Detectability of the spectral method for sparse graph partitioning

We show that modularity maximization with the resolution parameter offers a unifying framework of graph partitioning. In this framework, we demonstrate that the spectral method exhibits universal detectability, irrespective of the value of the resolution parameter, as long as the graph is partitioned. Furthermore, we show that when the resolution parameter is sufficiently small, a first-order phase transition occurs, resulting in the graph being unpartitioned.

💡 Research Summary

**
The paper presents a unified framework for graph partitioning based on modularity maximization with an adjustable resolution parameter θ. By incorporating θ, the authors show that three widely used objective functions—standard modularity, normalized cut (Ncut), and the log‑likelihood of the degree‑corrected stochastic block model—can all be expressed as a single modularity function
\