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Scale-Invariant Dynamics in a Purely Deterministic Game of Life Model

ArXiv, 2025-June

Abstract: Scale invariance is a key feature that characterizes criticality in complex dynamical systems, which often organize into structures exhibiting no typical size and/or lifespan. While random external inputs or tunable stochastic interactions are typically required for showcasing such criticality, the question of whether scale-invariant dynamics can emerge from purely deterministic interactions remains unclear. In this work, we discover highly affirmative signatures of critical dynamics in equal-state clusters that emerge in the \(\textit{logistic}\) Game of life (LGOL): an extension of Conway’s GOL into a Cantor set state space that is nevertheless deterministic. We uncover at least three types of asymptotic behavior, i.e. phases, that are separated by two fundamentally distinct critical points. The first critical point — associated with a peculiar form of self-organized criticality — defines the non-analytic boundary between a sparse-static and a sparse-dynamic asymptotic phase. Meanwhile, the second point marks an enigmatic deterministic percolation transition between the sparse-dynamic and a third, dense-dynamic phase. Moreover, we identify distinct power-law distributions of cluster sizes with unconventional critical exponents that challenge the current paradigms for critical behavior. Overall, our work concretely paves the way for studying emergent scale invariance in purely deterministic systems.

Recommended citation: Akgun, Hakan, Xianquan Yan, Tamer Taskiran, Muhamet Ibrahimi, Ching Hua Lee, and Seymur Jahangirov. “Scale-Invariant Dynamics in a Purely Deterministic Game of Life Model.” arXiv, June 7, 2025. https://doi.org/10.48550/arXiv.2411.07189.
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HSG-12M: A Large-Scale Spatial Multigraph Dataset

ArXiv, 2025-June

Abstract: Existing graph benchmarks assume non-spatial, simple edges, collapsing physically distinct paths into a single link. We introduce HSG-12M, the first large-scale dataset of \(\textbf{spatial multigraphs}\) — graphs embedded in a metric space where multiple geometrically distinct trajectories between two nodes are retained as separate edges. HSG-12M contains 11.6 million static and 5.1 million dynamic \(\textit{Hamiltonian spectral graphs}\) across 1401 characteristic-polynomial classes, derived from 177 TB of spectral potential data. Each graph encodes the full geometry of a 1-D crystal’s energy spectrum on the complex plane, producing diverse, physics-grounded topologies that transcend conventional node-coordinate datasets. To enable future extensions, we release \(\href{https://github.com/sarinstein-yan/Poly2Graph}{\texttt{Poly2Graph}}\): a high-performance, open-source pipeline that maps arbitrary 1-D crystal Hamiltonians to spectral graphs. Benchmarks with popular GNNs expose new challenges in learning from multi-edge geometry at scale. Beyond its practical utility, we show that spectral graphs serve as universal topological fingerprints of polynomials, vectors, and matrices, forging a new algebra-to-graph link. HSG-12M lays the groundwork for geometry-aware graph learning and new opportunities of data-driven scientific discovery in condensed matter physics and beyond.

Recommended citation: Yan, Xianquan, Hakan Akgün, Kenji Kawaguchi, N. Duane Loh, and Ching Hua Lee. “HSG-12M: A Large-Scale Spatial Multigraph Dataset.” arXiv, 2025. https://doi.org/10.48550/ARXIV.2506.08618.
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Precise Identification of Topological Phase Transitions with Eigensystem-Based Clustering

AI4X 2025 International Conference, 2025-July

Abstract: Recent advances in machine learning have spurred new ways to explore and classify quantum phases of matter. We propose an \(\textit{Eigensystem-based}\) representation, combined with a Gaussian Mixture Model (GMM), to unsupervisedly cluster Hamiltonians into distinct topological phases with minimal feature engineering. The method identifies different topological phases without any prior knowledge, pinpoints phase boundaries with remarkable precision \(\sim\mathcal{O}(10^{-5})\), remains robust under moderate noise, and scales efficiently via a simple dimensionality-reduction step. The success of GMM offers a novel physical insight — each phase forms a well-separated multivariate Gaussian in a high-dimensional “Eigensystem space.” We illustrate the approach on several 1D lattice models, all achieving near 100\% accuracies.

Recommended citation: Yan, Xianquan, and Jian-Song Pan. “Precise Identification of Topological Phase Transitions with Eigensystem-Based Clustering,” 2025.
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Automated Metric Discovery: Navigating Quantum Geometry with Symbolic Regression

AI4X 2025 International Conference, 2025-July

Abstract: I demonstrated how symbolic regression can \(\textit{automatically learn}\) quantum geometric metrics from numerical data, recovering known formulas (e.g. Fubini-Study and Bures) in a transparent, closed-form fashion. Beyond these canonical examples, the same approach opens the door to exploring \(\textit{customized}\) quantum metrics that maximize a problem-specific objective.

Recommended citation: Yan, Xianquan. “Automated Metric Discovery: Navigating Quantum Geometry with Symbolic Regression,” 2025.
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