We construct and analyze a random graph model for discrete choice with social interaction and several groups of equal size. We concentrate on the case of two groups of equal sizes and we allow the interaction strength within a group to differ from the interaction strength between the two groups. Given that the resulting graph is sufficiently dense we show that, with probability 1, the average decision in each of the two groups is the same as in the fully connected model. In particular, we show that there is a phase transition: If the interaction among a group and between the groups is strong enough the average decision per group will either be positive or negative and the decision of the two groups will be correlated. We also compute the free energy per particle in our model.

我们构建和分析随机图模型，用于具有社交互动和几组相等大小的离散选择。我们专注于大小相等的两组的情况，并允许一组内的交互强度不同于两组之间的交互强度。给定结果图足够密集，我们证明在概率为1的情况下，两组中的平均决策与完全连接模型中的决策相同。特别是，我们表明存在一个相变：如果一个组之间以及组之间的交互足够强大，则每个组的平均决策将为正或负，并且两个组的决策将相关。我们还计算了模型中每个粒子的自由能。