This paper is concerned with almost sure exponential synchronization of multilayer complex networks with Markovian switching via aperiodically intermittent discrete observation noise. First, Markovian switching and multilayer interaction factors are taken into account simultaneously, which make our system more general compared with the existing literature. Meanwhile, the network architecture may be undirected or directed, and consequently, the adjacency matrix is symmetrical and asymmetrical. Second, the control strategy is based on aperiodically intermittent discrete observation noise, where the average control rate is integrated to depict the distributions of work/rest intervals of the control strategy from an overall perspective. Third, different from the work about $p$th moment exponential synchronization of network systems, by utilizing M-matrix theory and various stochastic analysis techniques including the Itô formula, the Gronwall inequality, and the Borel-Cantelli lemma, some criteria on almost sure exponential synchronization of multilayer complex networks with Markovian switching have been constructed and the upper bound of the duration time has been also estimated. Finally, several numerical simulations are exhibited to validate the effectiveness and feasibility of our analytical findings.

This paper rigorously investigates the stochastic stability of time-varying neutral stochastic pantograph systems with Markovian switching. Primarily, multiple proportional delays and time-varying coefficients are taken into account in the neutral hybrid stochastic systems. In order to surmount the complexities stemming from those factors, the Lyapunov functional approach and several stochastic analytical techniques are utilized, and a modified version of pantograph delays integral inequality is developed. Moreover, various stability criteria such as $q$th moment stability, $q$th moment asymptotic stability, $q$th exponential stability, almost sure asymptotic stability and almost sure exponential stability are put forward, where the upper bound of the diffusion operator can be a function with sign-changed time-varying coefficients rather than negative constants. Eventually, through several numerical examples, the theoretical results are strictly verified, demonstrating the feasibility and effectiveness of the proposed method.