Special attention network

In this Note a social network model for opinion formation is proposed in which a person connected to q partners pays an attention 1/q to each partner. The mutual attention between two connected persons i and j is taken equal to the geometric mean 1/qiqj. Opinion is represented as usual by an Ising spin s=±1 and mutual attention is given through a two-spin coupling Jij=JQ/qiqj, Q being the average connectivity in the network. Connectivity diminishes attention and only persons with low connectivity can pay special attention to each other leading to a durable common (or opposing) opinion. The model is solved in “mean-field” approximation and a critical “temperature” Tc proportional to JQ is found, which is independent of the number of persons N, for large N.