Heckel, Reinhard; Schober, Steffen; Bossert, Martin (Institute of Telecommunications and Applied Information Theory, Ulm University, Albert-Einstein-Allee 43, 89081 Ulm, Germany)
Ensembles of Boolean networks using linear random threshold functions with memory are considered. Such ensembles have been studied previously by Szejka et al. They obtained analytical results for the order parameter which can be used to predict the expected behavior of a network randomly drawn from the ensemble. Using numerical simulations of randomly drawn networks, Szejka et al. found marked deviations from the predicted behavior. In this work improved analytical results are provided that better match up the numerical results. Furthermore, the critical point in their analysis is identified. In the model studied, each node is not only dependent on the K regular inputs, but also on the previous state of the node. The results show that this feedback loop accounts for the low order parameter and tolerance on random errors, even for networks with high in-degree.