Almost Sure Stability of Complex-Valued Complex Networks: A Noise-Based Delayed Coupling Under Random Denial-of-Service Attacks

This article is concerned with stability for stochastic complex-valued delayed complex networks under random denial-of-service (RDoS) attacks. Different from the existing literature on the stability of stochastic complex-valued systems that concentrate on moment stability, we investigate almost sure stability (ASS), where noise plays a stabilizing role. It is noted that, besides the vertex systems influenced by noise, the interactions among vertices are also at the mercy of noise. As a consequence, an innovative noise-based delayed coupling (NDC) in the presence of RDoS attacks is proposed first to accomplish the stability of complex-valued networks, where the RDoS attacks have a certain probability of triumphantly interfering with communications at active intervals of attackers. Namely, RDoS attacks considered are randomly launched at active periods, which is more realistic. In terms of the Lyapunov method and stochastic analysis theory, an almost sure exponential stability (ASES) criterion of the system discussed straightforwardly is developed by constructing a delay-free auxiliary system, while removing the traditional assumption of moment stability. The criterion strongly linked with topological structure, RDoS frequency, attack successful probability, and noise intensity reveals that the higher the noise intensity, the faster the convergence rate is for the stability of the network. In light of the criterion established, we present an algorithm that can be employed to analyze the tolerable attack parameters and the upper bound of the coupling delays, under the prerequisite of guaranteeing the stability of the network. Eventually, the theoretical results are applied to inertial complex-valued neural networks (ICNNs) and an illustrative example is presented to substantiate the efficiency of the theoretical works.