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PreprintCC BY 4.0arXiv

Spectrum of Directed Inhomogeneous Random Graphs

  • Rajat Subhra Hazra
  • Giacomo Passuello
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Abstract

We study the spectrum of the adjacency matrix An of directed inhomogeneous random graphs on n vertices. We assume that An has independent entries and diverging average degree scale sn. This framework includes, as special cases, the directed Chung--Lu random graph and directed stochastic block models. Assuming boundedness of the variance profile and that sn diverges faster than log(n)2, we show that the rank-one Chung--Lu model satisfies a non-homogeneous version of the circular law, which in some situations allows for an explicit expression. Moreover, under mild conditions, we identify the asymptotic singular value distribution using tools from free probability. Finally, for finite-rank directed models, we establish that eigenvalues outside the bulk have joint Gaussian fluctuations at the scale √(sn/n), with an explicit covariance matrix. These results extend the theory of spectral outliers and their fluctuations to directed inhomogeneous random graphs.

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Cite

BibTeXRIS
BibTeX
@misc{hazra2026spectrum,
  title = {{Spectrum of Directed Inhomogeneous Random Graphs}},
  author = {Rajat Subhra Hazra and Giacomo Passuello},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.08696},
  eprint = {2607.08696},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.08696},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Spectrum of Directed Inhomogeneous Random Graphs
AU  - Rajat Subhra Hazra
AU  - Giacomo Passuello
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2607.08696
UR  - https://arxiv.org/abs/2607.08696
AB  - We study the spectrum of the adjacency matrix A_n of directed inhomogeneous random graphs on n vertices. We assume that A_n has independent entries and diverging average degree scale s_n. This framework includes, as special cases, the directed Chung--Lu random graph and directed stochastic block models. Assuming boundedness of the variance profile and that s_n diverges faster than log(n)^2, we show that the rank-one Chung--Lu model satisfies a non-homogeneous version of the circular law, which in some situations allows for an explicit expression. Moreover, under mild conditions, we identify the asymptotic singular value distribution using tools from free probability. Finally, for finite-rank directed models, we establish that eigenvalues outside the bulk have joint Gaussian fluctuations at the scale √(s_n/n), with an explicit covariance matrix. These results extend the theory of spectral outliers and their fluctuations to directed inhomogeneous random graphs.
ER  -