Introduction

The imminent deployment of 5G and high-capacity access optical networks will stress the telecommunication infrastructure1. Two main approaches have been proposed to extend the wavelength-division multiplexing (WDM) capacity: spatial-division multiplexing and multi-band transmission. However, the spatial-division multiplexing needs the deployment of additional fibers, which requires huge a expenditure. By contrast,the multi-band transmission exploits the available low-loss optical spectrum using already deployed fibers2. Thus, the multi-band transmission is more economically feasible.

The multi-band transmission faces many challenges of physical impairment. The most important one is the stimulated Raman scattering (SRS)3. It results in the power transfer from higher-frequency channels to lower-frequency ones, hence producing an uneven power spectrum4. The maximum SRS efficiency occurs in a frequency separation of approximately 13 THz5, which is relevant for multi-band systems. It leads to a noticeable ripple and non-uniform performance in both the signal power and generalized signal-to-noise ratio (GSNR) at the receivers.

Research on the SRS effect in the past focused in the physical layer. The SRS effect is analyzed by obtaining the power profile, which is influenced by fiber attenuation and inter-channel power transfer. The power profile can be obtained by solving a set of coupled ordinary differential equations6,7. This is the principal method for analyzing the SRS effect. However, it is based on the information in the physical layer and overlooks the information in the network layer. In large-scale, multi-band optical networks with massive flows, this method can not capture the impact of SRS in the network layer and entails an insurmountable complexity. In the network layer, Guo et al.8 find that there is a small difference between maximizing the capacity and flatness in short-distance systems, and the difference becomes large when the system comprises long-distance links. Emanuele et al.9 evaluated the network performance using the Statistical Network Assessment Process and compared the capacity between spatial-division multiplexing and multi-band transmission in two topologies. However, they still used the method based in the physical layer information to estimate the GSNR, rather than considering it directly from the overall network. Zhang et al.10 claimed that considering only based on full-spectral loading is not flexible, and network deployments should consider partially loaded and dynamic scenarios. They proposed a power allocation scheme based on digital twin systems. However, the digital twin remains in the conceptual stage beyond deployment. Thus, proposing a method that can directly analyze the overall network and adapt to highly dynamic and partially loaded scenarios is significant.

In the literature on network optimization, launch power optimization is crucial. Classified by the number of variables, current literature primarily encompasses three directions: (1) Optimize a flat launch power for each band. This method pertains to a single variable for each band11,12,13,14. (2) Optimize the launch power for each channel. This method pertains to the same number of variables as the number of channels15,16,17,18,19. (3) Optimize the offset and tilt of launch power for each band. This method pertains to two variables for each band20,21,22,23,24. The most widely used method to achieve optimization is the meta-heuristic algorithm, including the swarm optimization algorithm15,16,22,25, genetic algorithm20,21, etc. Optimization for each channel requires too many variables in multi-band transmission, which is unaffordable. Thus, optimization for the offset and tilt in each band is the most common method. However, the state of flows is not solely determined by frequency. In the network layer, flows form complex interaction relationships that are non-negligible. However, the interactions between flows are too difficult to consider because of the randomness and dynamics in routing and wavelength assignment, especially in large-scale multi-band optical networks with numerous flows.

Currently, research on complex networks has attracted great attention. The complex network is an abstract representation of intricate relationships and has become an effective method to analyze the relationships between complex systems. In a complex system, each element is represented by a single node, and the interaction between nodes forms edges of the network. Researchers investigated complex networks in different fields, i.e., smart city26,27,28, social study29,30,31,32, power grid33,34,35,36, and biology37,38,39. Interactions between flows in the Internet have been investigated40,41,42. Existing research has demonstrated that the interactions among flows within a network can have significant impacts on network operations. In the optical network, flows form a complex relationship of power transfer due to the SRS effect. Complex networks have been widely used to study interacting and coupling patterns43. Thus, we introduce the complex network to analyze the flow interactions in the optical network.

