Dynamics of agents on random networks.

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.
In this figure, the average opinion (order parameter) <|O|> is plotted for different values of the noise r showing two transitions. Following a line defined by a constant order parameter: when p is very small, most agents have zero advisor, and information cannot percolate through the network. When p increases, there is a critical probability p where agents have enough connections in order to establish consensus among the entire network. If we increase again p, a second transition occurs where groups of advisors are so large that agents cannot change their opinion anymore.

The Java Applet
Try to vary the noise r, i.e. the number of people who are changing their opinion randomly: if the noise is too large opinions of agents are different and <|O|>=0. If r is small, then opinions are suddenly correlated below a certain critical value of r. This is similar to a ferromagnetic phase transition. Observe the number of advisor in the group at the transition.

Opinion of agents are represented on a circle (positions are arbitrary). Colors mean: Green = -1, Red = +1, White = 0.
For clarity, arrows represent active advisors only for one agent.

  • If advisors groups are small, then consensus is present: since agent a more likely to change their opinion when they have a small number of advisors, information can percolate through the network and consensus can be established.
  • If advisors groups are large, then consensus disappears: too many advisors block the decision processus since all advisors have rarely the same and opposite opinion as the agent.
  • At the transition, the number of advisors is independent of the system size (fix noise).
The critical connectivity (points) marks the transition between stable consensus and stable disorder phases. The transition coincides with the maximum of the standard deviation or activity $\sigma$ of the order parameter. Computer simulations have been done for different noise $r$ and probability $p$. The standard deviation, and therefore the activity or evolution, is maximum in the red region between the two phases where it is lower (blue).
The vertical dashed line is where the majority rule applies (this corresponds to occidental democratic model!). Surprisingly the line cross the maximum of the critical connectivty curve. This means that the majority rule is the most tolerant rule to noise where activity is still possible.


Optimal evolution of random networks: from social to airports networks.
Ph. Curty
pre-print physics/0509074 (arXiv.org)

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