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

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.

💡 Research Summary

The paper investigates how heterogeneity—differences in time scales, network structure, and node functions—affects the emergence of criticality and antifragility in Random Boolean Networks (RBNs), a classic model of discrete dynamical systems originally proposed to capture genetic regulatory networks. Traditional RBNs (often called Classical RBNs or CRBNs) assume four forms of homogeneity: each node receives exactly K inputs, connections are chosen uniformly at random, all nodes update synchronously, and every Boolean function has the same bias p (probability of outputting 1). Under these assumptions, analytical work by Derrida and Pomeau shows that a phase transition occurs at a critical connectivity Kc =