Origins of Instability in Dynamical Systems on Undirected Networks

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.

💡 Research Summary

This paper investigates the origins of instability in nonlinear dynamical systems defined on undirected, sparse networks. The authors focus on a broad class of systems described by the Barzel-Barabási family of equations, where the dynamics of a node depend on a self-interaction term and a sum of pairwise interactions modulated by the network adjacency matrix.

The core analytical effort revolves around studying the eigenspectrum of the Jacobian matrix linearized around a stationary state. The Jacobian exhibits a specific structure (J_ii = a_i, J_ij = A_ij b_i c_j) dictated by the underlying dynamics and network structure. By combining techniques from sparse random matrix theory and the cavity method, the authors derive analytical expressions for the spectral density.

A key finding is that the linear stability of the system is always governed by a spectral outlier eigenvalue lying outside the continuous bulk of the spectrum. This contrasts with some classical results from random matrix theory. More importantly, the nature of this stability-determining outlier can be distinctly categorized:

1. Delocalized Outliers: The corresponding eigenvector is spread across the entire network. In this regime, instability (or stability) is a collective phenomenon driven by the average properties of the network (e.g., mean degree) and dynamical parameters. The leading eigenvalue is given by λ_o ≈ ⟨a⟩ + ⟨b⟩⟨c⟩⟨d⟩.
2. Localized Outliers: The eigenvector is highly concentrated on a single node or a few nodes. Here, stability becomes sensitive to extremal nodes in the network, typically those with the minimum or maximum degree. The instability originates locally from these nodes. The expression for the eigenvalue λ_i is more complex and depends directly on the parameters (a_i, d_i) of the specific node.

The transition between these regimes is controlled by the detailed parameters of the dynamical model. The authors formalize this using a “dynamic Jacobian ensemble” characterized by exponents (µ, ν, ρ), presenting a phase diagram that shows how the stability boundary shifts between localized (on max/min degree nodes) and delocalized predictions.

The generality and practical relevance of the framework are demonstrated through applications to two well-known nonlinear systems:

• SIS Epidemic Model: Analysis of the endemic state reveals that for high infection rates, the least stable mode localizes on the nodes with the minimum degree, identifying them as potential origins of disease fade-out or resurgence.
• Gene Regulatory Network (Michaelis-Menten dynamics): The analysis uncovers a complex stability phase diagram for a biochemical system, showcasing the framework’s applicability beyond simple models.

The work highlights significant finite-size effects, where the stability of finite networks depends strongly on the actual maximum/minimum degrees, which scale with network size according to the degree distribution. This provides crucial guidance for interpreting results from finite network simulations or real-world data. Overall, the paper offers a refined lens to predict where and how instability might emerge in complex networked systems, with implications for monitoring and preventing large-scale failures in biological, social, and technological networks.