We study how multi-hop information impacts convergence in social influence networks. In influence networks, nodes have a choice between two options A and B, and each node prefers to end up choosing the option that a majority of its neighbors choose. We consider the case of innovation adoption in which nodes can only change from A to B, but not backwards. For this model, we ask the question, when is it safe for a node to switch from A to B? If nodes have multi-hop information about the network, rather than knowing only the state of their immediate neighbors, the answer to this question becomes complex. The reason is that a node needs to recursively reason about what its neighbors know, and whether given their knowledge they will also upgrade to B. In this paper, we assume that each node has complete knowledge about its k-hop neighborhood, but does not know anything about the network beyond k-hops. We study how different local decision algorithms achieve different properties in terms of safety and conversion ratio (how many nodes ultimately upgrade to B). We characterize the possible algorithms by classifying them into a hierarchy of algorithms. Each class of algorithms in this hierarchy is distinguished by a natural safety property that it guarantees. For each class, we give an optimal algorithm in terms of conversion ratio, and we show that each class is fully contained in the class of lower safety level. Conversely, each lower-safety class can achieve strictly higher conversion ratio than any algorithm in the safer class. Thus, our hierarchy reveals a strict trade-off between safety and conversion ratio. Finally, we show that each class of algorithms satisfies two natural closure properties.