In previous studies, complex networks have been employed to address problems in optical networks. For instance, analyze from the perspective of network structure. Community detection is used to identify nodes to be upgraded in the bearer network44. They achieve adjustable-scale community discovery while preserving tree/ring structures and hierarchical affiliations, providing a new framework for the application of complex network community detection in communication network upgrades. Besides, the invulnerability of optical networks is studied based on complex networks47,57. Complex networks are applied to the invulnerability research of optical fiber backbone networks. By constructing network models and conducting attack simulations, it analyzes the robustness performance of networks under different attack strategies. In addition to the topological structure, clustering is used for fault localization mechanisms.46 Aiming at the link fault localization problem in WDM optical networks, this paper applies the Minimum Dominating Set and clustering analysis from complex network theory to propose a fast fault localization mechanism based on Minimum Dominating Set Clustering. There are also studies analyzing optical networks from the perspective of resource allocation. Graph coloring technique is used for wavelength assignment in optical networks.45,58 The authors abstract optical paths in optical networks as vertices of a graph. If there is at least one overlapping physical link between two optical paths, an edge is connected between the corresponding vertices to construct a conflict graph. Through graph coloring, colors (corresponding to wavelengths) are assigned to each vertex to ensure that adjacent vertices have different colors and avoid wavelength conflicts.

Complex networks have been applied to analyze various problems in optical networks. Although some works, such as the conflict graph45,58, have considered from the perspective of flows, the transmission quality is not considered. In our work, we analyze the nonlinear effects in optical networks from the flow perspective. Nonlinear effects are originally characteristics of the physical layer. Through our modeling, we can intuitively analyze the impacts of nonlinear effects at the network layer, thereby guiding the optimization of the network layer, which will bring about significant computational simplification.

The coupling effect is employed in this letter to describe the complex interactions between flows in multi-band optical networks caused by the SRS effect from the perspective of flows. In the physical layer, the SRS effect leads to uneven signal power, while in the network layer, the coupling effect influences the network capacity. Based on the coupling effect, we can directly investigate the influence of the SRS effect in the network layer. To address the complexity of the coupling effect, we introduce the complex network and model the coupling effect with a directed weighted graph called the Flow Coupling Network (FCN) using the routing, propagation distance, and frequency information. Based on the FCN, we propose the flow coupling strength and the network coupling strength.

In response to the flow coupling strength, we conduct two investigations as follows:

  1. 1.

    We verify the model rationality based on the linear correlations between the flow coupling strength and signal power variation.

  2. 2.

    We proposed the power adjustment algorithm based on the flow coupling strength to optimize the network capacity and compare it with the optimization based on frequency.

In response to the network coupling strength, we conduct two investigations as follows:

  1. 1.

    We investigate the influence of the network coupling strength on the network capacity. We find that the optimization under stronger network coupling strength leads to greater gain in network capacity.

  2. 2.

    We investigate the influence of topological features on the network coupling strength among BA, ER, and WS networks. We find that the network coupling strength increases as the network scale increases in the WS network.

We conclude that the coupling effect can be a novel and effective perspective to analyze the influence of the SRS effect in the network layer. And the FCN can well model the coupling effect.

Methods

Coupling effect

Fig. 1
figure 1

The difference between the coupling effect and the SRS effect. The SRS effect only involves the physical layer, while the network layer information is considered in the coupling effect.

Full size image

The SRS effect exists between channels in the fiber and induces inter-channel power transfer. According to previous research6, a frequency-dependent signal power profile can be obtained by solving equations48, given by Formula 1.

$$\begin{aligned} \frac{{\partial {p_i}}}{{\partial z}} = - \sum \limits _{k = i + 1}^N {\frac{{{f_k}}}{{{f_i}}}*{g_r}(\Delta f)*{p_k}(z)*{p_i}(z)} + \sum \limits _{k = 1}^{i - 1} {{g_r}(\Delta f)*{p_k}(z)*{p_i}(z) - \alpha *{p_i}(z)} \end{aligned}$$
(1)

In the physical layer, signal power, distance, and relative frequency between channels affect the power transfer and result in uneven signal power. In the network layer, highly dynamic and random resource allocation (including routing, frequency, and power allocation) induces complex interactions between flows and consequently influences the network capacity.

