Abstract
This paper develops a mathematical framework to study signal networks, in which nodes can be active or inactive, and their activation or deactivation is driven by external signals and the states of the nodes to which they are connected via links. The focus is on determining the optimal number of key nodes (= highly connected and structurally important nodes) required to represent the global activation state of the network accurately. Motivated by neuroscience, medical science, and social science examples, we describe the node dynamics as a continuous-time inhomogeneous Markov process. Under mean-field and homogeneity assumptions, appropriate for large scale-free and disassortative signal networks, we derive differential equations characterising the global activation behaviour and compute the expected hitting time to network triggering. Analytical and numerical results show that two or three key nodes are typically sufficient to approximate the overall network state well, balancing sensitivity and robustness. Our findings provide insight into how natural systems can efficiently aggregate information by exploiting minimal structural components.