the network-size, since all spatial effects induced by the coupling-structure are contained in the delayed terms. The reduced characteristic equation for the strong spectrum can be formally obtained by setting $e^{-\lambda\tau}e^{iq_{+}}=0$ in (\[eq: SLststChEq-1\]): $$0=\lambda^{2}+2(2a^{2}-\alpha)\lambda+2a^{2}(a^{2}-\alpha)+R^{2}$$ Its solutions are $$\lambda_{\pm}=\alpha-2a^{2}\pm\sqrt{a^{4}+(a^{2}-\alpha)^{2}-R^{2}}.%\label{eq: SLpsolStrongEV}$$ Any of the solutions $\lambda_{\pm}$ with positive real part belongs to the strong spectrum. Simple calculations show that there exists at least one strong eigenvalue with positive real part if $$a^{2}<a_{S}^{2}(\alpha;R)=\begin{cases}
\alpha/2 & \mbox{for }-|R|\leq\alpha\leq\sqrt{2}|R|\\
\frac{1}{2}\left(\alpha+\sqrt{\alpha^{2}-2R^{2}}\right) & \mbox{for }\sqrt{2}|R|<\alpha,
\end{cases}$$ i.e. if the amplitude of the plane wave is smaller than the one determined by the curve $a_{S}(\alpha,R)$. Moreover, when the inequality $a^{4}+(a^{2}-\alpha)^{2}-R^{2}<0$ is satisfied, there are two complex conjugate unstable eigenvalues $\lambda_{+}=\lambda_{-}^{\ast}\in\mathbb{C}$ with $\operatorname{Re} \lambda_{\pm}=\alpha-2a^{2}$. The bifurcation diagram in Fig. \[fig:SLBD\] illustrates the regions of strong instability of plane waves (dark gray regions, labeled with $S$).

### Pseudo-continuous (weak) spectrum {#pseudo-continuous-weak-spectrum .unnumbered}

Besides the strong spectrum, there are infinitely many eigenvalues in the *weak* or *pseudo-continuous spectrum*. Similarly to Sec. \[sub:ststLong\], this spectrum can be found by substituting the ansatz $$\lambda=\frac{\gamma(\omega)}{\tau}+i\omega$$ into the characteristic equation (\[eq: SLpsolCE\]). In the limit of large delay, the terms of the order $\mathcal{O}(1/\tau)$ can be neglected, resulting in the following equation $$0=S(q_{-})Y^{2}-2\left[A(\omega,q_{-})+iB(\omega,q_{-})\right]Y+D(\omega)+iE(\omega),\label{eq:psolY}$$ with 