Emergent hierarchy through conductance-based node constraints

Abstract

The presence of hierarchy in many real-world networks is not yet fully explained. Complex interaction networks are often coarse-grain models of vast modular networks, where tightly connected subgraphs are agglomerated into nodes for simplicity of representation and feasibility of analysis. The emergence of hierarchy in growing complex networks may stem from one particular property of these ignored subgraphs: their graph conductance. Being a quantification of the main bottleneck of flow on the subgraph, all such subgraphs will then have a specific structural limitation associated with this scalar value. This supports the consideration of heterogeneous degree restrictions on a randomly growing network for which a hidden variable model is proposed based on the basic \textit{rich-get-richer} scheme. Such node degree restrictions are drawn from various probability distributions, and it is shown that restriction generally leads to increased measures of hierarchy, while altering the tail of the degree distribution. Thus, a general mechanism is provided whereby inherent limitations lead to hierarchical self-organization.