{
  "annotation_id": "eb83a5fa-5c08-427b-9a30-b314d7f8afc7",
  "date_created": "2026-03-02T18:01:18.087000Z",
  "date_modified": "2026-03-02T18:01:18.087000Z",
  "file_hash": "3070ccec8c46a98932ac0d1288b222f59f77c0ece6603197b888e4aab37bb14b",
  "private": false,
  "record": {
    "abstract": "We study the betweenness centrality of fractal and non-fractal scale-free\nnetwork models as well as real networks. We show that the correlation between\ndegree and betweenness centrality $C$ of nodes is much weaker in fractal\nnetwork models compared to non-fractal models. We also show that nodes of both\nfractal and non-fractal scale-free networks have power law betweenness\ncentrality distribution $P(C)\\sim C^{-\\delta}$. We find that for non-fractal\nscale-free networks $\\delta = 2$, and for fractal scale-free networks $\\delta =\n2-1/d_{B}$, where $d_{B}$ is the dimension of the fractal network. We support\nthese results by explicit calculations on four real networks: pharmaceutical\nfirms (N=6776), yeast (N=1458), WWW (N=2526), and a sample of Internet network\nat AS level (N=20566), where $N$ is the number of nodes in the largest\nconnected component of a network. We also study the crossover phenomenon from\nfractal to non-fractal networks upon adding random edges to a fractal network.\nWe show that the crossover length $\\ell^{*}$, separating fractal and\nnon-fractal regimes, scales with dimension $d_{B}$ of the network as\n$p^{-1/d_{B}}$, where $p$ is the density of random edges added to the network.\nWe find that the correlation between degree and betweenness centrality\nincreases with $p$.",
    "arxiv_id": "physics/0702001",
    "authors": [
      "Maksim Kitsak",
      "Shlomo Havlin",
      "Gerald Paul",
      "Massimo Riccaboni",
      "Fabio Pammolli",
      "H. Eugene Stanley"
    ],
    "categories": [
      "physics.soc-ph",
      "physics.data-an"
    ],
    "doi": "10.1103/PhysRevE.75.056115",
    "title": "Betweenness Centrality of Fractal and Non-Fractal Scale-Free Model   Networks and Tests on Real Networks",
    "url": "https://arxiv.org/abs/physics/0702001"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "4e26bcd1-9c06-4d24-8828-4c335f426df9",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}