Geometrical congruence and efficient greedy navigability of complex networks

TL;DR

This paper challenges the assumption that hyperbolic networks are geometrically congruent and efficiently navigable, revealing that such properties are not universal and depend on specific network parameters and real-world factors.

Contribution

It introduces a new measure called geometrical congruence (GC) and demonstrates its variability across different network types and real-world brain connectomes.

Findings

Hyperbolic networks often lack geometrical congruence and efficient greedy navigability.

GC varies significantly in real brain networks based on gender and age.

Efficient navigation emerges only near a power-law exponent of 2 in certain models.

Abstract

Hyperbolic networks are supposed to be congruent with their underlying latent geometry and following geodesics in the hyperbolic space is believed equivalent to navigate through topological shortest paths (TSP). This assumption of geometrical congruence is considered the reason for nearly maximally efficient greedy navigation of hyperbolic networks. Here, we propose a complex network measure termed geometrical congruence (GC) and we show that there might exist different TSP, whose projections (pTSP) in the hyperbolic space largely diverge, and significantly differ from the respective geodesics. We discover that, contrary to current belief, hyperbolic networks do not demonstrate in general geometrical congruence and efficient navigability which, in networks generated with nPSO model, seem to emerge only for power-law exponent close to 2. We conclude by showing that GC measure can impact also real networks analysis, indeed it significantly changes in structural brain connectomes grouped by gender or age.