Uncovering the non-equilibrium stationary properties in sparse Boolean networks

Dynamic processes of interacting units on a network are out equilibrium in general. In the case directed tree, dynamic cavity method provides an efficient tool that characterises trajectory process for linear threshold model. However, because computational complexity method, analysis has been limited to systems where largest number neighbours is small. We devise implementation which substantially reduces with discrete couplings. Our approach opens up possibility investigate properties networks fat-tailed degree distribution. exploit this new study non-equilibrium steady-state. extend dynamical calculate pairwise correlations induced by different motifs network. results suggest just two basic able accurately describe entire statistics observed correlations. Finally, we models defined containing bi-directional interactions. observe stationary state associated symmetric or anti-symmetric interactions biased towards active inactive respectively, even if independent interaction entries drawn from This phenomenon, can be regarded as form spontaneous symmetry-breaking, peculiar formulated terms Boolean variables, opposed Ising spins.