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Stay or Stray - A Dynamical Systems Viewpoint of Popularity Bias
Sarvesh Shashidhar, Lankireddy Prabhat, Arpit Agarwal, D. Manjunath, Karan Bhukar, Tanmay Khandelwal
https://arxiv.org/abs/2608.10474 https://arxiv.org/pdf/2608.10474 https://arxiv.org/html/2608.10474
arXiv:2608.10474v1 Announce Type: new
Abstract: Popularity bias in recommendation systems arises when a majority user class generates disproportionate interaction data, causing the system to increasingly favour it while degrading recommendation quality for niche users. While extensive empirical evidence of popularity bias exists, the dynamics leading to its emergence are not well understood. In this work, we study the coupled evolution of recommender model updates and user engagement through the lens of dynamical systems. We formulate a stochastic process and analyse its asymptotic behaviour through an ordinary differential equation (ODE) framework grounded in two-time-scale stochastic approximation. We characterise the equilibrium points of this dynamical system, and derive conditions under which popularity bias is provably emergent, as well as conditions under which symmetric retention of all user classes is possible. We conduct experiments on synthetic data and real-world production logs derived from a large-scale commercial music recommendation platform to validate our theoretical results.
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Towards Relating Ciao Assertions and LPTP Theorems
Marco P\'erez (IMDEA Software Institute, Madrid, Spain), Pedro L\'opez-Garc\'ia (IMDEA Software Institute, Madrid, Spain), Jose F. Morales (IMDEA Software Institute, Madrid, Spain), Manuel V. Hermenegildo (IMDEA Software Institute, Madrid, Spain), Fred Mesnard (Universit\'e de La R\'eunion)
https://arxiv.org/abs/2607.20249 https://arxiv.org/pdf/2607.20249 https://arxiv.org/html/2607.20249
arXiv:2607.20249v1 Announce Type: new
Abstract: Abstract interpretation-based verification is a central component of the Ciao Prolog system, enabling expressive specifications of properties of programs, predicates, and execution states. Independently, the LPTP (Logic Programming Theorem Proving) framework offers a first-order logical formalism for expressing and proving properties of predicates. In this paper, we address a fundamental issue in relating these two frameworks: studying the translation of Ciao assertions into LPTP formulae and identifying a partial correspondence between assertion-based and logic-based specifications. We introduce a systematic translation scheme, characterize assertion classes according to their logical encodability, and propose approximation strategies and auxiliary constructs for non-translatable cases, and finally analyze the resulting soundness and completeness trade-offs. We argue that our proposal enables a tight integration of Ciao's assertion checking with LPTP-based deductive verification, thereby leveraging their complementary capabilities.
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Integrability-breaking phase transitions in stadium-like billiards
Anne K\'etri P. da Fonseca, Edson D. Leonel
https://arxiv.org/abs/2607.16482 https://arxiv.org/pdf/2607.16482 https://arxiv.org/html/2607.16482
arXiv:2607.16482v1 Announce Type: new
Abstract: We investigate integrability-breaking transitions in two classes of stadium-like billiards with parabolic boundaries. While focusing boundaries generate a mixed phase space in which regular islands coexist with a chaotic sea, dispersing boundaries produce a fully chaotic phase space for any finite boundary deformation. By analyzing the scaling behavior of the roughness $\omega$, we identify two qualitatively distinct transitions: a continuous transition for the focusing geometry and a first-order transition for the dispersing one. We determine the corresponding critical exponents and establish the associated scaling laws. For the continuous transition, we further provide a complete characterization within the framework of critical phenomena by identifying the broken symmetry, the order parameter and its diverging susceptibility, the elementary excitations responsible for chaotic diffusion, and the topological defects governing transport. These results establish a statistical-mechanics framework for describing integrability-breaking transitions in Hamiltonian billiards and suggest that the concepts of critical phenomena naturally extend to deterministic nonlinear dynamical systems.
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