For inhomogeneous random matrices, the spectral radius is bounded by the variance row/column sums up to the optimal sparsity (log n)^{-1/2}.
The stability–complexity relationship at age 40: a random matrix perspective
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Spectral radius concentration for inhomogeneous random matrices with independent entries
For inhomogeneous random matrices, the spectral radius is bounded by the variance row/column sums up to the optimal sparsity (log n)^{-1/2}.