# Multi-group Binary Choice with Social Interaction and a Random Communication Structure -- a Random Graph Approach

## TL;DR

This paper introduces a random graph model for binary choices with social interactions across two groups, revealing phase transitions and decision correlations depending on interaction strengths, and analyzing the model's free energy.

## Contribution

It develops a random graph approach to model social interactions in binary choice scenarios, highlighting phase transitions and decision correlations between groups.

## Key findings

- Average decisions match fully connected models in dense graphs.
- Strong interactions lead to correlated group decisions.
- Computed free energy per particle for the model.

## Abstract

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 one, 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.