Many biological networks are characterized by directed edges that represent either activating (positive) or inhibiting (negative) regulation. Most graph-theoretical methods used to study biological networks either disregard this important feature, or study the role of edge sign only in the context of small subgraphs called motifs. Here, we develop path-based measures which capture, on continuous scales spanning negative and positive values, both the long- and short-range regulatory relationships among node pairs. These measures also allow the quantification of each node's overall influence on the whole network and its susceptibility to regulation by the rest of the network. We apply the measures to a network representation of the mammalian immune response to simultaneous attack by allergen and respiratory bacteria. Although allergen and bacteria elicit different immune pathways, there is significant overlap (cross-talk) and feedback between these pathways. We identify key immune components in this cross-talk; particularly revealing the importance of natural killer cells as a key regulatory target in the cross-talk.