A Mathematical Formalisation of the γ-contraction Problem

TL;DR

This work formalises the $\gamma$-contraction problem for vertex-coloured graphs, addressing the lack of a rigorous mathematical definition in prior studies. It introduces the weaker $\beta$-contraction as a greedy, variadic relaxation and provides constructive convergence analyses for the accompanying algorithm. A serial implementation is described as a two-phase process with explicit data structures and per-iteration complexity bounds, and the global convergence is analyzed with worst-case iteration counts tied to the golden ratio $\varphi$. The approach enables structure-preserving compression of large networks and lays groundwork for parallelisation and broader benchmarks.

Abstract

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.

Figures (7)

• Figure 1: \ref{['gt-fig:regular-contraction-a']} a graph $G$ and \ref{['gt-fig:regular-contraction-b']} the corresponding contraction $G' = G/\{u, v\}$. Contracted vertices $u, v \in V_G$ are highlighted in orange and the resulting blended vertex $w \in V_{G'}$ is highlighted in green.
• Figure 2: \ref{['gt-fig:variadic-contraction-a']} a graph $G$ and \ref{['gt-fig:variadic-contraction-b']} the corresponding contraction $G' = G/U$. The set of contracted vertices $U \subseteq V_G$ are highlighted in orange and the resulting blended vertex $w \in V_{G'}$ is highlighted in green.
• Figure 3: Example of a 1-iteration $\beta$-contraction of the graph $G$ to the graph $G' = G/\gamma$. \ref{['fig:contraction-example-a']} The graph $G = (V, E)$. The directed edges $B$ of the digraph $D = (V, B)$ are highlighted as bold arrows over $G$ (self loops are omitted). The map $\pi$ is represented with dashed bend arrows which link each node within a colour cluster to the corresponding tree root nodes where $\pi$ form a self loop. Numeric values of the map $\hat{\alpha}$ are reported on the $\pi$ self loops. Map $\beta$ is not reported for the sake of readability (see Figure \ref{['fig:caso-patologico']}). \ref{['fig:contraction-example-b']} The contracted graph $G' = (V', E') = G/\beta = G/\gamma$. Vertices in $V'$ ( the codomain of $\beta$) are enumerated according to values on self loops from $\pi$.
• Figure 4: A small example of graph $G = (V, E)$ for which $\beta$ determines two distinct colour clusters instead of a single colour component. The corresponding contracted graph $G' = G/\beta = (V', E')$ is also reported. The directed edges $B$ of the digraph $D = (V, B)$ are highlighted as bold arrows over $G$ (self loops are omitted). The map $\pi$ is represented with dashed bend arrows and numeric values of the map $\hat{\alpha}$ are reported on the $\pi$ self loops. Map $\beta$ is represented as dotted arrows connecting $G$ with $G'$.
• Figure 5: Description of the $\beta$-contraction algorithm, an iteration of our novel $\gamma$-contraction algorithm, sketched as a function tree. Each leaf summarises a different phase by means of a simplified math-oriented pseudo-code.

Theorems & Definitions (31)

• definition 1: Colour-preserving contraction
• proposition 1: Properties of colour-preserving contraction
• proof
• definition 2: $\gamma$-contraction
• definition 3: Colour neighbourhood
• remark 1: (On colour neighbourhoods)
• definition 4: Colour cluster and colour component
• remark 2: (On the connectivity of colour clusters)
• proposition 2: Colour contraction, variadic form
• proof