Soft happy colourings and community structure of networks

TL;DR

Several theorems are presented showing that communities of graphs in the stochastic block model induce soft happy colouring if certain conditions on the model parameters are satisfied, and heuristic polynomial-time algorithms for soft happy colouring that often correlate with the graphs' community structure are developed.

Abstract

For $0<ρ\leq 1$, a $ρ$-happy vertex $v$ in a coloured graph $G$ has at least $ρ\cdot \mathrm{deg}(v)$ 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.

Figures (15)

• Figure 1: A graph with 3 communities, represented by three colour classes. Dashed lines represent inter-community edges. Here the colouring is $0.5$-happy.
• Figure 2: An example of a $\rho$-happy colouring problem using $\rho=0.5$. (a) The vertex $u$ is $0.5$-happy because its degree is 8 and at least $\lceil 0.5\times 8\rceil=4$ of its neighbours have the same colour as its colour (black). (b) The vertex $v$ is not $0.5$-happy because its degree is 9 and $\lceil 0.5\times 9\rceil= 5$, but only $4$ of its neighbours have the same colour as its colour (black).
• Figure 3: This graph shows the minimum ratio of $\rho$-happy vertices when $\rho\leq\frac{\xi}{2}$ in the induced colouring by graphs communities. Tests were performed over 20,000 randomly generated graphs with $n\in\{$500, 1,000, 2,000, 3,000, 5,000$\}$ (100,000 graphs in total) and parameters randomly selected from $2\leq k\leq 20$, $0<p\leq 1$, and $0<q\leq \frac{p}{2}$.
• Figure 4: The minimum ratio of $\rho$-happy vertices of the induced colouring by communities of 100,000 randomly generated precoloured graphs with $n\in\{500, 1,000, 2,000, 3,000, 5,000\}$ (20,000 graphs for each of these numbers) while other parameters are randomly chosen from $2\leq k\leq 20$, $0<p\leq 1$, and $0<q\leq \frac{p}{2}$. It can be seen when $\rho<\xi$, the probability of having a large number of $\rho$-happy vertices is high (yellow areas).
• Figure 5: The charts show the effects of increasing the number of vertices on the average number of $\rho$-happy vertices in the colouring induced by communities for graphs in the SBM. Here, $k=20$, $p=0.7$ and $q=0.06$. The chart (a) shows an image of these effects when $200\leq n\leq$ 20,000 and $0.31\leq\rho\leq 0.4$. Part (b) shows these effects when $0.1\leq\rho\leq 1$. When $\rho\leq 0.37$ and $n$ grows, the average number of $\rho$-happy vertices in the colouring induced by communities also grows, while the graphs are entire $\rho$-happily coloured by their communities when $\rho\leq 0.35$ and $n\geq$ 10,000. Interestingly, this number decreases when $\rho>0.38$. When $\rho = 0.38$, the chart is still increasing, but at a very slow rate.

Theorems & Definitions (8)

• Definition 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Definition 4.1