  This paper presents a novel approach to estimating community membership
probabilities for network vertices generated by the Degree Corrected Mixed
Membership Stochastic Block Model while preserving individual edge privacy.
Operating within the $\varepsilon$-edge local differential privacy framework,
we introduce an optimal private algorithm based on a symmetric edge flip
mechanism and spectral clustering for accurate estimation of vertex community
memberships. We conduct a comprehensive analysis of the estimation risk and
establish the optimality of our procedure by providing matching lower bounds to
the minimax risk under privacy constraints. To validate our approach, we
demonstrate its performance through numerical simulations and its practical
application to real-world data. This work represents a significant step forward
in balancing accurate community membership estimation with stringent privacy
preservation in network data analysis.
