The development of robust Early Warning Signals (EWSs) is necessary to quantify the risk of crossing tipping points in the present-day climate change. Classically, EWSs are statistical measures based on time series of climate state variables, without exploiting their spatial distribution. However, spatial information is crucial to identify the starting location of a transition process and can be directly inferred by satellite observations. By using complex networks constructed from several climate variables on the numerical grid of climate simulations, we seek for network properties that can serve as EWSs when approaching a state transition. We show that network indicators such as the normalized degree, the average length distance, and the betweenness centrality are capable of detecting tipping points at the global scale, as obtained by the MIT general circulation model in a coupled-aquaplanet configuration for CO2 concentration-driven simulations. The applicability of such indicators as EWSs is assessed and compared to traditional methods. We also analyze the ability of climate networks to identify nonlinear dynamical patterns.