Phase Transitions of Diversity in Stochastic Block Model Dynamics

This paper proposes a stochastic block model with dynamics where the population grows using preferential attachment. Nodes with higher weighted degree are more likely to recruit new nodes, and nodes always recruit nodes from their own community. This model can capture how communities grow or shrink based on their collaborations with other nodes in the network, where an edge represents collaboration on a project. Focusing on the case of two communities, we derive a deterministic approximation to the dynamics and characterize the phase transitions for diversity, i.e. the parameter regimes in which either one of the communities dies out or the two communities reach parity over time. In particular, we find that the minority may vanish when the probability of cross-community edges is low, even when cross-community projects are more valuable than projects with collaborators from the same community.