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Bridging relationships between individuals situated in different parts of asocial network are important conduits for information and resources in socialand organizational settings. Dyadic tie strength has often been used as anindicator for whether a relationship is bridging, under the assumption thatbridging ties are always weak ties. However, recent empirical evidence suggeststhat bridging ties are often strong, forcing us to rethink the relationshipbetween social network structure and dyadic tie strength. Here, we provide ananalysis based on algebraic topology which clarifies this relationship betweennetwork structure and dyadic tie strength. Rather than model the network as agraph, we use a simplicial complex which can explicitly encode groupinteractions between three or more individuals. First, we show theoreticallyand empirically that Edge PageRank, an algebraic topological measure originallydefined as an extension of the classical PageRank measure, is a validcontinuous measure of how well a relationship acts as a bridge. Second, we usethe tool of Hodge Decomposition, which allows us to decompose any flow in asimplicial complex into three orthogonal components, to clarify therelationship between dyadic tie strength and network structure. We find thatindividuals invest less in relationships associated with topological holes inthe network, replicating and explaining recent empirical results that bridgingrelationships spanning short network distances tend to be weak, whereas thosespanning longer distances are strong. Our results are validated on 15 largescale datasets and suggest the value of algebraic topological methods inempirical network analysis.

Which Bridges Are Weak Ties? Algebraic Topological Insights on Network Structure and Tie Strength