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We could not directly compare our result with these methods, since most of the data we used in this work comes from discrete models for which they are not applicable. 2 Approach Our approach is based on our previous work on the Stochastic Logical Network model for regulatory networks . Here we provide new eﬃcient algorithm for the problem of network topology reconstruction. In this section we brieﬂy introduce the SLN modelling and present our algorithm. 1 Stochastic Logical Networks The SLN model extends the kinetic logic approach  by taking into account the stochastic nature of regulatory processes.
M } we have α∗m (x) = 0 ⇒ αm (x) = 0, x ∈ S, or equivalently, starting with the original propensity functions, αm (x) > 0 ⇒ α∗m (x) > 0, x ∈ S. Importance Sampling then generates trajectories according to the changed propensity functions and multiplies the results with the likelihood ratio to get unbiased estimates for the original system. g. g. [19,23,24] for formal deﬁnitions. Simultaneous Stochastic Simulation of Multiple Perturbations 25 path density as in (11). Thus, denoting by p∗(t0 ) the initial distribution for the states, the likelihood ratio becomes p(t0 ) (x0 ) · L(ω) = R αmi−1 (x(ti−1 )) exp (α0 (x(ti−1 ))τi−1 ) i=1 R .
Hence, all information on transition probabilities is covered by the single matrix Q. In terms of P and Q the Kolmogorov forward diﬀerential equations, the Kolmogorov backward diﬀerential equations, and the Kolmogorov global diﬀerential equations can be expressed by (from left to right): ∂ P(t) = P(t)Q, ∂t ∂ P(t) = QP(t), ∂t ∂ (t) p = p(t) Q, ∂t (6) where p(t) denotes the vector of the transient state probabilities corresponding to (2). Explicitly writing the Kolmogorov global diﬀerential equations in terms of the coeﬃcients and some algebra yields (t) ∂pi = ∂t (t) (t) pj qji − j:j=i (t) j:j=i (t) pj qji − pi qij .