Dominant vertices and attractors' landscape for Boolean networks

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.

💡 Research Summary

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This paper investigates how to compress the dynamics of Boolean networks by exploiting a structural property called dominant vertices. A dominant vertex set (U\subseteq V) is defined purely from the underlying directed graph: starting from (U), one repeatedly adds all vertices whose entire input set lies inside the current collection. This process yields a chain (U_0=U\subseteq U_1\subseteq\cdots\subseteq U_d=V); the length (d) is called the depth of the dominant set. Proposition 1 shows that a set is dominant exactly when every directed cycle of the graph contains at least one vertex of (U), and the depth equals the longest directed path that starts in (U).

The central dynamical result (Theorem 1) states that if two trajectories of the Boolean network coincide on (U) for a time interval of length (d), then they become identical on the whole vertex set for all later times. Consequently, the dominant set acts as an “observer”: the behavior of the whole system after a transient is fully determined by the evolution of (U). This observation yields immediate bounds on transient lengths, on the number of distinct basins of attraction, and on the size of basins.

Using the dominant set, the authors construct a reduced graph ((U,A_U)). An edge ((u_1,u)) belongs to (A_U) iff there exists a simple path from (u_1) to (u) in the original graph. For each such pair they collect all possible path lengths (\ell_{u_1,u}) and define the recursion length (\ell_U=\max_{u_1,u}\ell_{u_1,u}). By construction (\ell_U\le d+1). This reduced graph captures precisely the way dominant vertices influence each other through intermediate nodes.

On the reduced graph the authors define an induced automata network. The idea is to project a reversed segment of length (\ell_U) of an original orbit onto the coordinates in (U). Formally, a configuration (y\in{0,1}^{U}) evolves according to a map (\Phi) that recursively applies the original Boolean functions along the paths that connect dominant vertices. The resulting global update function (F) on ({0,1}^{U}) constitutes the induced system.

The main theoretical claim (Theorem 2) is that the original Boolean network and its induced automata network are asymptotically conjugate: when each system is restricted to its attractors, there exists a bijective, dynamics‑preserving correspondence between the two state spaces. Hence the two systems share exactly the same attractor set, the same periods, the same basin sizes, and the same maximal tree depths. In other words, the reduction is lossless with respect to the attractor landscape.

The paper also relates dominant sets to the well‑studied feedback vertex set (FVS). Proposition 1 shows that in graphs where every vertex has non‑zero indegree, a set is dominant if and only if it is a feedback vertex set. Since finding a minimum FVS is NP‑complete, identifying a minimum dominant set is computationally hard as well.

To illustrate the theory, the authors introduce clover networks: Boolean networks whose dominant set consists of a single vertex. In this case the reduced graph reduces to a single self‑loop, and the induced system is a one‑dimensional Boolean map that encapsulates the whole dynamics. They generate a parametrized ensemble of such networks, varying the Boolean functions randomly, and perform extensive simulations (thousands of instances). Measured quantities include the number of attractors, average period, basin sizes, and transient lengths. The empirical distributions respect the theoretical bounds derived earlier: networks with smaller depth (d) exhibit shorter transients and smaller basins, while the maximal period never exceeds the bound implied by the recursion length.

Overall, the paper contributes four major advances: (1) a graph‑theoretic definition of dominant vertices that guarantees observability of the whole dynamics; (2) a systematic construction of a reduced graph and an induced automata network; (3) a rigorous proof of asymptotic conjugacy, ensuring that the reduction preserves the full attractor structure; and (4) concrete examples and numerical validation on clover networks. This framework offers a powerful tool for analyzing large Boolean networks, especially in biological contexts where state‑space explosion hampers exhaustive simulation. By focusing on a small, structurally identified subset of nodes, researchers can obtain exact information about attractors, basins, and transients, facilitating both theoretical insight and practical applications such as network control, model reduction, and the design of synthetic regulatory circuits.