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## 5 Discussion 

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In the previous section, the statistical results showed that MRBNs are more robust than classic RBNs for the same K$_{T}$ values, since modularity reduces the probability of damage to spread. This can also be confirmed analytically. 

The method of Luque and Sol´ e [32] can be extended to MRBNs. Instead of focussing on a single node with a Boolean state, one can focus on a single module m with internal dynamics. The probability that the internal dynamics of m may be perturbed will depend mainly on 〈 L 〉 (as well as p ). If p = 0 . 5, then L < 2 implies that on average damage will not propagate between modules, even if the internal dynamics of m are chaotic, i.e. K > 2. Still, if K >> 2, i.e. m is deep within the chaotic phase because of a high number of intramodular links, then the fragility of the network will be noticeable in the MRBN (Δ H >> 0) even if damage does not spread outside m . It is clear that damage spreading will depend on the value of K as well, e.g. if K is small, damage will have a lower probability of propagating within modules, affecting the probability of spread across modules. 

Since damage across the whole MRBN can spread through internal ( K ) or external ( L ) links, it can be seen that there is the following relationship between critical L and K , extending equation 1: 

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