One-way dependent clusters and stability of cluster synchronization in directed networks

Cluster synchronization in networks of coupled oscillators is the subject of broad interest from the scientific community, with applications ranging from neural to social and animal networks and technological systems. Most of these networks are directed, with flows of information or energy that propagate unidirectionally from given nodes to other nodes. Nevertheless, most of the work on cluster synchronization has focused on undirected networks. Here we characterize cluster synchronization in general directed networks. Our first observation is that, in directed networks, a cluster A of nodes might be one-way dependent on another cluster B: in this case, A may remain synchronized provided that B is stable, but the opposite does not hold. The main contribution of this paper is a method to transform the cluster stability problem in an irreducible form. In this way, we decompose the original problem into subproblems of the lowest dimension, which allows us to immediately detect inter-dependencies among clusters. We apply our analysis to two examples of interest, a human network of violin players executing a musical piece for which directed interactions may be either activated or deactivated by the musicians, and a multilayer neural network with directed layer-to-layer connections.

1 Multilayer structure of a network Following [1], in order to better evidence the layers, and the role of intralayer and interlayer connections, the network nodes can be arranged in sets {V α , α = 1, . . . , M }, where each set (containing N α nodes) corresponds to a given layer of the multilayer network. The intralayer interactions inside layer α are described by an adjacency matrixÂ k,αα , a nonlinear coupling function h h h k,αα , and a coupling strength σ αα k . The interlayer interactions from layer β to layer α are described by an N α × N β adjacency matrixÂ k,αβ , a nonlinear coupling function h h h k,αβ , and a coupling strength σ αβ k . We now call x x x α i the state of node i inside layer α (i = 1, . . . , N α and α = 1, . . . , M ) and f f f α the dynamics of all uncoupled nodes in layer α. Then Eq. (1) in the main paper for node i in layer α can be rewritten as follows, The matrixÂ k is now the so-called supra-adjacency matrix for all the connections of type k of the multilayer network, which is different from the matrix A k of Eq. (1) in the main paper.
For example, for the network in Fig. 1 of the main paper, the matrices A k are For the same network, the matricesÂ k arê 4 5 6 7 8 11 12 9 10 0 1 0 2 Adjacency matrix of the network of Fig. 2 of the main paper 3 Matrices T and B for some of the considered examples As stated in the main paper, the structure of B ⊥ allows one to easily detect the presence of intertwined or one-way dependent clusters by inspection. The complete matrices T and B, whose structure is described in this section, are provided in datasets S1 and S2.

Violin players network in its undirected configuration (high δ)
We focus on the network shown in Fig In particular, B ⊥ has a block-diagonal structure with three blocks and the left-upper block (with red-green borders) evidences that the two clusters C 1 and C 2 are intertwined.

Violin players network in its directed configuration 'arrowhead'
We focus on the network shown in Fig. 6 of the main paper, with the non-minimal balanced coloring shown in panel i. Figure 2 below shows the structure of the matrices T (left) and B (right) for the violin players network in its directed configuration with 5 clusters.  Fig. 6i, in the main paper. Notice that C 1 and C 5 do not appear in T ⊥ because they are trivial.
In this case, B ⊥ is block-upper triangular, meaning that some clusters are one-way dependent. In particular, the stability of C 4 depends on that of C 3 (third-last row) and the stability of C 3 depends on that of C 2 (second-last row). The last row (related to cluster C 2 ) has only zero entries; this implies that the dynamics of each perturbation component η k depends only on η k through the matrix Ψ 1 in Eq. (4).

Two-layer network with interlayer connections oriented in both directions
We focus on the network shown in Fig. 7 of the main paper, with the minimal balanced coloring shown in panel b. We remark that the complete matrices are independent of σ 2 . Fig. 3 below shows the structure of the matrices T (left) and B (right). B ⊥ is a unique 18 × 18 full block (with red-green borders), which evidences that the two clusters C 1 and C 2 are intertwined.

Two-layer network with interlayer connections oriented in one direction
We now consider two networks obtained from the network shown in Fig. 7a of the main paper, by cutting the interlayer connections in one direction. We focus on the case of minimal balanced coloring (Fig. 7b). First, we keep only the thin red connections from layer I (red) to layer II (blue), removing the thick red connections from layer II to layer I. in the main paper, removing the connections from layer II to layer I, in the case of minimal balanced coloring (Fig. 7b).
In this case, the 18 × 18 submatrix B ⊥ is block-upper triangular and the structure of the corresponding matrix T evidences that cluster C 1 is one-way dependent on C 2 .
Conversely, if we keep only the thick red connections from layer II to layer I, removing the thin red connections from layer I to layer II, we obtain matrices with the structure shown in Fig. 6: T (left) and B (right). in the main paper, removing the connections from layer I to layer II, in the case of minimal balanced coloring (Fig. 7b).
B ⊥ is again a unique 18 × 18 block-upper triangular submatrix and the structure of the corresponding matrix T evidences that cluster C 2 is one-way dependent on C 1 .  we identify equitable clusters, ISSs, breaking vectors and intertwining indices. Based on these quantities, we can (a) find the Q synchronized motions and the small perturbations {w w w i } about them, (b) detect interrelations among clusters (easily only for simple networks) and (c) find the irreducible transformation matrix T . Starting from the network equations, we find the equations governing the dynamics of a small perturbation w w w(t) = x x x(t) − s s s(t) about a synchronous solution s s s(t). Through the matrix T we find (i) the optimal (irreducible) coordinate system η η η(t) to separate the perturbation modes and (ii) the matrix B ⊥ . The matrix B ⊥ evidences the inter-dependencies among clusters (easily also for complex networks) and the variational equations in the coordinate system η η η permits to analyze the stability of each cluster, through the MLEs.