Rules, agents and order

Abstract: Complex systems often exhibit highly structured network topologies that reflect functional constraints. In this work, we investigate how, under varying combinations of system-wide selection rules and special agents, different classes of random processes give rise to global order, with a focus restricted to finite-size networks. Using the large-$N$ Erdos-Renyi model as a null baseline, we contrast purely random link-adding processes with goal-directed dynamics, including variants of the chip-firing model and intracellular network growth, both driven by transport efficiency. Through simulations and structural probes such as $k$-core decomposition and $HITS$ centrality, we show that purely stochastic processes can spontaneously generate modest functional structures, but that significant departures from random behavior generically require two key conditions: critical topological complexity and dynamic alignment between topology and functionality. Our results suggest that the emergence of functional architectures depends not only on the presence of selection mechanisms or specialized roles, but also on the network's capacity to support differentiation and feedback. These findings provide insight into how topology-functionality relationships emerge in natural and artificial systems and offer a framework for using random graph baselines to diagnose the rise of global order in evolving finite-size networks.