Abstract
Networks and graphs are highly relevant in modeling real-life communities and their interactions. In order to gain insight in their structure, different roles are attributed to vertices, effectively clustering them in equivalence classes. A new formal definition of regular equivalence is presented in this paper, and the relation with other equivalence types is investigated and mathematically proven. An efficient algorithm is designed, able to detect all regularly equivalent roles in large-scale complex networks. We apply it to both Barabási–Albert random networks, as well as real-life social networks, which leads to interesting insights.
Highlights
The availability of large datasets, derived from many sources of real-life social networks from different kinds, allows deep research into the underlying structure behind these networks
We investigate social networks using minimal regular equivalence relations, attaching roles to vertices with a similar local neighborhood, which results in a good balance between being too strict or too loose when analyzing the network structure
We introduce a fast and efficient algorithm to calculate these minimal regular equivalences, which allows researchers to quickly understand the structure of a network
Summary
The availability of large datasets, derived from many sources of real-life social networks from different kinds, allows deep research into the underlying structure behind these networks. Networks generated through the procedure by Watts and Strogatz are not scale-free, a property that is sometimes observed in real-life networks: the distribution of vertex degrees in real-life networks might follow a power law [5,6,7], such that the probability that an arbitrary vertex has degree k is inversely proportional to kα for some α > 1 called the scaling exponent This behavior is modeled by the scale-free random network generator proposed by Barabási and Albert [8], which applies an iterative growth process combined with edge addition through preferential attachment [9,10]. The Barabási–Albert generator, together with its many variations, is used throughout the literature as it is the most realistic scale-free network generator available [11,12,13]
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