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 {\em attention} $1/q$ to each partner. The mutual attention between two connected persons $i$ and $j$ is taken equal to the geometric mean $1/\sqrt{q_iq_j}$. Opinion is represented as usual by an Ising spin $s=\pm 1$ and mutual attention is given through a two-spin coupling $J_{ij} = J Q/\sqrt{q_iq_j}$, $Q$ 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" $T_c$ proportional to $JQ$ is found, which is independent of the number of persons $N$, for large $N$.