Eigenvalue Spectrum and Synchronizability of Double-Layer Directed Ring Networks

The synchronizability of multiplex undirected regular networks has been intensively studied based on the study of the synchronizability of single-layer networks. However, most real networks are characterized by some degree of directionality. So multiplex directed networks can better explain the synchronizability phenomenon. Here, based on the theory of master stability function (MSF), we study the eigenvalue spectrum and synchronizability of double-layer inter-layer directed ring networks (Networks-A) and double-layer intra-layer directed ring networks (Networks-B). The eigenvalue spectrum of the supra-Laplacian matrix of the networks is rigorously derived, and the influence of the networks structure parameters on the network’s synchronizability is analyzed. The correctness of the theory is further verified by numerical simulation analysis. Finally, the synchronizability of four kinds of double-layer ring networks with different coupling modes, namely, Networks-A, Networks-B, Networks-C (double-layer undirected ring networks), and Networks-D (double-layer undirected inter-layer random-added-edge ring networks), is compared and the results can provide guidance for constructing the optimal synchronization network.


Introduction
In the last two decades, research on complex networks has developed rapidly and many important results have been achieved. Actual complex systems are composed of multiple networks coupled and interacting with each other [1] [2] [3], for example, interpersonal social networks, neural networks, ecological networks composed of species interactions, etc. [4] [5] [6]. The multiplex networks of the interconnection mode of a multiplex oscillator grid consisting of two subnetworks on the synchronization ability, which found that different interconnection modes affect the synchronization ability of the whole network. It revealed that increasing the number of inter-layer connections between ring networks of the same property facilitates the synchronization of double-layer networks [26].
In 2020, Zhang Li et al. derive an analytic expression for the eigenvalue spectrum of a multilayer k-nearest neighbor coupled network with one-to-one corresponding full connectivity between layers in terms of the master stability function and analyzed the effect of the network structure parameters on the synchronization ability of this network [27]. In 2021, Yang Feimei et al. rigorously derived the eigenvalue spectrum of two types of double-layers directed star-ring networks and analyzed the parameters that affect the synchronizability of the networks [28]. In summary, we find that the synchronizability of the ring networks is mostly studied by numerical simulation and comparison with other regular networks, and there is still a space for a rigorous theoretical derivation of the synchronization capability of multilayer directed ring networks. Studies on the synchronizability of multilayer networks have found that the inter-layer connec-Journal of Applied Mathematics and Physics tion density, inter-layer coupling weight, intra-layer coupling weight, and the scale of the network impact the synchronization ability of the networks. In the actual network, the direction and coupling weight is more sophisticated, following which we consider how the degree of influence of the directionality of the networks and the number of inter-layer connections on the synchronizability of the networks?
In this paper, we rigorously and strictly derive the eigenvalue spectrum of

The Dynamics Model of Multilayer Network
For a multiplex network consisting of M layers and N nodes each layer, the dynamics of i x α can be described as [29] [30]: x α ∈ ℜ is the state vector of the ith node in the αth layer.
: n n i f α ℜ → ℜ is the dynamic equation of the ith node in the αth layer, 1 : n n Γ ℜ → ℜ is the intra-layer coupling function defining the interaction between nodes in the same layer, and 2 : n n Γ ℜ → ℜ is the inter-layer coupling function defining the interaction between nodes on a separate layer. For simplic- Let  is the supra-Laplacian matrix of multiplex networks, L  is the in-Journal of Applied Mathematics and Physics tra-layer supra-Laplacian matrix, I  is the inter-layer supra-Laplacian matrix, then , , L I    can be written as: is the intra-layer Laplacian matrix of the αth layer. ⊕ is the direct sum operation, ( ) L α can be written as: (6) .
⊗ is the Kronecker product, identity matrix. Eigenvalues of the supra-Laplacian matrix of the networks are recorded as: 1 2 max 0 λ λ λ = < ≤ ≤ . based on the theory of master stability function (MSF), A network can be achieving synchronization when all eigenvalues of its Laplacian matrix fall within the synchronized region of that network. The synchronization region of a real network can be mainly divided into two kinds: unbounded synchronized region and bounded synchronized region (other cases such as concurrent synchronized region of multiple unconnected intervals and empty region rarely occur, and only two cases of unbounded and bounded synchronized region are studied in this paper). In general, the synchronizability of the network is determined by the minimum non-zero eigenvalue 2 λ or the ratio max 2 R λ λ = of the maximum eigenvalue to the minimum non-zero eigenvalue of the supra-Laplacian matrix  . When the network synchronization region is unbounded, the larger 2 λ is, the stronger the synchronizability of the network; when the network synchronized region is bounded, the smaller max 2 R λ λ = is, the stronger the synchronizability of the network.
To convenient the following theoretical derivation, two lemmas are given here: Lemma 1 ( [31]). If A is a square matrix of degree n, D is a square matrix of degree m, O is m n × zero matrices, and B is a n m

