content_image/317854.jpg content_image/317855.jpg content_image/317856.jpg
and for p = 0 . 5: 

and 

<img src='content_image/317854.jpg'>

<img src='content_image/317855.jpg'>

<img src='content_image/317856.jpg'>

A plot of equation 16 can be seen in Figure 14, with simulation averages for cases 1, 2, 3, and 5 $^{3}$. Even when the analysis presented above assumes infinitely-sized modules and networks, the simulation results with small networks match the theoretical analysis. 

It can be seen that a higher κ implies lower L , i.e. lower values in Figure 14. When κ = 1, L = 0, damage cannot propagate between modules, even for the highest local connectivities ( K = N ). Thus, in principle κ can be used to modulate the sensitivity to initial conditions of MRBNs [19]. 

There is a negative correlation between κ and Δ H , and a positive correlation between κ and A . However, the explanations for the effect on sensitivity to initial conditions and on the number of attractors seem to be also related to different effects of the topology of MRBNs. Still, it would be interesting to study whether it is always the case that RBNs with more attractors on average are more robust to damage spread, as it is the case for MRBNs. Alternatively, finding counterexamples would be illustrative as well. 

## 6 Conclusions and Future Work 

We have presented a generalization of random Boolean networks, where modules can be constructed. With statistical studies on ensembles of MRBNs and an analytical study, it could be seen that different parameters that define MRBNs affect most of the properties of the networks and their dynamics. Thus, it can be said that modularity is one way of guiding the self-organization of RBNs 

