Characterizing consensus in threshold Boolean networks

Consensus has become an active research topic in the field of social science, blockchain, and decision-making, to name a few. The study of how a group of people, entities, or agents generally reach an agreement is of interest. This paper studies how a consensus is reached using the threshold Boolean network model, where nodes represent agents taking on two possible values: 1 or 0. A threshold Boolean network is a directed graph with weights. It has typically been used as a model of gene regulatory networks. Each node has a Heaviside function depending linearly on its inputs and an updating scheme (in what order the nodes update their values). By using threshold Boolean networks, there are two possibilities of reaching a consensus. When all the possible configurations in the network converge to the fixed point attractor, all the nodes have only 1s or only 0s. We adopt a reverse engineering approach to study the characteristics of the networks that can model consensus. We search for such networks using evolutionary computation containing only one of the two mentioned attractors (consensus property). The search consists of finding the weights of the edges and the threshold value of each node. We characterize the resulting networks by the total number of edges, the number of positive edges, the number of negative edges, the average indegree, and the steps needed to reach consensus.