Abstract
The study of complex networks has pursued an understanding of macroscopic behaviour by focusing on power-laws in microscopic observables. Here, we uncover two universal fundamental physical principles that are at the basis of complex network generation. These principles together predict the generic emergence of deviations from ideal power laws, which were previously discussed away by reference to the thermodynamic limit. Our approach proposes a paradigm shift in the physics of complex networks, toward the use of power-law deviations to infer meso-scale structure from macroscopic observations.
Highlights
Degrees, until the process is stopped by node depletion
While our real-world examples are often related to biology, all of our arguments are immediately transferable to physical situations where previous analysis has generally stopped at the preferential attachment level
This provides an important input for the modelling of real world systems
Summary
We consider a novel generic network building algorithm (our ‘primary model’) that implements both principles at the most basic level as follows. We start from a connected network of N0 nodes. If an outside node is added, the new node connects to the network by m edges, where the target nodes are sampled according to their degree k Two nodes are independently chosen using preferential attachment. If the two chosen nodes are not identical and not already connected, an edge is established, which expresses the second fundamental principle in terms of an ‘edge saturation’ (at a level defined by p and m, implemented right from the start of the network’s growth). The process stops if the set of available nodes is depleted. The details of how this happens in time are outlined in our ‘Statistical modelling’ section)
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