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The increasing popularity of social networks has initiated a fertile research area in information extraction and data mining. Anonymization of these social graphs is important to facilitate publishing these data sets for analysis by external entities. Prior work has concentrated mostly on node identity anonymization and structural anonymization. But with the growing interest in analyzing social networks as a weighted network, edge weight anonymization is also gaining importance. We present Anónimos, a Linear Programming-based technique for anonymization of edge weights that preserves linear properties of graphs. Such properties form the foundation of many important graph-theoretic algorithms such as shortest paths problem, k-nearest neighbors, minimum cost spanning tree, and maximizing information spread. As a proof of concept, we apply Anónimos to the shortest paths problem and its extensions, prove the correctness, analyze complexity, and experimentally evaluate it using real social network data sets. Our experiments demonstrate that Anónimos anonymizes the weights, improves k-anonymity of the weights, and also scrambles the relative ordering of the edges sorted by weights, thereby providing robust and effective anonymization of the sensitive edge-weights. We also demonstrate the composability of different models generated using Anónimos, a property that allows a single anonymized graph to preserve multiple linear properties.