{
  "abstract": "In machine learning, a self-attention dynamics describes the attention mechanism of transformers, and is obtained treating the layer index of a transformer as time index. Even though naturally discrete-time, this dynamical system is most often studied in continuous-time. Scope of this paper is to provide a thorough investigation of the discrete-time version of the self-attention dynamics: a complete classification of its equilibria is given, and their stability properties are studied, in both the single-head and the multihead case. We show that in both cases the main attractor of the dynamics is a consensus equilibrium aligned with the principal eigenvector of the value matrix (in the single-head case, a corresponding matrix of weights in the multihead case), even though other locally asymptotically stable bipartite equilibria can be show to exist at least in the single-head situation. The behavior is confirmed by looking at the learned weight matrices of a real transformer.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13518",
  "author_total": 1,
  "authors": [
    {
      "name": "Claudio Altafini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9830xkpx84xc2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13518",
  "fulltext_url": null,
  "id": "9830xkpx84xc2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13518",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.SY"
    ],
    [
      "arxiv",
      "eess.SY"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7221572542",
  "pdf_url": "https://arxiv.org/pdf/2610.13518",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9830xkpx84xc2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13518",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13518"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Asymptotic Analysis of Discrete-Time Self-Attention Dynamics in Transformers",
  "title_text": "Asymptotic Analysis of Discrete-Time Self-Attention Dynamics in Transformers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/9830xkpx84xc2",
  "venue": "arXiv",
  "versions": []
}