Spectrum Optimization of Dynamic Networks for Reduction of Vulnerability Against Adversarial Resonance Attacks

TL;DR

The paper addresses vulnerability to adversarial resonance in networks with second-order dynamics by introducing a formal vulnerability measure under a stochastic forcing model. It develops two spectrum-based mitigation approaches: Network Graph Optimization, which redistributes edge weights to flatten the Laplacian spectrum, and Auxiliary Graph Optimization, which attaches a tunable auxiliary network to absorb resonant energy. The authors derive closed-form expressions for vulnerability, demonstrate convexity regimes, and validate the methods through extensive simulations on synthetic and social networks, including robot networks. The results show substantial reductions in resonance amplitudes and highlight practical considerations such as damping, connectivity, and the role of spectrum flattening, with implications for security and robustness in power systems, robotics, and social networks.

Abstract

Resonance is a well-known phenomenon that happens in systems with second order dynamics. In this paper we address the fundamental question of making a network robust to signal being periodically pumped into it at or near a resonant frequency by an adversarial agent with the aim of saturating the network with the signal. Towards this goal, we develop the notion of network vulnerability, which is measured by the expected resonance amplitude on the network under a stochastically modeled adversarial attack. Assuming a second order dynamics model based on the network graph Laplacian matrix and a known stochastic model for the adversarial attack, we propose two methods for minimizing the network vulnerability through optimization of the spectrum of the network graph. We provide extensive numerical results analyzing the effects of both methods.

Figures (17)

• Figure 1: Illustration of a network being attacked by an adversarial agent trying to cause resonance.
• Figure 2: Cauchy distributions centered at the natural frequencies $\omega_1=1$, $\omega_2=2$, and $\omega_3 = 4$ with a spread of $h=0.5$. The probability density function $\rho(\nu)= \frac{1}{3} \sum\limits_{i=1}^3 \rho_{\omega_i}(\nu)$ for the adversarial agent's choice of forcing frequency is obtained as the uniformly weighted sum of the Cauchy distributions each of which are centered at the natural frequencies of the network.
• Figure 3: Integration contour for equation \ref{['eq:g-interation']}. Poles $\nu_1$ to $\nu_4$ correspond to the forcing vector component $\mathbb{E}_{\mathbf{f}} \left(\|\mathbf{x}_s\|^2_2 \right)$ and they collapse on to the real line as $\gamma$ goes to zero. Poles $\nu_5$ and $\nu_6$ correspond to the forcing frequency component $\rho(\nu)$.
• Figure 4: Histograms of the Laplacian matrix eigenvalues for the initial network graph ${G}$ and optimized network graph ${G}^*$. The initial network is modeled by a complete graph, whose edge weights are perturbed away from a uniform distribution by a small amount. The corresponding spectrum (on the left) is peaky, whereas as a result of the spectrum optimization, the spectrum (on the right) has become flatter.
• Figure 5: Illustration of an auxiliary graph ${\widetilde{G}}$ attached to the original graph ${G}$ with an aim to decrease vulnerability against adversarial attacks. The auxiliary graph is of type mirrored (has the same connectivity as the main graph). Green lines indicate the inter-graph connections with weights $c$.

Theorems & Definitions (10)

• Definition 1: Network Vulnerability to Adversarial Resonance Attack
• Proposition 1: Network vulnerability
• Lemma 1
• Lemma 2
• proof
• Proposition 2
• proof
• Proposition 3: Network vulnerability with attached auxiliary network
• proof
• proof