# Explaining the emergence of complex networks through log-normal fitness in a Euclidean node similarity space

## TL;DR

This paper introduces a novel log-normal fitness and Euclidean space-based model that better explains the architecture of diverse complex networks, reconciling power-law and log-normal degree distributions.

## Contribution

It proposes a new theory modeling link probability with log-normal fitness and Euclidean similarity, outperforming existing models across various network types.

## Key findings

- Model outperforms power-law and hyperbolic models on 110 networks.
- Degree distributions show power-law at low densities and log-normal at high densities.
- Surface factor inversion improves geometric alignment in real-world networks.

## Abstract

Networks of disparate phenomena-- be it the global ecology, human social institutions, within the human brain, or in micro-scale protein interactions-- exhibit broadly consistent architectural features. To explain this, we propose a new theory where link probability is modelled by a log-normal node fitness (surface) factor and a latent Euclidean space-embedded node similarity (depth) factor. Modelling based on this theory considerably outperforms popular power-law fitness and hyperbolic geometry explanations across 110 networks. Importantly, the degree distributions of the model resemble power-laws at small densities and log-normal distributions at larger densities, posing a reconciliatory solution to the long-standing debate on the nature and existence of scale-free networks. Validating this theory, a surface factor inversion approach on an economic world city network and an fMRI connectome results in considerably more geometrically aligned nearest neighbour networks. This establishes new foundations from which to understand, analyse, deconstruct and interpret network phenomena.