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Figure 2 | Cell Communication and Signaling

Figure 2

From: Recent development and biomedical applications of probabilistic Boolean networks

Figure 2

An example of truth table, state transition diagram, and transition probability matrix of a PBN. The truth table, the state transition diagram, and the transition probability matrix A of a PBN without perturbations consisting of three variables V = {x1,x2,x3} andF=( F 1 , F 2 , F 3 ), where F 1 ={ f 1 ( 1 ) }, F 2 ={ f 1 ( 2 ) , f 2 ( 2 ) }, and F 3 ={ f 1 ( 3 ) , f 2 ( 3 ) }. Since there is one predictor function for node x1 and two predictors for nodes x2 and x3, there are 1 · 2 · 2 = 4 realisations of the PBN given by four network transition functions f 1 =( f 1 ( 1 ) , f 1 ( 2 ) , f 1 ( 3 ) ), f 2 =( f 1 ( 1 ) , f 1 ( 2 ) , f 2 ( 3 ) ), f 3 =( f 1 ( 1 ) , f 2 ( 2 ) , f 1 ( 3 ) ), and f 4 =( f 1 ( 1 ) , f 2 ( 2 ) , f 2 ( 3 ) ) with associated probabilities c1 = 0.12, c2 = 0.18, c3 = 0.28, and c4 = 0.42, respectively. For example, c 3 = c 1 ( 1 ) · c 2 ( 2 ) · c 1 ( 3 ) =1·0.7·0.4=0.28. The edges in the state transition diagram are labelled with the transition probabilities. As can be seen from the state transition diagram, the underlying Markov chain is irreducible and aperiodic, thus ergodic. The steady-state (limiting) distribution for the chosen c i values, i = 1..4, is given by[ 7 1609 , 3640 14481 , 49 4827 , 716 4827 , 175 4827 , 238 4827 , 2548 14481 , 4696 14481 ] (the states are considered in the lexicographical order from 000 to 111).

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