We uncovered the underlying energy landscape of the mitogen-activated protein kinases signal transduction cellular network by exploring the statistical natures of the Brownian dynamical trajectories. We introduce a dimensionless quantity: The robustness ratio of energy gap versus local roughness to measure the global topography of the underlying landscape. A high robustness ratio implies funneled landscape. The landscape is quite robust against environmental fluctuations and variants of the intrinsic chemical reaction rates. As the environmental fluctuations or the variances of the inherent chemical reaction rates increase further more, the landscape becomes less robust and more flatter. We also show that more robust network has less dissipation costs. Our approach is quite general and can be applied to other cellular networks.