To investigate the influence of the SRS effect in the network layer, we define the interactions between flows in the network layer as the coupling effect, as depicted in Figure 1.

Unlike Formula 1, we do not mean to get the specific power profile, but to investigate the interactions between flows. We start with the coupling effect between two channels, as \({\varepsilon _{ij}}\) in Formula 2, and extract the influence of power, frequency, and propagation distance from Formula 1. The \({S_i}\) is the i-th flow in the optical network. The \({f_i}\) is the frequency of flow \({S_i}\). The \({p_i}\) is the power in \({f_i}\). Higher value of \({\varepsilon _{ij}}\) leads to stronger power transfer between channel in \({f_i}\) and channel in \({f_j}\).

$$\begin{aligned} {\varepsilon _{ij}} = {p_i}*{p_j}*{g_r}(|{f_i} - {f_j}|) \end{aligned}$$
(2)

The coupling effect between flows, as \({\eta _{ij}}\) in Formula 3, can be considered as the sum of the coupling effect between the channels in their common links. The \({P_i}\) is the launch power of \({S_i}\). Higher \({\eta _{ij}}\) indicates stronger interactions between \({S_i}\) and \({S_j}\).

$$\begin{aligned} {\eta _{ij}} = {P_i}*{P_j}*{g_r}(|{f_i} - {f_j}|)*\sum \nolimits _{t \in D_{ij}^C} t \end{aligned}$$
(3)

The \(L = \{ {l_1},{l_2},...,{l_k}\}\) is the set of links in the optical network and the \(D = \{ {d_1},{d_2},...,{d_k}\}\) is the set of their corresponding link length. The \({R_i} = \{ {r_1},{r_2},...,{r_M},r \in L\}\) is the routing of \({S_i}\), the \(L_{ij}^C = {R_i} \cap {R_j}\) and the \(D_{ij}^c\) are the common links and corresponding length that \({S_i}\) and \({S_j}\) share together. The \({g_r}\) is the Raman gain coefficient, which is a function of the frequency interval between channels49.

It is necessary to emphasize the differences between \({P_i}\) and \({p_i}\) and the approximate assumptions we have made. The \({P_i}\) is the launch power of flow \({S_i}\) while the \({p_i}\) is the signal power in frequency \({f_i}\). The difference lies in that as the signal propagates in the optical fiber,\({p_i}\) will change due to the characteristics of the physical layer, while \({P_i}\) is a fixed value, which is the launch power set manually when the flow is requested. When the SRS effect occurs, the power transfer intensity is actually calculated by \({p_i}\), which requires complex computations. In the modeling of the coupling effect, we use \({P_i}\) to approximately substitute for \({p_i}\).

Flow coupling network

We construct a weighted directed network to investigate the coupling effect among all flows in the optical network, called the Flow Coupling Network (FCN). The FCN is described as \(G = (V,E)\), where \(V = \{ {S_1},{S_2},...,{S_N}\}\) denotes flows in the optical network, \(E \in V \times V\)represents edges between flows. Routing, frequency, and propagation distance are all included in the information for constructing the FCN. The construction rules are outlined as follows:

  1. 1.

    Edges in the FCN indicate the coupling effect between two flows. If flows \({S_i}\) and \({S_j}\) share common links in the optical network (Fig. 2a), the corresponding edge exists between the corresponding nodes in the FCN (Fig. 2c). To distinguish from topology nodes in optical networks, we refer to the node in the FCN as the “flow node”.

  2. 2.

