  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.
