Stability of Boolean networks: The joint effects of topology and update rules

We study the stability of orbits in large Boolean networks. We treat the case in which the network has a given complex topology, and we do not assume a specific form for the update rules, which may be correlated with local topological properties of the network. While recent past work has addressed the separate effects of complex network topology and certain classes of update rules on stability, only crude results exist about how these effects interact. We present a widely applicable solution to this problem. Numerical simulations confirm our theory and show that local correlations between topology and update rules can have profound effects on the qualitative behavior of these systems.

Figure 1

Normalized average Hamming distance $\langle H\rangle$ and ${\lambda }_R$ for a network with xor, or, and and update rules (described in Sec. 5a). Filled markers are averaged over 50 quenched realizations of the update rules, while hollow markers show a single quenched realization. Red squares, blue circles, and green triangles represent different correlations between network topology and update rules; see text for details. (a) When the fraction of xor rules $\alpha$ is used as a tuning parameter, the stability transitions for the three cases are far apart. Locations where ${\lambda }_R=1$ are marked by arrows and agree with the observed transitions. (b) ${\lambda }_R$ plotted as a function of $\alpha$. (c) Plotting $\langle H\rangle$ against ${\lambda }_R$ directly, we see that the transition for all three curves occurs at ${\lambda }_R=1$.

Figure 2

Normalized average Hamming distance $\langle H\rangle$ and ${\lambda }_R$ for a threshold network (described in Sec. 5b). The notation and results are analogous to Fig. 1, except that threshold rules are used and $\overline{\theta }$ is used as a tuning parameter in (a) and (b). Once again, the theoretical predictions [arrows in panels a and c] agree with the observed transitions.