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Fixed points are fundamental states in any dynamical system. In the case ofgene regulatory networks (GRNs) they correspond to stable genes profilesassociated to the various cell types. We use Kauffman's approach to model GRNswith random Boolean networks (RBNs). We start this paper by proving that, if wefix the values of the source nodes (nodes with in-degree 0), the expectednumber of fixed points of any RBN is one (independently of the topology wechoose). For finding such fixed points we use the {\alpha}-asynchronousdynamics (where every node is updated independently with probability 0 <{\alpha} < 1). In fact, it is well-known that asynchrony avoids the cycleattractors into which parallel dynamics tends to fall. We perform simulationsand we show the remarkable property that, if for a given RBN with scale-freetopology and {\alpha}-asynchronous dynamics an initial configuration reaches afixed point, then every configuration also reaches a fixed point. By contrast,in the parallel regime, the percentage of initial configurations reaching afixed point (for the same networks) is dramatically smaller. We contrast theresults of the simulations on scale-free networks with the classicalErdos-Renyi model of random networks. Everything indicates that scale-freenetworks are extremely robust. Finally, we study the mean and maximum time/workneeded to reach a fixed point when starting from randomly chosen initialconfigurations.

Fixed-points in Random Boolean Networks: The impact of parallelism in the scale-free topology case