A Mathematical Formalisation of the γ-contraction Problem

Networks play an ubiquitous role in computer science and real-world applications, offering multiple kind of information that can be retrieved with adequate methods. With the continuous growing in the amount of data available, networks are becoming larger day by day. Consequently, the tasks that were easily achievable on smaller networks, often becomes impractical on huge amount of data, either due to the high computational cost or due to the impracticality to visualise corresponding data. Using distinctive node features to group large amount of connected data into a limited number of clusters, hence represented by a representative per cluster, proves to be a valuable approach. The resulting contracted graphs are more manageable in size and can reveal previously hidden characteristics of the original networks. Furthermore, in many real-world use cases, a definition of cluster is intrinsic with the data, eventually obtained with the injection of some expert knowledge represent by a categorical function. Clusters then results in set of connected vertices taking the same values in a finite set C. In the recent literature, Lombardi and Onofri proposed a novel, fast, and easily parallelisable approach under the name of $γ$-contraction to contract a graph given a categorical function. In this work, we formally define such approach by providing a rigorous mathematical definition of the problem, which, to the best of our knowledge, was missing in the existing literature. Specifically, we explore the variadic nature of the contraction operation and use it to introduce the weaker version of the colour contraction, under the name of $β$-contraction, that the algorithmic solution exploits. We finally dive into the details of the algorithm and we provide a full assesment on its convergence complexity relying on two constructive proofs that deeply unveil its mode of operation.