Fixed points of Boolean networks with sparse connections

TL;DR

This work considers several models, calculating the first and second moments of the number of fixed points, and finds that these moments remain finite in the large $N$ limit, except at the transitions where they become singular.

Abstract

We study fixed points of cellular automata with $N$ sites on random sparse graphs. In the large $N$ limit such models are known to exhibit phase transitions, from a ``frozen'' phase, where at most a finite number of sites fluctuate at long times, to a ``fluctuating'' phase where a finite fraction of sites fluctuate. We consider several models, calculating the first and second moments of the number of fixed points, and find that these moments remain finite in the large $N$ limit, except at the transitions where they become singular. The singularities can take several forms, including divergence of the mean or variance of the number of fixed points, on one or both sides of the transition. The type of singularity is related to properties of the mean field dynamics or dynamics of the distance between copies of the system. In configuration space, we find that fixed points are organized into clusters, each consisting of sets of fixed points that agree with one another except for on a finite number of sites. In the frozen phase there is only one cluster, while in the fluctuating phase there may be multiple clusters. If there are multiple clusters, the distance between fixed points in different clusters is extensive. We show that the differences within the clusters correspond to local changes near short cycles in the directed graph of connections whose influence is eventually limited. In the frozen phase, we calculate the full distribution of the number of fixed points.

Figures (6)

• Figure 1: (A) $\left\langle \Omega^{2}\right\rangle$ for the Kauffman model. The dashed vertical line marks the transition at $C=2$. (B) The function $A(\phi)$ for two values of $C$, below and above the transition. It shows the saddle point contributions to $\left\langle \Omega^{2}\right\rangle$ at $\phi=1$ for both values of $C$, and for $C>2$ the contribution at $0<\phi^{*}<1$.
• Figure 2: Distribution of the number of FPs in the frozen phase. (A) Example of the subset $I$, of sites on which assignments change between FPs in the frozen phase. (B) The final $P(\Omega)$, theory (blue bars) against exhaustive searches after graph simplification (red points). Both panels are obtained by first applying the simplification algorithm (see main text) for instances with $C=1.5$ and $N=5000$ sites.
• Figure 3: Examples of cycles that may affect the FP number in the Kauffman model. The functions are labeled on the arrows: The constant function $f(x)=1$ is labeled as 1; the "copy" function $f(x)=x$ as "C", and the "reverse" function $f(x)=1-x$ as "R." (A) A cycle with any constant function has a unique fixed point and is removed from the reduced graph: in this example the function 1 is contant so, starting from it, all the sites along the cycle are determined. (B) A cycle with two fixed points. (C) A cycle with no consistent fixed points.
• Figure 4: The excitatory model. (A) The function $q(\psi)$ for different values of $C$, above and below $C_{\text{crit}}=1$. (B) The first moment $\left\langle \Omega\right\rangle$. (C) SCCs in the phase $C<C_{\text{crit}}$. Each circle denotes an SCC, each containing a finite number of sites. (D) SCCs in the phase $C>C_{\text{crit}}$ include a giant SCC comprised of a finite fraction of the sites (large circle), and possibly a finite number of SCCs with a finite number of sites (small circles). They may be disconnected from the giant component, or with directed paths to or from it. Arrows denote paths of links between the SCCs.
• Figure 5: The double-excitatory model. (A) The saddle-point function for $\left\langle \Omega\right\rangle$ as a function of $\psi$, at $C=2,3.35,4$. (B) $\left\langle \Omega\right\rangle$ for the Double Excitatory model (black curve), and finite-$N$ behavior. Solid curves are calculated from an exact evaluation of $\left\langle \Omega\right\rangle$ when expressed as a sum, and crosses (with 1SE error bars) are the results of exhaustive searches at $N=5,15$, each averaged over 1000 disorder realizations. Dashed line: transition at $C_{\text{crit}}$.