Embedded dimensionality as minority group size tends to zero in two-block stochastic block models

Determine whether, in the two-block stochastic block model used for the group-prevalence experiments (with fixed in-group and between-group link probabilities and Random Dot Product Graph embedding dimension estimated via truncated singular value decomposition with elbow-based selection), the optimal embedding dimension d-hat remains low or rebounds to a higher value as the size of the smaller block approaches zero relative to the overall network size.

Background

The paper investigates how polarization relates to the embedded dimensionality of social networks modeled as Random Dot Product Graphs. Using two-block stochastic block models (SBMs), the authors varied group sizes while keeping link probabilities fixed to explore how group imbalance affects the optimal embedding dimension d-hat estimated via the elbow method on singular values.

In simulations with group-size ratios progressing from balanced (50/50) to highly imbalanced (99/1), the authors observed that d-hat initially increased slightly as one group became predominant but then decreased sharply at extreme imbalance. They attribute this drop partly to sparsity effects in the very small group and note the real-world relevance for contexts with very small minority groups (e.g., refugees). However, they did not extend the simulations to smaller minority sizes, leaving open whether d-hat remains low or rebounds as the minority group’s size tends toward zero.