Label propagation is frequently encountered in machine learning and data mining applications on graphs, either as a standalone problem or as part of node classification. Many label propagation algorithms utilize random walks (or network propagation), which provide limited ability to take into account negatively-labeled nodes (i.e., nodes that are known to be not associated with the label of interest). Specialized algorithms to incorporate negatively labeled samples generally focus on learning or readjusting the edge weights to drive walks away from negatively-labeled nodes and toward positively-labeled nodes. This approach has several disadvantages, as it increases the number of parameters to be learned, and does not necessarily drive the walk away from regions of the network that are rich in negatively-labeled nodes. We reformulate random walk with restarts and network propagation to enable "variable restarts", that is the increased likelihood of restarting at a positively-labeled node when a negatively-labeled node is encountered. Based on this reformulation, we develop CusTaRd, an algorithm that effectively combines variable restart probabilities and edge re-weighting to avoid negatively-labeled nodes. In addition to allowing variable restarts, CusTaRd samples negatively-labeled nodes from neighbors of positively-labeled nodes to better characterize the difference between positively and negatively labeled nodes. To assess the performance of CusTaRd, we perform comprehensive experiments on four network datasets commonly used in benchmarking label propagation and node classification algorithms. Our results show that CusTaRd consistently outperforms competing algorithms that learn/readjust edge weights, and sampling of negatives from the close neighborhood of positives further improves predictive accuracy.