$k$-local Graphs
Christian Beth, Pamela Fleischmann, Annika Huch, Daniyal Kazempour, Peer Kröger, Andrea Kulow, Matthias Renz
TL;DR
This work generalizes the $k$-locality concept from strings to coloured graphs, defining locality via colour-marking sequences and induced subgraphs $G_i$, and showing how to determine if a graph is $k$-local. It presents a prefix-expansion optimal priority-search algorithm to compute the minimum $k$-locality more efficiently than exhaustive search, with empirical validation on scale-free graphs and a DBLP subgraph case study that reveals hub-like venues driving locality. Theoretical contributions include NP-completeness for general graphs, polynomial-time solvability for $\textsf{Loc}_{G,1}$, and detailed locality bounds for several graph classes, including bipartite graphs where $\operatorname{loc}(G,c)=\min\{|V_1|,|V_2|\}$. The results support practical applications in knowledge discovery and data exploration, illustrating how $k$-locality can guide motif finding, co-location pattern mining, and understanding the organising principles of complex networks.
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.