A surface-depth theory of the emergence of complex networks
The broadly general characteristics of complex networks found across disciplines-- such as high clustering coefficients and heavy-tailed degree distributions-- has long invited the question of whether there are general generating mechanisms behind them. Here, we propose a theory of such mechanisms and undertake several experiments which validate it. This theory proposes that there are two key principles at work in the emergence of a network, constituting a `surface' factor and a `depth' factor making up the existence probability of network edges. The surface factor describes nodes as having tendencies for attachment which follow a log-normal distribution. The deep factor suggests that beneath these potentials for attachment there exist any number of important latent variables describing the nodes. These variables are formulated as a high-dimensional manifold and the `distances' between pairs of nodes on this manifold constitute a similarity weighting informing on the probability that any two nodes are connected. Using standard network measures, the topology of 110 networks across a variety of disciplines shows agreement with a simple two-parameter model based on this theory. Importantly, we show that the log-normal surface factor can explain the power-law-like degree distributions of sparse networks and, more strongly, the variety of degree distributions found across networks of different densities. We also demonstrate how inverting an estimated surface factor of a complete weighted world city network provides more plausible clusters of nodes than the original network based on geometric and cultural considerations. This theory proposes a new fundamental formulation of complex networks with wide reaching consequences throughout the multidisciplinary domain of complex systems.