[Paper Review] Revisiting the Common Neighbour Analysis and the Centrosymmetry Parameter

Peter Mahler Larsen|arXiv (Cornell University)|Mar 19, 2020

Graph theory and applications21 citations

TL;DR

This paper proposes improved variants of two widely used structural analysis methods in molecular dynamics: the Interval Common Neighbour Analysis (i-CNA) and the Minimum-Weight Matching Centrosymmetry Parameter (MWM-CSP). i-CNA enhances robustness by searching over a range of thresholds to avoid misclassification under atomic disorder, while MWM-CSP uses graph matching to ensure consistent, continuous, and physically intuitive centrosymmetry calculations—both methods improve accuracy and reliability without adding parameters.

ABSTRACT

We review two standard methods for structural classification in simulations of crystalline phases, the Common Neighbour Analysis and the Centrosymmetry Parameter. We explore the definitions and implementations of each of their common variants, and investigate their respective failure modes and classification biases. Simple modifications to both methods are proposed, which improve their robustness, interpretability, and applicability. We denote these variants the Interval Common Neighbour Analysis, and the Minimum-Weight Matching Centrosymmetry Parameter.

Motivation & Objective

• To address the limitations of conventional CNA and CSP methods in classifying crystal structures under atomic disorder and varying length scales.
• To resolve classification bias and inconsistent results in existing CNA and CSP implementations, especially in polycrystalline or disordered systems.
• To develop robust, parameter-free alternatives that maintain interpretability while improving structural recognition accuracy.
• To ensure continuity and consistency in centrosymmetry measurement under small atomic displacements, a key requirement for reliable simulation analysis.

Proposed method

• Proposes Interval Common Neighbour Analysis (i-CNA), which evaluates structural signatures across a range of neighbor thresholds instead of a single fixed threshold, improving robustness to thermal or structural fluctuations.
• Introduces a local length scale calculation for each atom using first- or second-shell neighbor distances to define atom-specific thresholds, enabling scale-invariance in multi-phase systems.
• Replaces the greedy or heuristic matching in CSP with a Minimum-Weight Matching (MWM) on a complete graph of atoms, using squared distances as edge weights to find the optimal pairing of opposite neighbors.
• Ensures each atom has exactly one opposite neighbor and minimizes the total weight sum, producing a continuous, non-differentiable but stable CSP function under small perturbations.
• Employs a hybrid strategy: first apply the faster Greedy Edge Selection (GES) method; only if it fails to produce a valid matching is the full MWM algorithm invoked, improving computational efficiency.
• Validates the new methods using test cases including Bain transformations, ideal crystal structures, and polycrystalline Ru simulations to assess classification bias and consistency.

Experimental results

Research questions

• RQ1 How can the Common Neighbour Analysis (CNA) be made more robust to atomic displacements and varying local length scales in molecular dynamics simulations?
• RQ2 What causes failure in existing CSP implementations, and how can the calculation of centrosymmetry be made continuous and physically meaningful under small atomic distortions?
• RQ3 Can a parameter-free, robust variant of CNA be developed that avoids misclassification due to threshold sensitivity?
• RQ4 Does the proposed Minimum-Weight Matching CSP method produce consistent and intuitive results across different levels of structural symmetry and disorder?
• RQ5 How do the new methods compare to conventional CNA and CSP in terms of classification accuracy, bias, and computational efficiency?

Key findings

• The Interval Common Neighbour Analysis (i-CNA) significantly improves structural recognition rates in disordered or perturbed local environments by searching over a range of neighbor thresholds instead of relying on a single fixed threshold.
• The i-CNA method successfully passes the Bain transformation test, demonstrating it is free from classification bias, unlike conventional CNA variants.
• The Minimum-Weight Matching Centrosymmetry Parameter (MWM-CSP) produces a continuous, stable, and intuitive CSP value that correctly reflects the degree of centrosymmetry, even under small atomic displacements.
• Greedy-based CSP methods (GES and GVM) fail to produce consistent results: GVM is discontinuous under rotation, and GES assigns similar CSP values to very different structures, leading to poor separation of centrosymmetric from non-centrosymmetric environments.
• In polycrystalline hcp Ru simulations, the MWM-CSP produces a single, broad peak in the CSP distribution, while GVM generates spurious secondary peaks and GES yields an artificially narrow peak, indicating poor resolution and overfitting.
• The hybrid MWM-CSP approach achieves ~30,000 CSP calculations per second on a single CPU thread, and by default uses the faster GES method with fallback to MWM only when needed, balancing speed and accuracy effectively.