  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.
