Aneuploidy and structural variations (SVs) generate cancer genomes containing a mixture of rearranged genomic segments with extensive somatic copy number alterations. However, existing methods can identify either SVs or allele-specific copy number alterations but not both simultaneously, which provides a limited view of cancer genome structure. Here, we introduce Weaver, an algorithm for the quantification and analysis of allele-specific copy numbers of SVs. Weaver uses a Markov random field to estimate joint probabilities of allele-specific copy numbers of SVs and their inter-connectivity based on paired-end whole-genome sequencing data. Weaver also predicts the timing of SVs relative to chromosome amplifications. We demonstrate the accuracy of Weaver using simulations and findings from whole-genome optical mapping. We apply Weaver to generate allele-specific copy numbers of SVs for MCF-7 and HeLa cell lines and identify recurrent SV patterns in 44 TCGA ovarian cancer whole-genome sequencing datasets. Our approach provides a more complete assessment of the complex genomic architectures inherent to many cancer genomes.