Bridging relationships between individuals situated in different parts of a social network are important conduits for information and resources in social and organizational settings. Dyadic tie strength has often been used as an indicator for whether a relationship is bridging, under the assumption that bridging ties are always weak ties. However, recent empirical evidence suggests that bridging ties are often strong, forcing us to rethink the relationship between social network structure and dyadic tie strength. Here, we provide an analysis based on algebraic topology which clarifies this relationship between network structure and dyadic tie strength. Rather than model the network as a graph, we use a simplicial complex which can explicitly encode group interactions between three or more individuals. First, we show theoretically and empirically that Edge PageRank, an algebraic topological measure originally defined as an extension of the classical PageRank measure, is a valid continuous measure of how well a relationship acts as a bridge. Second, we use the tool of Hodge Decomposition, which allows us to decompose any flow in a simplicial complex into three orthogonal components, to clarify the relationship between dyadic tie strength and network structure. We find that individuals invest less in relationships associated with topological holes in the network, replicating and explaining recent empirical results that bridging relationships spanning short network distances tend to be weak, whereas those spanning longer distances are strong. Our results are validated on 15 large scale datasets and suggest the value of algebraic topological methods in empirical network analysis.