Properties of networks with partially structured and partially random connectivity

We provide a general formula for the eigenvalue density of large random N× N matrices of the form A = M + LJR, where M, L and R are arbitrary deterministic matrices and J is a random matrix of zero-mean independent and identically distributed elements. For A nonnormal, the eigenvalues do not suffice to specify the dynamics induced by A, so we also provide general formulae for the transient evolution of the magnitude of activity and frequency power spectrum in an N-dimensional linear dynamical system with a coupling matrix given by A. These quantities can also be thought of as characterizing the stability and the magnitude of the linear response of a nonlinear network to small perturbations about a fixed point. We derive these formulae and work them out analytically for some examples of M, L and R motivated by neurobiological models. We also argue that the persistence as Nrightarrow∞ of a finite number of randomly distributed outlying eigenvalues outside the support of the eigenvalue density of A, as previously observed, arises in regions of the complex plane Omega where there are nonzero singular values of L-1 (z1 - M) R-1 (for z∈Omega) that vanish as Nrightarrow∞. When such singular values do not exist and L and R are equal to the identity, there is a correspondence in the normalized Frobenius norm (but not in the operator norm) between the support of the spectrum of A for J of norm σ and the σ-pseudospectrum of M.