Structural Model of Double-Layer Ring Networks
In this paper, we focus on the synchronizability of inter-layer directed doublelayer ring networks and intra-layer directed double-layer ring networks. It is supposed that the topology of each layer of the double-layer ring network is identical. The number of nodes within each layer N , the inter-layer coupling weight d , and the intra-layer coupling weight a is all the same in the networks. Number each layer node orderly and form node pairs with the same number. The layer interconnection method of double-layer inter-layer directed ring networks is a unidirectional connection between layer node pairs as shown in Figure 1(a), double-layer intra-layer directed ring networks means that the nodes of the two layers are connected in opposite sequence as shown in Figure   1(b), double-layer undirected ring networks as shown in Figure 1(c), doublelayer undirected inter-layer random-added-edge ring networks is based on the double-layer undirected ring networks with randomly connected undirected edges of interlayer nodes, the interlayer connected edge probability is ( ) Figure 1(d).
For clarity of description, we denote the double-layer inter-layer directed ring networks as Networks-A, double-layer intra-layer directed ring networks as Networks-B, double-layer undirected ring networks as Networks-C, and double-layer undirected inter-layer random-added-edge ring networks as Networks-D. Additionally, Syn A denotes the synchronizability of Networks-A, and so forth.

The Eigenvalue Spectrum and Synchronizability of Networks-A
From the structural model in Figure 1(a), the supra-Laplacian matrix corresponding to Networks-A can be expressed as: According to Lemma 1, the Characteristic polynomials of 1  is: , the eigenvalues of 1  can be written as [27]: when N is odd, at 1 1 2 when N is even, at 1 2 In practice, the number of network sizes is huge and for simplicity we take: According to the MSF theory, the relationship between the synchronizability of Networks-A and the structural parameters is shown in Table 1.

Numerical Simulation of the Synchronizability of Networks-A
In this paper, the values of various parameters verified by numerical simulation are set within the allowed range.
From Figure 2 The case of the synchronized region bounded -: unchanged; ↑: strengthen; ↓: weaken. As is shown in Figure 3, when increases with the increase of N, so the value of R is less and less affected by N being odd and even. Hence, when the synchronized region is bounded, the value of R first remains invariant ( According to Lemma 2, we can get the Characteristic polynomials of 2  is: , the eigenvalues of 2  can be written as [32]: According to the MSF theory, the relationship between the synchronizability of Networks-B and the structural parameters is shown in Table 2.
In a d = , the relationship between the synchronizability of Networks-B and the structural parameters can be transformed from Table 2 to Table 3.

Numerical Simulation of the Synchronizability of Networks-B
When a d = , with the unbounded synchronized region, Figure 5(a) shows that the synchronizability of Networks-B is strengthened because 2 λ becomes larger with increases in , a d . Figure 6(a) shows that the synchronizability of Networks-B is weakened because 2 λ becomes smaller with increases N. When the bounded synchronized region, as is shown in Figure 5(b) and Figure 6(b), the synchronizability of Networks-B is only determined by the number of nodes within     1013.212 R N ≈ π ≈ is shown in Figure 5(b).
When d a > it is observed that the value 2 λ increases with the increase of a (Figure 7(a)) and the value R decreases exponentially with the increase of a (Figure 7(b)). This means that, with the bounded or unbounded synchronized region, the synchronizability of Networks-B is enhanced continuously with an increase in a . Figure 8 and Figure 9 show the change of synchronizability of

