor & Francis, London. (Standard reference for percolation exponents and backbone dimension.) - Deng, Y. & Blöte, H. W. J. (2005). *Monte Carlo study of the site-percolation model in two and three dimensions.* Phys. Rev. E **72**, 016126. [DOI](https://doi.org/10.1103/PhysRevE.72.016126) (Numerical values for $d_B$ and other universal exponents.)