What are the necessary conditions in order to have consensus among an assembly of people? Why, sometimes, do all people share the same opinion in a short time? Is consensus more difficult to achieve in our complex and connected society than in the less connected society of the beginning of the 20th century?
Consider for example, a school class of students who want to make an important travel. They have to choose a destination among two possible choices, Alaska or Rio de Janeiro, and every student has naturally a strong preference for one of the destinations. If students are isolated then nobody changes his opinion, and no consensus will be found. When students take advice from one or two friends in the class, they may change their opinion quickly and a consensus will be achieved. Now if each student has many friends, he will always find a small group among his friends that share his opinion and support his choice. Hence students will keep their opinion and no consensus is found although every student is connected to a large number of friends.
A new model of basic dynamics is introduced where agents have a three state opinion: yes, no or no opinion. Each agent has a group of G advisors taken at random among other agents. When the group of advisors is large, opinions are no more correlated and are random. A phase transition occures between the two states. Many real systems like social networks, neural networks, gene networks and supply chains have an optimal evolution at the maximum of activity, precisely at the critical connectivity depending on their intrinsic noise.
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