Our world is abundant with interdependent interactions occurring at all levels-- be it in the global ecology, human social institutions, within the human brain, or in micro-scale protein interactions. When mapped as networks, connectivity patterns across such different phenomena show broadly consistent features, yet an accurate universal theory to explain this remains elusive. Here, we pose a new theory which considerably outperforms current mechanistic theories of complex network emergence in network modelling accuracy. Here, link probability is defined by a log-normal attachment (surface) factor and a Euclidean space-embedded node similarity (depth) factor. Topological modelling based on this theory strongly outperforms power-law and hyperbolic geometry explanations across 110 networks. A surface factor inversion approach on an economic world city network and an fMRI connectome results in considerably more geometrically aligned nearest neighbour networks. The proposed theory establishes new foundations from which to understand, analyse, deconstruct and interpret network phenomena.