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Structural crossover of complex networks: bridging degree correlation and fractality

  • Shogo Mizutaka
  • Jun Yamamoto
  • Kousuke Yakubo
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Abstract

We investigate the relationship between long-range degree correlations and fractality in scale-free random networks. By analyzing degree correlations in scale-free random networks at the percolation threshold, at which a giant component emerges, we derive the ball volume ν̃k(l), defined as the average number of nodes within distance l from a root node with degree k. The resulting expression predicts a degree-dependent structural crossover. For l≪ k1/(df−1), the ball volume grows as ν̃k(l)∼ kl, whereas for l≫ k1/(df−1), it crosses over to the global fractal scaling ν̃k(l)∼ ldf. Here, df is the fractal dimension of critical scale-free random networks. We further show that the structural crossover can be naturally understood in terms of critical branching processes and is also observed in empirical fractal networks.

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Cite

BibTeXRIS
BibTeX
@misc{mizutaka2026structural,
  title = {{Structural crossover of complex networks: bridging degree correlation and fractality}},
  author = {Shogo Mizutaka and Jun Yamamoto and Kousuke Yakubo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02883},
  eprint = {2610.02883},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02883},
}
RIS
TY  - UNPB
TI  - Structural crossover of complex networks: bridging degree correlation and fractality
AU  - Shogo Mizutaka
AU  - Jun Yamamoto
AU  - Kousuke Yakubo
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.02883
UR  - https://arxiv.org/abs/2610.02883
AB  - We investigate the relationship between long-range degree correlations and fractality in scale-free random networks. By analyzing degree correlations in scale-free random networks at the percolation threshold, at which a giant component emerges, we derive the ball volume ν̃_k(l), defined as the average number of nodes within distance l from a root node with degree k. The resulting expression predicts a degree-dependent structural crossover. For l≪ k^1/(df−1), the ball volume grows as ν̃_k(l)∼ kl, whereas for l≫ k^1/(df−1), it crosses over to the global fractal scaling ν̃_k(l)∼ l^df. Here, df is the fractal dimension of critical scale-free random networks. We further show that the structural crossover can be naturally understood in terms of critical branching processes and is also observed in empirical fractal networks.
ER  -