Origins of Instability in Dynamical Systems on Undirected Networks

TL;DR

The paper investigates the onset of instability in large, sparse networked dynamical systems by analyzing the Jacobian spectrum of Barzel-Barabási-type dynamics. It develops a cavity-method approach within sparse random matrix theory to show that the rightmost eigenvalue is always an outlier and can be controlled by global Jacobian statistics or by anomalous nodes, with distinct localization regimes. The authors derive analytic outlier expressions using the cavity formalism and validate them across SIS epidemic dynamics and gene regulatory networks, demonstrating the practical utility of monitoring spectral outliers for network stability. This framework provides a principled way to predict, monitor, and potentially prevent catastrophic failures in complex networks by linking degree heterogeneity to stability signals.

Abstract

Robustness to perturbation is a key topic in the study of complex systems occurring across a wide variety of applications from epidemiology to biochemistry. Here we analyze the eigenspectrum of the Jacobian matrices associated to a general class of networked dynamical systems, which contains information on how perturbations to a stationary state develop over time. We find that stability is always determined by a spectral outlier, but with pronounced differences to the corresponding eigenvector in different regimes. We show that, depending on model details, instability may originate in nodes of anomalously low or high degree, or may occur everywhere in the network at once. Importantly, the dependence on extremal degrees results in considerable finite-size effects with different scaling depending on the ensemble degree distribution. Our results have potentially useful applications in network monitoring to predict or prevent catastrophic failures, and we validate our analytical findings through applications to epidemic dynamics and gene regulatory systems.

Figures (8)

• Figure 1: Stability of the dynamic Jacobian ensembles on a Barabási-Albert (BA) network with two different network sizes: $N=200$, and $N=1500$, with $\langle d \rangle=10$ and with a fixed minimum degree of $5$. The maximum degree is $\approx 53$ for $N=200$ and $179$ for $N = 1500$. In BA networks, increasing the network size results in a higher maximum degree, which consequently reduces the extent of the stable regime, as indicated by the bounded region between the vertical black lines. The numerical and theoretical $\lambda_{\rm max}$ have been plotted in the (Appendix \ref{['additional figures']}, and, Fig. \ref{['fig:enter-label']}), for three $\beta$ values: $-2, 0, 2$.
• Figure 2: Principal eigenvector of the dynamic Jacobian ensemble on a BA network of size $N=200$ (same BA graph as in Figure 1), with dynamical exponents $\mu=1,\nu=-1$ and $\rho=0$, for the fixed $\beta=0$ and three values of $\chi$ = -0.1, 0.01, 0.3 (left to right). The size of the circles is scaled with their degree, and the color is scaled with the value of the elements of the leading eigenvector, from high (red) to low (blue).
• Figure 3: The largest eigenvalue is plotted as a function of the infection parameter $\beta$ for the random network of SIS dynamics. The maximum degree of the network is $35$, the minimum degree is $10$, and the average degree ($\langle d \rangle$) is fixed at $20$. The black circles are obtained from the numerical data. The zero fixed point is stable until $0.05$, and the associated theoretical line is predicted by $\lambda_{\text{ext}}=\beta\langle d\rangle-1$, and marked by the red line. Beyond $\beta=0.05$, the largest localized (blue line) and delocalized (red line) eigenvalues were predicted by the Eqn. (\ref{['SIS_lambda']}).
• Figure 4: Eigenvalue boundary of the regulatory dynamics. The fixed points become imaginary at the green regime. In the blue regime, the fixed points are unstable as $\lambda_{\rm max}$ is positive. The lines are marked according to the theoretically predicted Eqns. (\ref{['lambda_zero_Regu']}-\ref{['lambda_i_Regu']}). In the blue and deep green regime, the system is unstable. The other colors (yellow to red) are marked according to the weighted sum of the degree vector and the leading eigenvector. In the deep red regime, the large degree ($d_{\rm max}$) plays the key role in determining the largest eigenvalue. In the yellow regime, the smallest degree ($d_{\rm min}$) determines the $\lambda_{\max}$. In the middle, the average degree ($\langle d \rangle$) plays the key role.
• Figure 5: SIS dynamics. (a)-(d) Stability ($\lambda_{\rm max}$) is plotted with a gradual increament of $\beta$. (e)-(h) The mean infection ($\langle x^* \rangle$) is reported as a function of $\beta$. $\langle x^* \rangle$ is obtained numerically (red dots in (e-h)) from the SIS model using the Runge-Kutta 4 algorithm. (a),(e) Network BA1 ($\langle d\rangle=6$) is used here. Here the critical onset of infection does not map with the theoretically obtained value (blue lines), but in higher infection, they almost match each other. (b), (f) BA network with average $\langle d\rangle =12$. The theoretical and analytical results are closely matched here. In (c), (g), and (d), (h) networks ER1 and ER2 are used here. (a)-(d) The black circles represent results from numerical simulations, while the red and blue lines correspond to analytical estimates using the average and minimum degrees of the network, respectively, based on $\lambda_o$ and $\lambda_i$ in Eqn. \ref{['SIS_lambda']} in the main text. Also see Eqn. (\ref{['sis_z0']}), and (\ref{['sis_zi']}) of Appendix. \ref{['SIS']}.