Critical Percolation as a Synthetic Data Model for Interpretability

A new family of synthetic datasets built on critical percolation clusters has been proposed to serve as a more realistic testbed for evaluating neural network interpretability methods, according to a paper submitted to arXiv on 18 Jun 2026 [1]. The model, introduced in a preprint posted to the open-access repository arXiv, generates hierarchical functions defined on critical mean-field percolation clusters embedded in a high-dimensional data space [1]. The resulting data consists of sparse, low-dimensional fractal clusters with a power-law size distribution [1]. The authors note that the model is analytically tractable, with known critical exponents that fix its properties without requiring hyperparameter tuning [1]. This stands in contrast to many existing synthetic datasets, which the paper argues lack the hierarchical, multi-scale structure characteristic of natural data [1]. A key contribution is a proposed algorithm for data generation. By leveraging a mapping between percolation clusters, random trees, and additive coalescence, the researchers outline an almost linear-time method to jointly sample a random tree and its hierarchical latent decomposition, enabling data generation at arbitrary scale [1]. Probing experiments detailed in the paper found that the model's ground-truth latent variables could be linearly decoded from neural network activations [1]. The work arrives as the arXiv platform itself continues to scale. As of November 2024, the submission rate to the repository was about 24,000 articles per month, and the total corpus had surpassed two million articles by the end of 2021 [9]. The paper was posted under the Machine Learning category and is accessible through the standard abstract page, which also features a series of experimental community tools under the arXivLabs framework [1][6]. arXivLabs, launched as a formalized framework in 2020, allows individuals and organizations to develop and share new features directly on the article record page [7]. These integrations, which appear as tabs below the abstract, include tools such as the Bibliographic Explorer for navigating citation trees and the CORE Recommender for discovering related open-access papers [8]. The framework operates under guidelines that require partners to share arXiv's values of openness, community, excellence, and user data privacy, with third-party collaborators receiving only minimal and anonymized user data [7]. The arXiv team has temporarily paused new Labs proposals while it focuses on modernizing and migrating its systems to the cloud, though this hiatus does not affect existing Labs or proposals already submitted [6]. The concept of emergence, where a complex entity exhibits properties its parts do not have on their own, is central to the behavior of such complex systems [4]. The new percolation model's combination of sparsity, self-similarity, power-law statistics, and analytical tractability is intended to make it a principled testbed for interpretability research [1].