A self-consistent approach to measure preferential attachment in networks and its application to an inherent structure network

TL;DR

This paper introduces a self-consistent method to detect and quantify preferential attachment in networks, applying it to an energy landscape network to understand its growth dynamics and the emergence of scale-free properties.

Contribution

A novel self-consistent approach to measure preferential attachment in static networks and its application to energy landscape networks.

Findings

Preferential attachment is detectable in the energy landscape network.

The network exhibits scale-free degree distribution influenced by preferential attachment.

Other factors besides preferential attachment affect the network's growth and structure.

Abstract

Preferential attachment is one possible way to obtain a scale-free network. We develop a self-consistent method to determine whether preferential attachment occurs during the growth of a network, and to extract the preferential attachment rule using time-dependent data. Model networks are grown with known preferential attachment rules to test the method, which is seen to be robust. The method is then applied to a scale-free inherent structure network, which represents the connections between minima via transition states on a potential energy landscape. Even though this network is static, we can examine the growth of the network as a function of a threshold energy (rather than time), where only those transition states with energies lower than the threshold energy contribute to the network.For these networks we are able to detect the presence of preferential attachment, and this helps to explain the ubiquity of funnels on energy landscapes. However, the scale-free degree distribution shows some differences from that of a model network grown using the obtained preferential attachment rules, implying that other factors are also important in the growth process.