Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples - adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erdős-Rényi graphs, and adjacency matrices of weighted oriented Erdős-Rényi graphs - we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors.

Original languageEnglish
Article number434003
JournalJournal of Physics A: Mathematical and Theoretical
Issue number43
Early online date26 Apr 2019
Publication statusPublished - 1 Oct 2019

Bibliographical note

Invited Topical Review


  • cond-mat.stat-mech
  • cond-mat.dis-nn
  • math-ph
  • math.MP
  • non-Hermitian matrices
  • random matrix theory
  • complex networks
  • sparse matrices

ASJC Scopus subject areas

  • General Physics and Astronomy
  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Mathematical Physics
  • Modelling and Simulation


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