This paper examines the relationship between sparse random network architectures and neural network stability by examining the eigenvalue spectral distribution. Specifically, we generalize classical eigenspectral results to sparse (not fully connected) connectivity matrices obeying Dale's law: neurons function as either excitatory (E) or inhibitory (I). By defining α as the probability that a neuron is connected to another neuron, we give explicit formulas that show how sparsity interacts with the E-I population statistics to scale key features of the eigenspectrum in both the balanced and unbalanced cases. Our results show that the eigenspectral outlier is linearly scaled by α, but the eigenspectral radius and density now depend on a nonlinear interaction between α and the E-I population means and variances. Contrary to previous results, we demonstrate that a nonuniform eigenspectral density results if any of the E-I population statistics differ, not just the variances. We also find that local eigenvalue outliers are present for sparse random matrices obeying Dale's law, and demonstrate that these eigenvalues can be controlled by a modified zero row-sum constraint for the balanced case, however, they persist in the unbalanced case. We examine all levels of connection sparsity 0≤α≤1 and distributed E-I population weights to describe a general class of sparse connectivity structures which unifies all the previous results as special cases of our framework. Sparsity and Dale's law are both fundamental anatomical properties of biological neural networks. We generalize their combined effects on the eigenspectrum of random neural networks, thereby gaining insight into network stability, state transitions, and the structure-function relationship.
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