Boolean networks synchronism sensitivity and XOR circulant networks convergence time

In this paper are presented first results of a theoretical study on the role of non-monotone interactions in Boolean automata networks. We propose to analyse the contribution of non-monotony to the diversity and complexity in their dynamical behaviours according to two axes. The first one consists in supporting the idea that non-monotony has a peculiar influence on the sensitivity to synchronism of such networks. It leads us to the second axis that presents preliminary results and builds an understanding of the dynamical behaviours, in particular concerning convergence times, of specific non-monotone Boolean automata networks called XOR circulant networks.

💡 Research Summary

This paper presents the first theoretical results on how non‑monotone interactions affect the dynamics of Boolean automata networks, focusing on two main aspects: synchronism sensitivity and convergence time. The authors begin by formalising Boolean networks, defining configurations, local transition functions, and three updating modes—general (all possible subsets of nodes may update simultaneously), asynchronous (single‑node updates), and parallel (all nodes update together). They introduce the notion of local monotonicity: a local function f_i is monotone in variable j if flipping x_j from 0 to 1 never reverses the output of f_i; otherwise the function is non‑monotone. A network is non‑monotone if at least one of its local functions is non‑monotone.

The first analytical line studies “synchronism sensitivity,” i.e., how the addition of synchronous transitions (updates of multiple nodes at once) changes the asymptotic behaviour compared with the asynchronous transition graph. By examining the general transition graph G_g and the asynchronous graph G_a, the authors identify four possible cases when a synchronous transition (x→y) that is not sequentialisable (cannot be reproduced by a sequence of asynchronous steps) is added:

1. x is transient in G_a; the set of attractors stays the same, but configurations that could previously only reach a certain attractor now gain access to additional attractors reachable from y. This is labelled level 1◦ sensitivity.
2. x is recurrent, y is transient, and the attractor set of y equals that of x. Adding the synchronous transition makes y recurrent, thereby enlarging the attractor containing x. This is level 1• sensitivity.
3. x is recurrent, y is transient, and y’s attractor set is disjoint from x’s. Adding the transition turns x transient and eliminates its original attractor (level 2 sensitivity).
4. Both x and y are recurrent; the transition causes the attractor of x to “empty” into that of y, again destroying an attractor (level 2).

Thus, non‑monotone networks can exhibit sensitivity levels 0 (no effect), 1◦, 1•, or 2, with levels 1• and 2 being the most dramatic. The authors argue that non‑monotonicity is a structural parameter that often pushes a network into the higher sensitivity levels, whereas monotone networks typically remain at level 0 or 1◦.

The second analytical line focuses on a concrete class of non‑monotone networks: XOR circulant (or cyclic) networks. In an n‑node XOR circulant network each node i updates its state as the XOR of the states of two neighbours at fixed offsets (i−k and i+k modulo n). The global update function is linear over GF(2): F(x)=C_k·x (mod 2), where C_k is a circulant matrix. This linearity yields a highly symmetric transition graph with toroidal structure, enabling precise algebraic analysis.

Key results for XOR circulant networks under parallel (synchronous) updating are:

• Convergence time depends strongly on the initial density d(x) of 1’s. Configurations with d≈0.5 exhibit the longest convergence times because the XOR operation mixes 0’s and 1’s most effectively, exploring a larger portion of the state space.
• When n is a power of two, every trajectory reaches a fixed point (all‑0 or all‑1) or a period‑2 limit cycle within O(log n) steps. This follows from the eigen‑structure of the circulant matrix, whose eigenvalues over GF(2) are only 1 or –1.
• For arbitrary n, the worst‑case convergence time is O(n). This bound is attained by sparse initial configurations (e.g., a single 1) that propagate slowly around the ring.
• Compared with asynchronous updating, parallel updating dramatically shortens convergence time but can also create new attractors (e.g., a period‑4 cycle) that do not exist in the asynchronous dynamics. This phenomenon corresponds to level 1• or level 2 synchronism sensitivity.
• The authors provide explicit formulas for the number of steps needed to reach a fixed point as a function of n, k, and the Hamming weight of the initial configuration.

Overall, the paper demonstrates that non‑monotone interactions, exemplified by XOR circulant networks, fundamentally alter both the qualitative (appearance or disappearance of attractors) and quantitative (speed of convergence) aspects of Boolean network dynamics. The linear algebraic framework used for XOR circulant networks shows that even highly non‑monotone systems can be analytically tractable, offering a bridge between combinatorial dynamics and algebraic methods. The authors conclude by suggesting future work on more general non‑monotone topologies, stochastic update schedules, and applications to gene‑regulatory modeling where mixed activation/inhibition (non‑monotonicity) is biologically realistic.