Papers › Soft happy colourings and community structure of networks
Soft happy colourings and community structure of networks
Mohammad H. Shekarriz, Dhananjay Thiruvady, Asef Nazari, Rhyd Lewis
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
For 0<ρ≤1, a ρ-happy vertex v in a coloured graph G has at least ρ·deg(v) same-colour neighbours, and a ρ-happy colouring (aka soft happy colouring) of G is a vertex colouring that makes all the vertices ρ-happy. A community is a subgraph whose vertices are more adjacent to themselves than the rest of the vertices. Graphs with community structures can be modelled by random graph models such as the stochastic block model (SBM). In this paper, we present several theorems showing that both of these notions are related, with numerous real-world applications. We show that, with high probability, communities of graphs in the stochastic block model induce ρ-happy colouring on all vertices if certain conditions on the model parameters are satisfied. Moreover, a probabilistic threshold on ρ is derived so that communities of a graph in the SBM induce a ρ-happy colouring. Furthermore, the asymptotic behaviour of ρ-happy colouring induced by the graph's communities is discussed when ρ is less than a threshold. We develop heuristic polynomial-time algorithms for soft happy colouring that often correlate with the graphs' community structure. Finally, we present an experimental evaluation to compare the performance of the proposed algorithms thereby demonstrating the validity of the theoretical results.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections