Stability of Boolean networks

The stability of Boolean networks depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes ($K_{c}$), and can be characterized by the Hamming distance as distance measure. In the unstable regime, the distance between two initially close states on average grows exponentially in time, while in the stable regime it decreases exponentially. In the critical regime, the Hamming distance is small compared with the number of nodes ($N$) in the network.

In the case of $N-K$ Boolean networks, the network is

• stable if & $K<K_{c}$,
• critical if & $K=K_{c}$,
• unstable if & $K>K_{c}$.

The state of a given node $n_{i}$ is updated according to its truth table, whose outputs are randomly populated. $p_{i}$ denotes the probability of assigning an off output to a given series of input signals.

If $p_{i}=p=const.$ for every node, the transition between the stable and chaotic range depends on $p$. The critical value of the average number of connections is $K_{c}=1/[2p(1-p)]$.

If $K$ is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by $\langle K^{in}\rangle$

The network is

• stable if & $\langle K^{in}\rangle <K_{c}$,
• critical if & $\langle K^{in}\rangle =K_{c}$,
• unstable if & $\langle K^{in}\rangle >K_{c}$.

The conditions of stability are the same in the case of networks with scale-free topology, where the in-and out-degree distribution is a power-law distribution: $P(K)\propto K^{-\gamma }$, and $\langle K^{in}\rangle =\langle K^{out}\rangle$, since every out-link from a node is an in-link to another.

Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks, $q_{i}=2p_{i}(1-p_{i})$. In the general case, stability of the network is governed by the largest eigenvalue $\lambda _{Q}$ of matrix $Q$, where $Q_{ij}=q_{i}A_{ij}$, and $A$ is the adjacency matrix of the network.

The network is

• stable if & $\lambda _{Q}<1$,
• critical if & $\lambda _{Q}=1$,
• unstable if & $\lambda _{Q}>1$.