Random maps and attractors in random Boolean networks

Despite their apparent simplicity, random Boolean networks display a rich variety of dynamical behaviors. Much work has been focused on the properties and abundance of attractors. The topologies of random Boolean networks with one input per node can be seen as graphs of random maps. We introduce an approach to investigating random maps and finding analytical results for attractors in random Boolean networks with the corresponding topology. Approximating some other non-chaotic networks to be of this class, we apply the analytic results to them. For this approximation, we observe a strikingly good agreement on the numbers of attractors of various lengths. We also investigate observables related to the average number of attractors in relation to the typical number of attractors. Here, we find strong differences that highlight the difficulties in making direct comparisons between random Boolean networks and real systems. Furthermore, we demonstrate the power of our approach by deriving some results for random maps. These results include the distribution of the number of components in random maps, along with asymptotic expansions for cumulants up to the fourth order.

Figure 1

The average number of proper $L$ cycles as a function of $N$ for different $L$, for networks with single-input nodes. $r=1$ in (a), and $r=\frac{3}{4}$ (solid lines), and $r=\frac{1}{2}$ (dotted lines) in (b). In (a), $L$ is indicated in the plot. In (b), $L$ is 3, 5, 7, 1, 2, 4, 6, and 8 for $r=\frac{3}{4}$ and 7, 5, 3, 8, 6, 4, 2, and 1 for $r=\frac{1}{2}$, in that order, from bottom to top along the right boundary of the plot area. In (b), the curves for $L=3$ and $L=5$ for $r=\frac{3}{4}$ essentially coincide at the right side of the plot, whereas they split up to the left, with $L=3$ as the upper curve there.

Figure 2

The average number of proper $L$ cycles for networks (with one-input nodes) with $N=100$ and $r=\frac{3}{4}$, as function of $L$. $\Delta r=\frac{3}{4}$ (thin solid line), $\Delta r=0$ (thick solid line) and $\Delta r=-\frac{3}{4}$ (dotted line). Note the importance of what numbers divide $L$.

Figure 3

Arithmetic and geometric means of the number of states, ${\Omega }_0$, in attractors of networks with one input per node. ${⟨{\Omega }_0⟩}_N$ (solid lines) and ${⟨{\Omega }_0⟩}_N^{\mathrm{G}}$ (dotted lines) for $r=\frac{1}{2}$, $r=\frac{3}{4}$ and $r=1$, in that order, from the bottom to the top of the plot. Note that both ${⟨{\Omega }_0⟩}_N$ and ${⟨{\Omega }_0⟩}_N^{\mathrm{G}}$ are independent of $\Delta r$.

Figure 4

The arithmetic mean of the number of attractors with lengths $L⩽L_{\max }$ in networks with $N$ single-input nodes, for different values of $N$. In (a) $N=10,{10}^2,\dots ,{10}^5$ for $r=1$ (thin solid lines) and $N=10,\dots ,{10}^4$ for $r=\frac{3}{4}$ (thin dotted lines). In (b) $N=10,{10}^2,{10}^3$ (thin solid lines) for $r=\frac{1}{2}$ and $N=10$ for $r=\frac{1}{4}$ (thin dotted line). For all cases, $\Delta r=0$. The thick lines in (a) and (b) show the limiting number of attractors when $N\to \infty$. The arrowhead in (b) marks this limit for $L_{\max }={10}^7$ for $r=\frac{1}{2}$. The small increase in the number of attractors when $L_{\max }$ is changed from ${10}^3$ to ${10}^7$ indicates that ${⟨C⟩}_N$ converges when $N\to \infty$. Note the drastic change in the $y$-scale between the case $r>\frac{1}{2}$ and $r⩽\frac{1}{2}$.

Figure 5

A comparison between simulations for power law in-degree networks of size $N=20$ (bold lines) and the corresponding networks with single-input nodes (thin lines). The fitted networks have identical values for $r$, $\Delta r$, and $N$. The solid lines show the number of fixed points, whereas the dashed and dotted lines show the number of two cycles plus fixed points and the total number of attractors, respectively. The probability distribution of in-degrees satisfies $p_K\propto K^{-\gamma }$, where $K$ is the number of inputs. The power law networks use the nested canalyzing rule distribution presented in Ref. 9.

Figure 6

A cross-section of Fig. 5 at $\gamma =2.5$, with simulation results for the power law in-degree networks (bold lines), and the corresponding single-input networks (thin lines). The distributions of the number of attractors of different types are presented with cumulative probabilities, along with the corresponding means (short vertical lines at the bottom of the plot). The solid lines show the number of fixed points, whereas the dashed and dotted lines show the number of two cycles plus fixed points and the total number of attractors, respectively. Note that the medians are found where the curves for the probability distributions intersect $\frac{1}{2}$ on the $y$-axis.

Figure 7

A comparison between critical $K=2$ Kauffman networks (thick lines) and the corresponding networks of single-input nodes (thin lines). The size of the single-input networks is set to $N^{\prime }=N^{2∕3}$. $r=1$ and $\Delta r=0$, consistent with the Kauffman model. (a) The number of proper $L$ cycles for the $L$ indicated in the plot. For the Kauffman networks, the numbers have been calculated from Monte Carlo summation for those network sizes where could could not be calculated exactly (see Ref. 13). The number of fixed points is one, independently of $N$, for both network types. (b) Total number of attractors. This quantity has been calculated analytically for the single-input networks, and estimated by simulations for the Kauffman networks using ${10}^2$, ${10}^3$, ${10}^4$, and ${10}^5$ random starting configurations per network.

Figure 8

A cross-section of Fig. 7 at $N=125$, with simulation results for the Kauffman networks (bold lines), and the corresponding single-input networks (thin lines). For the Kauffman networks, we use ${10}^5$ random starting configurations for 1600 network realizations. The corresponding single-input node networks have only $N^{\prime }=N^{2∕3}=25$ nodes, and we perform exhaustive searches through the state space of the relevant nodes in ${10}^6$ such networks. The distributions of the number of attractors of different types are presented with cumulative probabilities, along with the corresponding averages (short vertical lines at the top and bottom of the plot). Corresponding analytical averages, for the Kauffman networks, are marked with arrowheads. The solid lines show the number of fixed points, whereas the dashed and dotted lines show the number of two cycles plus fixed points and the total number of attractors, respectively. Note that the medians are found where the curves for the probability distributions intersect $\frac{1}{2}$ on the $y$-axis.