
TL;DR
This paper investigates the relationship between network curvature, logical centers, and tree representations, extending known connections to weighted graphs and clarifying the equivalence of different construction methods.
Contribution
It extends the link between Gromov's hyperbolicity and logical centers to weighted graphs and proves the asymptotic equivalence of key methods for tree representation.
Findings
Extended hyperbolicity-center connection to weighted graphs
Counterexamples show other curvature definitions lack logical centers
Proved asymptotic equivalence of tree construction methods
Abstract
Many applications in network science have recently been discovered for the "curvature" of a network, but there is no consensus on the definition for this term. A common approach in these applications is to derive from the curvature either a "logical center" of the network or a tree representation of the network (these would only exist when the curvature is negative), but that such structures can be extracted using curvature alone remains largely conjectural. A connection between one type of curvature -- Gromov's hyperbolicity -- and a tree representation has been known for decades, and recently it has also been connected for unweighted graphs to a logical center. We extend the connection between Gromov's hyperbolicity and a logical center to weighted graphs, and we construct counterexamples showing that no other proposed definition for curvature implies the existence of a logical…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Complex Network Analysis Techniques · Theoretical and Computational Physics
