A fast algorithm for 2D Rigidity Percolation

Rigidity Percolation is a crucial framework for describing rigidity transitions in amorphous systems. We present a new, efficient algorithm to study central-force Rigidity Percolation in two dimensions. This algorithm combines the Pebble Game algorithm, the Newman-Ziff approach to Connectivity Percolation, as well as novel rigorous results in rigidity theory, to exactly identify rigid clusters over the full bond concentration range, in a time that scales as $N^{1.02}$ for a system of $N$ nodes. We perform extensive numerical simulations with systems larger than $500$ million nodes, far beyond the previous limitations. We obtain new, precise estimates for the critical exponents, $ν=1.1694(8)$ and $D_f=1.8423(7)$, and locate the critical threshold at $p_c = 0.6602741(4)$. Besides opening the way to further accurate numerical studies of Rigidity Percolation, our work provides new rigorous theoretical insights on specific cluster merging mechanisms that distinguish it from the standard Connectivity Percolation problem.

💡 Research Summary

The authors present a groundbreaking algorithm for studying two‑dimensional central‑force rigidity percolation (RP), dramatically improving on the classic Pebble Game approach. By marrying the combinatorial Pebble Game (which implements Laman’s theorem) with the dynamic Newman‑Ziff (NZ) scheme originally devised for ordinary connectivity percolation, and by proving three rigorous theorems that describe how a single newly activated bond changes the rigidity landscape, they achieve an almost linear runtime of O(N^1.02). The key insight is that, unlike ordinary percolation where a new bond either does nothing or merges two clusters, a new bond in RP can be either an independent constraint (creating or merging rigid clusters) or a redundant constraint (leaving the rigidity structure unchanged). The three theorems give precise criteria for these cases and show how to update the set of rigid clusters with minimal pebble‑search operations.

Two types of pebble searches are employed. Type I moves pebbles across the auxiliary pebble graph to keep the distribution of degrees of freedom up‑to‑date; Type II tests mutual rigidity of two bonds without moving pebbles, using temporary freezing of pebbles and marking visited nodes as “floppy” or “rigid”. Failed Type II searches trigger a triangulation step that shortcuts future searches, a technique inherited from the original Jacobs‑Hendrickson algorithm but now used within the NZ incremental framework.

Cluster management uses a Union‑Find data structure with path compression, exactly as in NZ for connectivity percolation. When a bond joins two distinct rigid clusters, the smaller tree’s root is redirected to the larger one, achieving constant‑time merging. An adapted Mac‑Thal algorithm detects whether a rigid cluster wraps around periodic boundaries, allowing the authors to study percolation on toroidal lattices.

Performance tests on triangular lattices up to N≈5×10^8 nodes confirm the predicted scaling; the algorithm runs roughly 2–3 times faster than the best previous implementations (which scaled as N^1.15 near the critical point). This speedup enables high‑precision measurements of critical properties. By sampling the entire bond‑filling fraction in a single run, the authors obtain the percolation threshold p_c=0.6602741(4), the correlation‑length exponent ν=1.1694(8), and the fractal dimension of the spanning rigid cluster D_f=1.8423(7). These values improve substantially on earlier estimates (ν≈1.21, D_f≈1.86) and provide strong evidence that 2D central‑force rigidity percolation belongs to a distinct universality class from ordinary percolation.

The paper is organized as follows: Section II reviews the Jacobs‑Hendrickson Pebble Game and details pebble‑search mechanics; Section III recaps the Newman‑Ziff algorithm for connectivity percolation; Section IV states and proves the three theorems governing single‑bond activation in RP (proofs are in the Supplemental Material); Section V outlines the overall NZ‑style algorithmic strategy; Section VI gives implementation specifics, including the modified Mac‑Thal wrapping detection; Section VII presents benchmarking results and the precise critical exponent measurements; and Section IX discusses extensions to three dimensions, frictional networks, and non‑regular lattices.

In summary, this work transforms rigidity percolation from a computationally intensive, multi‑run problem into an efficiently incremental process that can handle hundreds of millions of nodes with near‑linear effort. The combination of rigorous theoretical insights with clever data‑structure engineering opens the door to systematic, high‑precision studies of rigidity transitions in a wide range of soft‑matter and amorphous systems.