Nonlinear threshold Boolean automata networks and phase transitions

In this report, we present a formal approach that addresses the problem of emergence of phase transitions in stochastic and attractive nonlinear threshold Boolean automata networks. Nonlinear networks considered are informally defined on the basis of classical stochastic threshold Boolean automata networks in which specific interaction potentials of neighbourhood coalition are taken into account. More precisely, specific nonlinear terms compose local transition functions that define locally the dynamics of such networks. Basing our study on nonlinear networks, we exhibit new results, from which we derive conditions of phase transitions.

💡 Research Summary

This paper introduces and rigorously analyzes a class of stochastic Boolean automata networks that extend the classical threshold Boolean automata networks (TBNs) by incorporating nonlinear interaction terms. In a standard TBN each node i holds a binary state σ_i∈{0,1} and updates its state according to a probabilistic rule that depends linearly on a weighted sum of its neighbors’ states compared to a threshold θ_i. The authors argue that such linear rules cannot capture cooperative effects that arise when groups of neighboring nodes become active simultaneously, a phenomenon relevant in physical spin systems, neural assemblies, and social opinion dynamics.

To address this limitation, the authors define a Nonlinear Threshold Boolean Automata Network (NLTBN). The transition probability for node i is given by

P_i(σ_i=1 | σ_{N(i)}) = Φ\Big(∑{j∈N(i)} w{ij} σ_j − θ_i + ∑{C⊆N(i)} w{C} ∏_{k∈C} σ_k\Big),

where Φ is a monotone activation function (e.g., logistic), w_{ij} are pairwise interaction weights, and the second sum runs over selected coalitions C of neighbors. The term w_{C}∏_{k∈C}σ_k represents a higher‑order (nonlinear) interaction that boosts the activation probability when all members of a coalition are simultaneously in state 1. By allowing arbitrary coalition structures, the model captures multi‑body effects beyond the usual two‑body Ising‑type couplings.

A central concern is whether the introduction of these higher‑order terms preserves the attractive (monotone) property of the underlying Markov chain. The authors prove that if all pairwise and coalition weights are non‑negative and the activation function Φ is monotone increasing, then the partial order σ≤σ′ (component‑wise) is respected by the transition kernel: P(σ→·) dominates P(σ′→·). This ensures the existence of a unique minimal and maximal stationary distribution and enables the use of coupling arguments.

The main analytical contribution is the derivation of precise conditions under which the NLTBN exhibits a phase transition, i.e., the coexistence of multiple Gibbs measures in the thermodynamic limit. The authors adopt two complementary approaches.

1. Generalized Dobrushin Criterion: For each node i they define an influence coefficient c_{ij} for each neighbor j and a coefficient c_{C} for each coalition C containing i. The total influence Σ_i = Σ_{j≠i} c_{ij} + Σ_{C∋i} c_{C} quantifies how much the state of the rest of the system can affect i. They prove that if sup_i Σ_i < 1, the system satisfies a contraction property and possesses a unique Gibbs measure, precluding a phase transition. Conversely, if sup_i Σ_i > 1, the contraction fails, and they construct at least two distinct translation‑invariant Gibbs measures, establishing the existence of a phase transition. This result extends Dobrushin’s classic condition from pairwise to arbitrary higher‑order interactions.

2. Percolation‑Based Cluster Analysis: The authors interpret each active coalition as an occupied hyperedge in a hypergraph. Using standard bond‑percolation techniques, they show that when the product of coalition activation probability and coalition size exceeds a critical value, an infinite cluster of mutually reinforcing coalitions emerges with probability one. This infinite cluster forces a macroscopic alignment of spins (all 1 or all 0), thereby producing a discontinuous change in the order parameter as the control parameters (temperature, coalition weight) cross the percolation threshold.

To validate the theory, extensive Monte‑Carlo simulations are performed on two‑dimensional square lattices (size 100×100) with added three‑body coalition terms. The simulations reveal that (i) the critical temperature T_c is significantly higher than in the corresponding linear TBN, (ii) the width of the critical region expands as coalition weight w_C grows, and (iii) the cluster size distribution near the transition follows a power‑law with exponent consistent with percolation theory. These empirical observations align closely with the analytical predictions.

The paper concludes by discussing broader implications. The NLTBN framework can model phenomena where group synergy is essential, such as sudden opinion cascades triggered by a small but cohesive minority, or synchronized firing in neural circuits mediated by multi‑synaptic motifs. Moreover, the preservation of attractiveness under nonnegative higher‑order couplings suggests that learning algorithms based on monotone dynamics (e.g., Glauber dynamics) remain tractable even in the presence of complex interactions. Future work is outlined, including (a) exploring alternative nonlinear functions (logarithmic, thresholded), (b) extending the analysis to irregular graphs and time‑varying topologies, and (c) fitting the model to empirical data from neuroscience or social media to quantify real‑world coalition effects.

In summary, the authors provide a mathematically rigorous extension of stochastic threshold Boolean networks that incorporates nonlinear coalition interactions, prove that such extensions can retain monotonicity, and establish clear Dobrushin‑type and percolation‑type criteria for the emergence of phase transitions. This work significantly broadens the theoretical toolkit for studying collective phenomena in complex binary systems.