Temporal, structural, and functional heterogeneities extend criticality and antifragility in random Boolean networks

TL;DR

The paper investigates how heterogeneity in time, structure, and function expands criticality and antifragility in random Boolean networks (RBNs). It generalizes RBNs to include structural (degree distributions), temporal (updating cadence), and functional (node-wise activation probabilities) heterogeneity, and uses a Shannon-entropy-based complexity $C$ with $I = -K \sum p_i \log p_i$ and $C = 4 I (1 - I)$, along with an antifragility measure $\oint = -\Delta\samekh \Delta x$ to quantify performance under perturbations. The findings show additive-like extension of the critical region with more heterogeneity, with triple heterogeneity (3He) providing the strongest extension across a range of connectivities $K$, while maximum antifragility tends to occur in homogeneous networks, highlighting a nontrivial balance between homogeneity and heterogeneity. These results imply heterogeneity as a natural mechanism to widen criticality and antifragility in complex systems, with potential implications for natural and engineered networks where robustness, adaptability, and evolvability are desirable.

Abstract

Most models of complex systems have been homogeneous, i.e., all elements have the same properties (spatial, temporal, structural, functional). However, most natural systems are heterogeneous: few elements are more relevant, larger, stronger, or faster than others. In homogeneous systems, criticality -- a balance between change and stability, order and chaos -- is usually found for a very narrow region in the parameter space, close to a phase transition. Using random Boolean networks -- a general model of discrete dynamical systems -- we show that heterogeneity -- in time, structure, and function -- can broaden additively the parameter region where criticality is found. Moreover, parameter regions where antifragility is found are also increased with heterogeneity. However, maximum antifragility is found for particular parameters in homogeneous networks. Our work suggests that the "optimal" balance between homogeneity and heterogeneity is non-trivial, context-dependent, and in some cases, dynamic.

Figures (3)

• Figure S1: Illustration of a random Boolean network with $N = 9$ nodes and $K = 2$ inputs per node (self-connections are allowed). The node rules are commonly represented by lookup-tables, which associate a $1$-bit output (the node's future state) to each $2^K$ possible $K$-bit input configuration. The out-column is commonly called the "rule" of the node.
• Figure S2: Phase diagram described by Derrida and Pomeau DerridaPomeau1986. The solid curve represents the critical connectivity $K_c = [2p(1-p)]^{-1}$. When $K<K_c$ the network exhibits ordered dynamics (small shaded area to the left of $K_c$), while for $K>K_c$ the network exhibits chaotic dynamics (large shaded area to the right of $K_c$).
• Figure S3: Example of three regimes of CRBN using 50 nodes ($N=50$) with 200 steps each. (time flows downwards). The resulting complexities $C$ (see Eq. \ref{['eq:C']}) are: for $K=1$, $C=0.0513$, for $K=2$, $C=0.8651$, and for $K=5$, $C=0.3079$.