Social network

Social network is a structure of relationships — ties, connections, interactions — among a set of agents, whether individuals, organizations, or institutions. The study of social networks focuses on how the pattern of ties shapes the flow of information, influence, and resources, producing outcomes that cannot be predicted from the properties of the agents in isolation. A social network is not merely a collection of dyads; it is a system-level structure whose topology determines reachability, diffusion speed, and the concentration of power.\n\nThe foundational insight of social network analysis is that "structure matters more than attributes." Who you are — your skills, beliefs, resources — matters less for many outcomes than where you are in the network: your centrality, your structural holes, your clustering coefficient. The same individual in a different network position experiences a different information environment, a different set of social pressures, and a different distribution of opportunities. This makes social networks a primary mechanism of emergence in social systems.\n\n\n

Network Geometry

Social networks are not merely combinatorial graphs; they are geometric objects whose large-scale curvature constrains their function. The field of network geometry — drawing on Ricci curvature, hyperbolic embedding, and discrete differential geometry — has shown that the curvature of a social network predicts phenomena that local structural measures cannot.

Networks with negative curvature, such as those with hierarchical or tree-like structure, exhibit distinct information-flow patterns: they form natural bottlenecks and echo chambers because geodesics diverge. Networks with positive curvature, like dense clusters with high triadic closure, promote rapid consensus but also groupthink. The Ollivier-Ricci curvature and Forman-Ricci curvature provide computable discrete analogs of classical curvature that have been applied to detect communities, predict viral spread, and identify vulnerable nodes in collaboration networks.

This geometric perspective resolves a limitation of the foundational claim that "structure matters more than attributes." Structure itself has geometry, and geometry is not reducible to adjacency lists. The curvature of a social network is an emergent property of its local connectivity patterns, just as the curvature of a Riemannian manifold emerges from its metric. To study social networks without geometry is to study topology without continuity: technically possible, conceptually impoverished.