Soft happy colourings and community structure of networks

For $0<\rho\leq 1$, a $\rho$-happy vertex $v$ in a coloured graph $G$ has at least $\rho\cdot \mathrm{deg}(v)$ same-colour neighbours, and a $\rho$-happy colouring (aka soft happy colouring) of $G$ is a vertex colouring that makes all the vertices $\rho$-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 $\rho$-happy colouring on all vertices if certain conditions on the model parameters are satisfied. Moreover, a probabilistic threshold on $\rho$ is derived so that communities of a graph in the SBM induce a $\rho$-happy colouring. Furthermore, the asymptotic behaviour of $\rho$-happy colouring induced by the graph's communities is discussed when $\rho$ 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.