Properties of networks with partially structured and partially random connectivity

摘要

We provide a general formula for the eigenvalue density of large random$N\times N$ matrices of the form $A = M + LJR$, where $M$, $L$ and $R$ arearbitrary deterministic matrices and $J$ is a random matrix of zero-meanindependent and identically distributed elements. For $A$ nonnormal, theeigenvalues do not suffice to specify the dynamics induced by $A$, so we alsoprovide general formulae for the transient evolution of the magnitude ofactivity and frequency power spectrum in an $N$-dimensional linear dynamicalsystem with a coupling matrix given by $A$. These quantities can also bethought of as characterizing the stability and the magnitude of the linearresponse of a nonlinear network to small perturbations about a fixed point. Wederive these formulae and work them out analytically for some examples of $M$,$L$ and $R$ motivated by neurobiological models. We also argue that thepersistence as $N\rightarrow\infty$ of a finite number of randomly distributedoutlying eigenvalues outside the support of the eigenvalue density of $A$, aspreviously observed, arises in regions of the complex plane $\Omega$ wherethere are nonzero singular values of $L^{-1} (z\mathbf{1} - M) R^{-1}$ (for$z\in\Omega$) that vanish as $N\rightarrow\infty$. When such singular values donot exist and $L$ and $R$ are equal to the identity, there is a correspondencein the normalized Frobenius norm (but not in the operator norm) between thesupport of the spectrum of $A$ for $J$ of norm $\sigma$ and the$\sigma$-pseudospectrum of $M$.