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This article delves into the issue of the pinning synchronization of complex dynamical networks with Markov topologies. First, to reduce communication demands and implement easily in practice, an adaptive event-triggered protocol for complex networks is developed. Later, to reduce conservatism, the T-S fuzzy model is introduced to linearize adaptive triggering condition, and the corresponding fuzzy controller is designed. This method can be applied to a triggered condition with a bound and nonlinear adaptive law, and the better synchronization performance can be arrived with the assistance of the fuzzy controller. Second, the switched pinning strategy is designed, which further saves the energy and avoids the occurrence of communication congestion. Then, by introducing a hidden Markov chain, an asynchronous pinning strategy is further developed. With the assistance of convex optimal techniques, some sufficient criteria are presented to reach the desired synchronization. Finally, the examples on circular trail and Chua's circuits are exploited to demonstrate the feasibility of the above synchronization laws. • The T-S fuzzy model is introduced to cope with the adaptive process of AETM for the first time. • A Markov switched pinning strategy is constructed to decline communication congestion and cost. • Asynchronous phenomenon of pinning control is also considered, and the presented example is practical. [ABSTRACT FROM AUTHOR]