    Edge direction indicates the direction of power transfer caused by the SRS effect between two flows. Frequency determines the edge direction, which flows from higher frequency to flows in lower frequency.

  3. 3.

    Edge weight \({W_{ij}}\) indicates the coupling effect strength between flow \({S_i}\) and flow \({S_j}\). Wider frequency interval between two flows and farther common propagation distance result in a stronger coupling effect. The edge weight is obtained by Formula 4.

$$\begin{aligned} {W_{ij}} = {g_r}(\left| {{f_i} - {f_j}} \right| )*\sum \nolimits _{t \in D_{ij}^C} t \end{aligned}$$
(4)

If a flow traverses several links, edges exist between it and other flows that also share these links in the FCN. A specific example for constructing an FCN is shown in Fig. 2. Figure 2a exhibits the optical network topology. Figure 2b exhibits the channel occupancy in each link. Figure 2c exhibits the corresponding FCN for Fig. 2a, b.

Flow coupling strength

Identifying crucial nodes in a complex network plays a significant role in preventing disease spreading and cascading faults50,51. But traditional key node identification is not suitable for FCN. We intend to establish the coupling effect strength of a single flow, which is denoted as \(\delta _{{S_i}}^{SRS}\), to reflect the coupling effect that a flow is undergoing.

Flows lose or gain power from their first-order neighbor flow nodes in the FCN. From Formula 1, the power reduction term is \(\frac{{{f_k}}}{{{f_i}}}*{g_r}(\Delta f)*{p_k}(z)*{p_i}(z)\), and the power increase term is \({g_r}(\Delta f)*{p_k}(z)*{p_i}(z)\). For the convenience of analysis, we ignore \(\frac{{{f_k}}}{{{f_i}}}\) in the power reduction term, and the coupling strength can be approximately fitted by multiplying \({g_r}(\Delta f)\),\({p_k}(z)\) and \({p_i}(z)\). First, we consider the simplest case of the SRS effect, where there is only the interaction between two flows, as Formula 5.

$$\begin{aligned} \frac{{\partial {p_i}}}{{\partial z}} = {g_r}(\Delta f)*{p_k}(z)*{p_i}(z) \end{aligned}$$
(5)

We attempt to understand Formula 5 from physical meaning. \({p_i}\) and \({p_k}\) are the variables we care about, and the physical meaning of the term \({g_r}(\Delta f)*{p_k}(z)*{p_i}(z)\) can be regarded as the coupling strength. The reason why the SRS effect is difficult to solve is that the coupling strength varies with the signal power, and the signal power is in turn affected by the coupling strength. As a result, an analytical solution can not be obtained, and only a numerical solution is achievable. When building the model, we try to separate the coupling strength and signal power. When studying the signal power change of \(S_i\), we obtain the Formula 6.

$$\begin{aligned} \frac{{\partial {p_i}}}{{\partial z}} = \theta ({p_i}(z),{p_k}(z),{g_r}(\Delta f))*{p_i}(z) \end{aligned}$$
(6)

We separate the coupling strength from the signal power in this homogeneous differential equation. The \(\theta ({p_i}(z),{p_k}(z),{g_r}(\Delta f))\) is the accurate coupling strength, which is determined by \({p_i}(z)\), \({p_k}(z)\) and \({g_r}(\Delta f)\) simultaneously. But \(\theta\) remains an unknown quantity, we substitute \({\varepsilon _{ij}}\) from Formula 2 to approximately replace \(\theta\), and then get Formula 7. As mentioned before, we use \({P_i}\) to approximate substitute for \({p_i}\). By solving this homogeneous differential equation, we obtain Formula 8.

