{
  "abstract": "We study a discrete-time stochastic model for the spiking activity of a network of $n$ interacting neurons partitioned into $k$ communities. The interaction network is a random directed weighted graph generated by a Stochastic Block Model with $k+1$ edge types ($k$ intra-community, $1$ inter-community). Each edge is present independently with a probability that depends only on its type. When present, an edge carries a type-dependent deterministic weight scaled by $n^{-1}$, which is excitatory or inhibitory depending on an independent random sign. Given the interaction network, the spiking process is a stationary Markov chain. Our results describe how the network's community structure manifests itself in the first- and second-order statistics of this chain, as $n \\to \\infty$ with $k$ fixed. Representing these statistics as explicit functions of the random interaction network, we prove that, with high probability: (i) the vectors of mean firing activity and firing variances are approximated in supremum norm by deterministic vectors that are constant within each community, with an error of order $\\sqrt{\\log(n)/n}$; (ii) the off-diagonal part of the $0$-lagged firing covariance matrix is approximated in max norm, with an error of order $\\sqrt{\\log(n)/n^3}$, by a deterministic community-block matrix (constant on each block defined by a pair of communities), with block values determined by the unique solution to a $k \\times k$ Stein matrix equation; (iii) in contrast, the $t$-lagged covariance matrix, for every lag $t \\ge 1$, is approximated in max norm, again with an error of order $\\sqrt{\\log(n)/n^3}$, by a random matrix built from two components: a random matrix capturing the interaction network and a deterministic community-block matrix. Finally, we discuss conditions on the model parameters under which these limiting statistics enable the identifiability of the community structure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41706",
  "author_total": 5,
  "authors": [
    {
      "name": "Thiago H. N. Coelho"
    },
    {
      "name": "Daniel Figueiredo"
    },
    {
      "name": "Giulio Iacobelli"
    },
    {
      "name": "Guilherme Ost"
    },
    {
      "name": "Conrado C. Pinto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7gt4yv9jb2y3s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41706",
  "fulltext_url": null,
  "id": "7gt4yv9jb2y3s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41706",
  "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",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41706",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7gt4yv9jb2y3s.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41706",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41706"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "First and second order firing statistics of large neuronal networks with community structure",
  "title_text": "First and second order firing statistics of large neuronal networks with community structure",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/7gt4yv9jb2y3s",
  "venue": "arXiv",
  "versions": []
}