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.

翻译：对于 $0<\rho\leq 1$，着色图 $G$ 中的一个 $\rho$-快乐顶点 $v$ 至少拥有 $\rho\cdot \mathrm{deg}(v)$ 个同色邻居，而 $G$ 的一个 $\rho$-快乐着色（亦称软快乐着色）是一种使所有顶点都成为 $\rho$-快乐顶点的顶点着色。社区是指其顶点内部连接比与外部顶点连接更紧密的子图。具有社区结构的图可以通过随机图模型（如随机块模型（SBM））来建模。在本文中，我们提出了若干定理，表明这两个概念是相互关联的，并具有众多实际应用。我们证明，在满足模型参数的特定条件下，随机块模型中的图社区以高概率诱导出所有顶点的 $\rho$-快乐着色。此外，我们推导了 $\rho$ 的一个概率阈值，使得 SBM 中图的社区能够诱导出 $\rho$-快乐着色。进一步地，我们讨论了当 $\rho$ 小于该阈值时，由图的社区诱导的 $\rho$-快乐着色的渐近行为。我们开发了用于软快乐着色的启发式多项式时间算法，这些算法通常与图的社区结构相关。最后，我们通过实验评估比较了所提出算法的性能，从而验证了理论结果的有效性。