  Modularity maximization has been a fundamental tool for understanding the
community structure of a network, but the underlying optimization problem is
nonconvex and NP-hard to solve. State-of-the-art algorithms like the Louvain or
Leiden methods focus on different heuristics to help escape local optima, but
they still depend on a greedy step that moves node assignment locally and is
prone to getting trapped. In this paper, we propose a new class of
low-cardinality algorithm that generalizes the local update to maximize a
semidefinite relaxation derived from max-k-cut. This proposed algorithm is
scalable, empirically achieves the global semidefinite optimality for small
cases, and outperforms the state-of-the-art algorithms in real-world datasets
with little additional time cost. From the algorithmic perspective, it also
opens a new avenue for scaling-up semidefinite programming when the solutions
are sparse instead of low-rank.
