<p>Centrality measures quantify node importance, but they typically evaluate it globally across an entire network. However, nodes often belong to distinct categories, and a user analyzing influence may prioritize certain categories over others. It is therefore natural to evaluate node importance according to a specific preference ranking of these categories. To address this, we extend standard core decomposition and the well-known peeling algorithm to a lexicographic setting, introducing <i>lexicographic core decomposition</i> and <i>lexipeeling</i>, which we show is a special case of weighted node core decomposition. We evaluate lexipeeling against 79 baselines on 7 real-world networks. Lexipeeling ranks the true top spreader higher than any other baseline on 5 of 7 datasets, with MRR ranging from 0.478 on iPhone-Samsung to 1.0 on PolBlogs, PolBooks, and LastFM-Asia. However, like standard coreness, it assigns the same rank to many nodes, so weak spreaders may appear among the highest ranked ones. To address this, we propose composite methods that break lexipeeling’s ties using a secondary measure. Among these, lexipeeling-degree most reliably places truly influential nodes among its highest ranked ones across all datasets. Our code is publicly available: <a href="https://github.com/rdenni/Lexicoreness">https://github.com/rdenni/Lexicoreness</a></p>