The problem of Markovian jumping complex dynamical networks with random coupling and mode-dependent time-varying delays is studied in this paper. The randomness in the network coupling and uncertainties in the control design are made to satisfy the Bernoulli sequence. An augmented Lyapunov Krasovskii function (LKF) is formulated, which considered the full information on the mode-dependent time-varying delays. Also, the involvement of triple integral terms and use of the extended Jensen's integral inequalities with the reciprocal convex method guaranteed some sufficient conditions for the error system synchronization. A non-fragile control is designed and proposed in terms of linear matrix inequalities (LMIs) which can easily be verified for feasibility with LMIs toolbox in Matlab. To indicate the effective performance of our proposed control scheme, one numerical example with simulations is given.