Before revision:  However, quasi-potential landscapes that have been developed for continuous systems by multiple groups are not suitable for discrete networks , which are generally used to study large networks or ensembles thereof. Here we introduce the relative stability of network states to Boolean networks by using the noise-perturbed Markov matrices. With an ensemble approach of a minimal gene network for pancreas development , we show that Boolean networks with canalized / sign-compatible Boolean functionscan capture essential features of cell fate dynamics and allow the estimation of relative stabilities of network states and, hence, of transition barriers.
After revision:  However, quasi-potential landscapes that have been developed for continuous systems by multiple groups are not suitable for discrete-valued networks which are important tools to study complex systems. In this paper, we provide a framework to quantify the landscape for discrete Boolean networks (BNs). We apply our framework to study pancreas cell differentiation where an ensemble of BN models is considered based on the structure of a minimal GRN for pancreas development , we show that Boolean networks with canalized / sign-compatible Boolean functionscan capture essential features of cell fate dynamics and allow the estimation of relative stabilities of network states and, hence, of transition barriers.