$$\begin{aligned} \frac{{\partial {p_i}}}{{\partial z}} \approx {\varepsilon _{ij}}*{p_i}(z) = {p_i}(z)*{p_j}(z)*{g_r}(|{f_i} - {f_j}|)*{p_i}(z) \approx {P_i}*{P_j}*{g_r}(|{f_i} - {f_j}|)*{p_i}(z) \end{aligned}$$
(7)
$$\begin{aligned} {p_i}(z) = C*\exp ({P_i}*{P_j}*{g_r}(|{f_i} - {f_j}|)*z) \end{aligned}$$
(8)
$$\begin{aligned} \Delta {p_i}(z) = {p_i}(z) - {p_i}(0) = C*\exp ({P_i}*{P_j}*{g_r}(|{f_i} - {f_j}|)*z) - C \end{aligned}$$
(9)

The \(\Delta {p_i}(z)\) is the variation of signal power caused by the SRS effect after the signal propagates a distance of z.Based on Formula 9, a linear correlation can be observed between \(\Delta {p_i}(z)\) and \(\exp ({P_i}*{P_j}*{g_r}(|{f_i} - {f_j}|)*z)\).

From the perspective of flow, the total power transfer accumulated during propagation can be regarded as the accumulation of multiple instances of formula 9 (where \(\Delta {p_i}\) can be either positive or negative). Considering all the coupling effects during propagation, \(\delta _{{S_i}}^{SRS}\) can be obtained from Formula 10. \({N_f}\) is the set of in-neighbor flow nodes, \({N_s}\) is the set of out-neighbor flow nodes.

$$\begin{aligned} \delta _{{S_i}}^{SRS} = \sum \limits _{j \in {N_f}} {\exp ({P_i}*{P_j}*{W_{ij}})} - \sum \limits _{k \in {N_s}} {\exp ({P_i}*{P_k}*{W_{ik}})} \end{aligned}$$
(10)

The \(\delta _{{S_i}}^{SRS}\) lies within \((-\infty ,+\infty )\). When the \(\delta _{{S_i}}^{SRS}\) is negative, the signal power of the \(S_i\) at the receiver is lower than the launch power. Conversely, when the \(\delta _{{S_i}}^{SRS}\) is positive, the signal power of \(S_i\) at the receiver is higher than the launch power. The greater the absolute value of \(\delta _{{S_i}}^{SRS}\), the more pronounced the power transfer of \(S_i\).

Network coupling strength

Fig. 2
figure 2

A simple example for the transformation from the optical network (a) to the Flow Coupling Network (c). The node in (a) represents the ROADM in the optical network, and (b) shows the correspondence between flows and frequency in links. Flow nodes in (c) are in accord with flows in (a).

Full size image

We intend to establish the network coupling strength to synthetically reflect the integrated nature of the coupling effect between all flows in the optical network.

Unlike the flow coupling strength, both power gain and power loss contribute to the network coupling strength. We define \(\sigma _{{S_i}}^{SRS}\) to describe the coupling effect contribution of flow \({S_i}\). The \(\sigma _{{S_i}}^{SRS}\) can be obtained from Formula 11.

$$\begin{aligned} \sigma _{{S_i}}^{SRS} = \sum \limits _{j \in {N_f}} {{P_i}*{P_j}*{W_{ij}}} + \sum \limits _{k \in {N_s}} {{P_i}*{P_k}*{W_{ik}}} \end{aligned}$$
(11)

Further, we use the average contribution of all flows to describe the network coupling strength. We define the network coupling strength as \({C_p}\) by Formula 12.

$$\begin{aligned} {C_p} = \sum \limits _{{S_i} \in V} {\sigma _{{S_i}}^{SRS}} /N \end{aligned}$$
(12)

The \(\sigma _{{S_i}}^{SRS}\) and the \({\mathrm{{C}}_P}\) lies within \([0,+\infty )\). The greater the \(C_p\), the stronger the coupling effect strength among flows in the optical network.

Power adjustment algorithm based on flow coupling strength

To optimize the launch power based on the flow coupling strength, we propose the power adjustment algorithm (PAA) based on the FCN. We turn up the launch power of flows with high \(\delta _{{S_i}}^{SRS}\) while turning down the launch power of flows with low \(\delta _{{S_i}}^{SRS}\).

