Dominant vertices and attractors' landscape for Boolean networks

TL;DR

A reduced graph on the dominant vertices and an induced dynamics on this graph is defined, which is asymptotically equivalent to the original Boolean dynamics, which is proved to be asymptotically equivalent to the original Boolean dynamics.

Abstract

In previous works, we introduced the notion of dominant vertices in the context of dynamical systems on networks. This is a set of nodes in the underlying network whose evolution determines the whole network's dynamics after a transient time. In this paper, we focus on the case of Boolean networks. We define a reduced graph on the dominant vertices and an induced dynamics on this graph, which we prove is asymptotically equivalent to the original Boolean dynamics. Asymptotic conjugacy ensures that the systems, restricted to their respective attractors, are dynamically equivalent. For a significant class of networks, the induced dynamics is indeed a reduction of the original system. In these cases, the reduction, which is obtained from the structure of dominant vertices, supplies a more tractable system with the same structure of attractors as the original one. Furthermore, the structure of the induced system allows us to establish bounds on the number and period of the attractors, as well as on the reduction of the basin's sizes and transient lengths. We illustrate this reduction by considering a class of networks, which we call clover networks, whose dominant set is a singleton. To get insight into the structure of the basins of attraction of Boolean networks with a single dominant vertex, we complement this work with a numerical exploration of the behavior of a parametrized ensemble of systems of this kind.

Figures (4)

• Figure 1: A five-vertex network with a dominant set of two vertices colored in blue, and depth $d=2$. The reduced graph is shown aside.
• Figure 2: Above, the transition diagram $\mathcal{T}_F$, corresponding to the Boolean network $(\mathtt{B}^{\{1,\ldots,5\}},F)$. Below, the transition diagram $\mathcal{T}_\mathcal{F}$, defined by the induced logic network $(\mathtt{B}_2^{\{1,2\}},\mathcal{F})$. The eventual conjugacy $\mathtt{h}$, is such that $\mathtt{h}(1)=(2,2)$ and $\mathtt{h}(3)=(2,0)$, which establishes the equivalence between the two periodic attractors of $(\mathtt{B}^{\{1,\ldots,5\}},F)$ with those of $(\mathtt{B}_2^{\{1,2\}},\mathcal{F})$.
• Figure 3: Above is a Boolean network with the dominant set $U=\{1\}$, and the corresponding induced logic network. Below is the transition diagram of the Boolean network with states in $\mathtt{B}^{\{1,\ldots,5\}}$ ordered lexicographically, and the transition diagram of the induced logic network. The eventual conjugacy $\mathtt{h}:\mathtt{B}^{\{1,\ldots,5\}}\to\mathtt{B}_2$ is codified by the correspondence of colors.
• Figure 4: Above is a clover network with signed majority rule defined by the sings depicted next to the arrows. The the corresponding induced automata network is a sigle loop at vertex 1. Below is the transition diagram of the original network with states in $\mathtt{B}^{\{1,\ldots,5\}}$ ordered lexicographically, and the transition diagram of the induced automata network. The eventual conjugacy $\mathtt{h}:\mathtt{B}^{\{1,\ldots,5\}}\to\mathtt{B}_2$ is codified by the correspondence of colors.

Theorems & Definitions (15)

• Proposition 1: Characterization of Dominant sets
• proof
• Example 1
• Theorem 1: Dominance and Dynamics
• proof
• Example 2
• Theorem 2: Eventual conjugacy
• proof
• Example 3
• Example 4