Clustering Does Not Always Imply Latent Geometry

The latent space approach to complex networks has revealed fundamental principles and symmetries, enabling geometric methods. However, the conditions under which network topology implies geometricity remain unclear. We provide a mathematical proof and empirical evidence showing that multiscale self-similarity in complex networks is a crucial factor of latent geometry. Using degree-thresholding renormalization, we prove that a general class of ensembles of random scale-free networks in Riemannian manifolds of constant curvature and of any dimension are self-similar when interactions are pairwise. Hence, both nonvanishing local clustering in the thermodynamic limit and self-similarity are required to imply geometricity. Our findings highlight that correlated links can lead to a finite clustering coefficient without self-similarity, and therefore without inherent latent geometry. The implications are significant for network mapping and ensemble equivalence between graphs and continuous spaces.