Fig. 3

Demonstration of MCMC sampling. For the directed network in Fig. 2a, after two iterations, we first obtain the set of matched links (red edges) which form a Markov chain M ₁ in the “linking and dynamic graph”. Based on the Markov chain, we identify the target controllable sunspace of each node in the network; Then, we get the driver nodes {v₃} by solving the problem (5) with integer linear programming (ILP), which result in the weight of the driver nodes W(M₁) = 2.Then we generate a new Markov chain M ₂ by replacing the maximum matching in the t = 0 updated bipartite graph (supplementary note 2 of Additional file 1) and obtaining the new maximum matching in the later updated bipartite graph after t = 0. Based on the new Markov chain, we can identify the driver nodes {v₁} with the weight W(M₂) = 1. Finally according to the Metropolis-Hastings algorithm, we will accept the markov chain M ₂ with the probability p(M _t ,M_t + 1) = min[1,exp.(c*W(M ₂ )-c*W(M ₁ )) for some c > 0, here we set c = 10