We study the problem of modeling multiple symmetric, weighted networks defined on a common set of nodes, where networks arise from different groups or conditions. We propose a model in which each network is expressed as the sum of a shared low-rank structure and a node-sparse matrix that captures the differences between conditions. This formulation is motivated by practical scenarios, such as in connectomics, where most nodes share a global connectivity structure while only a few exhibit condition-specific deviations. We develop a multi-stage estimation procedure that combines a spectral initialization step, semidefinite programming for support recovery, and a debiased refinement step for low-rank estimation. We establish minimax-optimal guarantees for recovering the shared low-rank component under the row-wise $\ell_{2,\infty}$ norm and elementwise $\ell_{\infty}$ norm, as well as for detecting node-level perturbations under various signal-to-noise regimes. We demonstrate that the availability of multiple networks can significantly enhance estimation accuracy compared to single-network settings. Additionally, we show that commonly-used methods such as group Lasso may provably fail to recover the sparse structure in this setting, a result which might be of independent interest.