$k$-local Graphs

TL;DR

This paper extends the concept of locality to coloured graphs, introduces an efficient algorithm for computing $k$-locality, and demonstrates its usefulness through a case study on a bibliographic network.

Contribution

It introduces a new $k$-locality measure for coloured graphs, along with an optimal priority search algorithm for its computation and a practical case study.

Findings

The algorithm is optimal in the number of prefix expansions.

The algorithm outperforms exhaustive search by orders of magnitude.

Case study shows potential for knowledge discovery using $k$-locality.

Abstract

In 2017 Day et al. introduced the notion of locality as a structural complexity-measure for patterns in the field of pattern matching established by Angluin in 1980. In 2019 Casel et al. showed that determining the locality of an arbitrary pattern is NP-complete. Inspired by hierarchical clustering, we extend the notion to coloured graphs, i.e., given a coloured graph determine an enumeration of the colours such that colouring the graph stepwise according to the enumeration leads to as few clusters as possible. Next to first theoretical results on graph classes, we propose a priority search algorithm to compute the $k$-locality of a graph. The algorithm is optimal in the number of marking prefix expansions, and is faster by orders of magnitude than an exhaustive search. Finally, we perform a case study on a DBLP subgraph to demonstrate the potential of $k$-locality for knowledge discovery.