For the static scenario where the number of flows does not change, flows are first requested. After the injection of flows is completed, launch power is allocated to flows, and the performance is evaluated. The PAA comprises three steps. In step I, we assume the same launch power \({P_{origin}}\) for all existing flows in the network and get the \(\delta _{{S_i}}^{SRS}\) of each flow. In step II, we pick a \(\Delta {P_{\max }}\), which is the maximum power adjustment. Then the power adjustment \(\Delta {P_i^{ad}}\) for flow \({S_i}\) is obtained from the Formula 13.

$$\begin{aligned} \Delta {P_i^{ad}} = (\frac{{ - \delta _{{S_i}}^{SRS}}}{{1500}})*\Delta {P_{\max }} \end{aligned}$$
(13)

In step III, the final launch power \({P_i}\) is obtained by adding \({P_{origin}}\) and \(\Delta {P_i^{ad}}\) with Formula 14.

$$\begin{aligned} {P_i} = {P_{origin}} + \Delta {P_i^{ad}} \end{aligned}$$
(14)

In this paper, all the experiments were carried out in static scenarios. But PAA adapts to the dynamic scenario in the optical network when any flow changes (including requesting new flows, ending existing flows, and adjusting the launch power of existing flows). Launch power is allocated to it while any flow is changed, and the performance is evaluated simultaneously. When a flow \({S_i}\) changes, step I, refresh flow nodes, edges, and edge weights in the FCN. In step II, get the launch power of \({S_i}\) with Formula 13 and Formula 14. In step III, the first-order neighbor flow nodes of \({S_i}\) become newly changed flow nodes and start the next iteration (second-order effect) from step I. By extension, calculate the n-th order effect. In the dynamic scenario, the PAA stops by setting n, and it stops when the n-th iteration is completed.

In both static and dynamic scenarios, the PAA is to adjust the launch power of flows according to the flow coupling strength. The PAA proposed for dynamic scenarios aims to illustrate how to adjust the launch power of flows in real-time based on the flow coupling strength. And demonstrate the interpretability of our methods.

It is worth noting that this method, regardless of the number of channels, only requires the adjustment of two parameters to optimize all flows in the optical network, thus making it adaptable to large-scale networks.

Results

Model rationality validation based on the flow coupling strength

Firstly, we intend to verify the model rationality with linear correlations. The linear correlation is measured with the Pearson coefficient. Experiments are conducted on the sixteen artificial datasets. We consider two different situations: (1) all flows are assigned with 0 dBm, (2) all flows are assigned with random power from -1 to 3 dBm.

As depicted before, we define the flow coupling strength as \(\delta _{{S_i}}^{SRS}\). Flows with higher \(\delta _{{S_i}}^{SRS}\) will be affected more significantly by the coupling effect.

In point-to-point systems, the flatness is measured in terms of \(\Delta GSNR\), which is the difference between the maximum and minimum GSNR20. However, it is not suitable for describing the power variation of a single flow. As mentioned before, we assume that the signal loss is completely compensated by optical amplifiers. Thus, the received power is only influenced by the SRS effect (the SRS effect is still affected by attenuation). We define the \(\Delta {P_i}\) to describe the signal power variation of a single flow. \(\Delta {P_i}\) is obtained by Formula 15. The \(P_i\) is the launch power of \({S_i}\), and the \(P_i^{received}\) is the signal power at the receiver.

$$\begin{aligned} \Delta {P_i} = P_i - P_i^{received} \end{aligned}$$
(15)

Flows that are more affected by the SRS effect will experience greater power variation. Thus, we verify the model rationality with the linear correlation between \(\delta _{{S_i}}^{SRS}\) and \(\Delta {P_i}\).

In this work, we generate flow datasets in four network topologies as Fig. 3 to conduct experiments:

  1. 1.

