Fig. 2. Illustrative example with all balanced partitions.

Balanced partitions (within the dark blue box) of a simple directed network with N = 5 nodes. The partition $\mathcal{P}$ corresponds to the minimal balanced coloring, with two clusters C₁ (containing nodes 1 and 2, colored blue) and C₂ (containing nodes 3–5, colored red): each node of cluster C₂ receives one arrow from one node of C₁; each node of cluster C₁ receives one arrow from one node of C₁ and two arrows from nodes of C₂. Breaking these clusters, we can obtain all the other balanced partitions, which are non-minimal, because they contain a higher number of clusters. Each partition ${\mathcal{P}}^1$, ${\mathcal{P}}^2$, and ${\mathcal{P}}^3$ (within red boxes) contains three clusters, which are obtained from $\mathcal{P}$ by breaking the largest cluster into two smaller ones. For instance, ${\mathcal{P}}^1$ contains $C_1^1$ (with nodes 1 and 2, colored blue), $C_2^1$ (with nodes 4 and 5, colored red), and $C_3^1$ (with node 3, colored green). This partition is balanced, because each node of $C_1^1$ receives one arrow from one blue node, each node of $C_2^1$ receives one arrow from a blue node, one arrow from a red node and one from a green node, and each node of $C_3^1$ receives one arrow from a blue node and two arrows from red nodes. Similarly, partitions ${\mathcal{P}}^4$ and ${\mathcal{P}}^5$ (within green boxes) contain four clusters each and are obtained from the partitions with three clusters by breaking the largest cluster into two smaller clusters: the partition ${\mathcal{P}}^4$ is obtained by breaking the red clusters of ${\mathcal{P}}^1$, ${\mathcal{P}}^2$, or ${\mathcal{P}}^3$; the partition ${\mathcal{P}}^5$ is obtained by breaking the blue cluster of ${\mathcal{P}}^3$. Finally, the partition ${\mathcal{P}}^6$ (within orange box) contains five trivial clusters (each one includes only one node).