The Comparison of Synchronizability of Four Kinds of Double-Layer Ring Networks
In For comparison purposes, the relationship between the synchronizability of the Networks-C and the structural parameters is shown in following Table 4 [27].
As illustrated in Figure 10, it is intuitive to show that the value 2 λ increases rapidly (Figure 10(a)) the value R decreases swiftly (Figure 10(b)) with the increase of the inter-layer random-added-edge probability p. So, the synchronizability of Networks-D is continuously strengthened, whether the synchronized region is bounded or unbounded. Naturally, it can be observed that the larger the interlayer random edge addition probability p, the faster the synchronizability of Networks-D optimization.
As is shown in Figure 11(a), when d a > , with the unbounded synchronized region, for all four kinds of double-layer ring networks, the value 2 λ increases with the increase of a, so the synchronizability is strengthened with the increase of a. Then the size of the synchronizability of these four kinds of double-layer ring networks is: With the bounded synchronized region, it is observed from Figure 11(b) that the value R decreases with the increase of a all four kinds of double-layer ring networks. Then the size of the synchronizability of these four kinds of double-layer ring networks is: . In sum, whether the synchronized region is bounded or unbounded, the synchronizability of the four kinds of double-layer ring networks is enhanced continuously with the increase of intra-layer coupling weight a .
The case of the synchronized region bounded   . Figure 12(b), it is clear that the R value of all four kinds of networks increases with the increase of d. Hence, the synchronizability of the four kinds of networks is weakened with the increase of d (bounded synchronized region). So, it is obtained from Figure 12(b) that the size of synchronizability of these four kinds of double-layer ring networks is weakened with inter-layer coupling weight d in the following order: the synchronized region is unbounded, it can be seen from Figure 13(a) that the 2 λ values of these four kinds of double-layer ring networks all have a slowly decreasing trend with the increase of N. Then the 2 λ value (   2  2  2 4a N λ = π ) of Networks-A and Networks-C have the same trend with the increase of N. So, the synchronizability of the four kinds of double-layer ring networks is diminished with the increase N. In particular, it is found that the order of these four kinds of double-layer ring network synchronizability from largest to smallest is the same as in Figure 11(a) and Figure 12(a). With the synchronized region is bounded, from Figure 13(b), it is clear that for any the number of nodes within each layer N. Then R value (    Figure 14(b), so the synchronizability of all four kinds of networks is monotonically weakened.
Combining Figure 13 and Figure 14, the four kinds of double-layer ring networks have a different ranking of synchronizability with N for different a and d. However, in general, the synchronizability of all four kinds of networks is in the following order: Syn Syn Syn Syn = > > , with the synchronized region is unbounded. It is easy to conclude that the ranks for synchronizability are: Syn Syn Syn > > > , with the synchronized region is bounded.

Conclusions
For Networks-A and Networks-B, firstly, we investigate the synchronizability of inter-layer directed double-layer ring networks (Networks-A) and intra-layer directed double-layer ring networks (Networks-B), and rigorously derive the effects of each parameter on the synchronizability of the two kinds of networks, giving the analytical expressions for the eigenvalues. Finally, the theory is verified by numerical simulation analysis. The results show that the R values of the ring network are affected by both odd and even numbers in a small network range, which is similar to the results of the analytical expression for the eigenvalues of the network derived in [28]. By varying a single parameter, an optimal value of the parameter was found, which led to the optimal synchronizability of the networks. The effects of changes in inter-layer coupling weight a , intra-layer coupling weight d , the number of nodes within each layer N upon the network synchronizability for Networks-A and Networks-B are similar. To sum up, the larger the intra-layer coupling weight, the smaller the inter-layer coupling weight, and the smaller the number of nodes is more preferable to the synchronization of Network-A and Networks-B. Then, we also find that Networks-A, Networks-B, and Networks-C have the same minimum non-zero eigenvalue 2 λ when the inter-layer coupling weight d and intra-layer coupling weight a are equal, indicating that these three kinds of networks have the same synchronizability when the synchronized region is unbounded. In addition, the synchronizability of Networks-D increases as the probability of randomly adding edges between layers increases. More importantly, we compare the synchronizability of four kinds of double-layer ring networks and find that the synchronizability of Networks-D is the best for each parameter variation, while it is remaining three kinds of ring networks are comparable, and the synchronizability of Networks-A is stronger compared to Networks-B and Networks-C when both intra-layer coupling weight a and inter-layer coupling weight d are affected. Recently, the diffusion of networks is an interesting and challenging topic. The spectrum of eigenvalues of multilayer directed networks is in general complex; it is part of our future work to investigate the diffusion dynamics of multiplex directed networks.