The Curvature of Culture and Society: Analyzing meaning and connection through curved embeddings of text and networks

Text and network embedding models have become a powerful tool for the study of culture and society, respectively, in recent years. Word embeddings may be understood as representing semantic relations between words as relationships between vectors in a high-dimensional semantic space, specifying a relational model of meaning consistent with many contemporary theories of culture. Similarly, network embeddings may be understood as representing social relations between persons or collectives as relationships between vectors in a high-dimensional space. In this paper, we theorize the independent ontology and value of discrete textual expressions and network ties and curved manifolds built from them, where linguistic expression trace communicative action and network topologies trace social interaction, while continuous manifolds capture the space of probable actions and interactions. Then we introduce a unified, geometric characterization of system-level structure in textual and social networks that simplifies description while improving prediction of system-level network processes. By viewing words in an expression and nodes in graphs as embedded in a latent manifold, geometric curvature of the manifold shapes action and interaction patterns. Regions embedded in positive curvature manifest dense thickets of words and ties through which information pools and cycles; regions embedded in negative curvature are characterized by infrequent communication and sparse ties representing ridges over which information spreads and flows. In this way, curvature combines local and global perspectives, providing a continuous characterization of semantic and network structure that eliminates the need for discrete distinctions between roles and communities to enable accurate modeling of system-level processes like semantic and network evolution and information diffusion with increased accuracy. We develop statistical tests for network curvature estimation, show how they link and validate emerging approaches to curvature measurement in discrete mathematics and machine learning, and demonstrate the utility of our approach for characterizing semantic and social networks, predicting network evolution and deepening geometric understanding of semantic and social constraint using simulated and observed networks across domains of multi-ethnic community life, adolescent school interactions, and scientific innovation.