Boolean Networks with Time Delays: A Condition for Fastest Consensus

Boolean networks consist of nodes that adopt discrete state values based on inputs from other nodes within the network. The study of these networks—focusing on their stability, convergence, and control—spans various disciplines including genetics, communication systems, social and neural networks, and computer science. One of the challenges in these networks is the presence of time delays, which arise from communication and information processing. Time delays often complicate analytical approaches, making their impact hard to measure and predict.



This paper considers Boolean networks, initialized with random states, and investigates how long they take to achieve consensus in the presence of communication delays. We reveal a straightforward principle that relates the volume of information in transit to the network dynamics. Remarkably, this principle holds across a diverse array of network types. As a practical application of our findings, we determine the optimal rate of communication that enhances network performance while minimizing consensus time.