When networks are generated from a degree-corrected stochastic block model (DCSBM), the theoretical results regarding the mis-classification rates and performance of sub-sampling routines may change significantly. The DCSBM allows for varying degrees among nodes, which introduces additional complexity in the network structure compared to the standard stochastic block model (SBM).
In the context of the divide-and-conquer algorithms discussed, the expected mis-classification rates would need to account for the degree distribution of nodes. Specifically, the probability of sampling core nodes may be influenced by their degree, as higher-degree nodes are more likely to be sampled under degree-based sub-sampling routines. This could lead to a more nuanced understanding of how well the sub-sampling methods capture the underlying community or core-periphery structures.
The theoretical bounds derived for the mis-classification rates would also need to incorporate the degree distribution, potentially leading to different expressions for the second term in the error bounds. For instance, if the degree distribution is heavy-tailed, the probability of sampling high-degree nodes could be significantly higher, which may improve the performance of certain sub-sampling methods. Consequently, the analysis would need to explore how these degree variations affect the coverage of core nodes and the overall accuracy of community detection or core-periphery identification.