Fig. 5. Summarizing illustrative example.
a The minimal balanced partition for the network of Fig. 1a, which has Q = 5 clusters: C1 = {1, 2, 3} (colored blue), C2 = {4, 5, 6} (colored red), C3 = {7, 8} (colored green), C4 = {9, 10} (colored orange), and C5 = {11, 12} (colored purple). b Structure of the corresponding matrix T. The T⊥ rows are color-coded as the corresponding clusters. c Complete matrix B. The transverse block B⊥ (within the dashed borders) has an upper triangular structure, containing three irreducible subblocks (color-bordered blocks). The purple-bordered subblock characterizes the only perturbation affecting cluster C5 (see the corresponding row in the T structure), which is independent (single diagonal entry). The green-orange-bordered 2 × 2 subblock evidence the two perturbations affecting clusters C3 (second row of the block) and cluster C4, which are intertwined. The blue-red-bordered upper triangular subblock is concerned with the two perturbations affecting clusters C1 and C2: both of them depend only on C1, indicating that C2 is one-way dependent on C1.