    GERMAN network consists of 17 nodes and 26 edges with an average node degree of 3.1, average link length of 170 km.

  2. 2.

    COST239 network consists of 11 nodes and 26 edges, with an average node degree of 4.7, average link length of 356 km.

  3. 3.

    USNET network consists of 24 nodes and 43 edges, with an average nodal degree of 3.6, average link length of 997 km.

  4. 4.

    NSF network consists of 14 nodes and 21 edges, with an average node degree of 3, average link length of 1080 km.

Fig. 3
figure 3

Topologies for constructing datasets: (a) GERMAN, (b) USNET, (c) COST239, (d) NSF.

Full size image
Table 1 Linear correlation between \(\delta _{{S_i}}^{SRS}\) and \(\Delta {P_i}\).
Full size table

For all network topologies, we consider every fiber span in the amplified lines to have an identical length of 75 km.

We consider two kinds of multi-band WDM transmission systems on the ITU-T 50 GHz spectrum grid. The C+L multi-band WDM system and the S+C+L multi-band WDM system. Every flow occupies a single channel with the same symbol rate. In the C+L system, there are 120 channels for both C- and L- bands with 50 GHz channel spacing. A total bandwidth of 12 THz with 240 channels. In the S+C+L system, there are 96 channels for the C-band, 96 channels for the L-band, and 192 channels for the S-band. And a frequency interval of 500 GHz is set between each band. A total bandwidth of 20 THz with 384 channels.

For each flow, we randomly select the source and destination. The K-shortest path algorithm (k=10) is implemented for routing, and first-fit is applied for wavelength assignment. This resource allocation strategy is consistent with other research52. For each topology, we generate datasets with two different loads, which block percentage (BP) = \({10^{ - 2}}\) and \({10^{ - 1}}\).

Based on these, we generate eight datasets within four different topologies and two different loads for each multi-band system.

Table 1 exhibits the results of the linear correlation between \(\Delta {P_i}\) and \(\delta _{{S_i}}^{SRS}\). A strong linear correlation can be observed among all datasets, and the flow coupling strength can well reflect the signal power variation of each flow. The main reason resulting in the inaccuracy of the FCN is the sequence of the coupling effect. When a couple of flows meet and generate a coupling effect, because of the distance already spread, the signal power has changed and is no longer equal to the initial launch power. However, we use the original launch power for all the coupling effects of a single flow as Formula 10. This model neglects the signal power variation during propagation and does not correspond to the actual process of signal propagation.

Although the model accuracy can not be perfect, the FCN can well model the overall interaction correlation between all flows in optical networks. It can be implemented to investigate the coupling effect between flows in general.

Power optimization with flow coupling strength

To optimize the optical network capacity based on the FCN, we propose the PAA. We conduct an experiment to verify the performance of the PAA on artificial datasets. This experiment corresponds to the application of the PAA in the static scenario. We calculate the network capacity after requesting all the flows.

The achievable network capacity is obtained based on the quality of transmission of each flow. And the quality of transmission is measured by the GSNR as Formula 16:

$$\begin{aligned} GSNR = \frac{{{P_S}}}{{{P_{ASE}} + {P_{NLI}}}} \end{aligned}$$
(16)

The \({P_s}\) represents the signal power, which is affected by attenuation and the SRS effect. A complete fiber loss compensation by optical amplifiers is assumed9. The \({P_{ASE}}\) is the power of amplifier spontaneous emission noise, which is accumulated after passing through each EDFA amplifier. The \({P_{ASE}}\) is obtained by Formula 17. The \({P_{NLI}}\) is the nonlinear noise caused by the nonlinear effect, which is calculated using the closed-form approximation of the ISRS-GN model53. The specific physical layer parameters for simulation are displayed in Table 2. We keep the physical layer parameters consistent with the existing study8.

$$\begin{aligned} {P_{ASE}} = n*2*{N_{sp}}(G - 1){h_v}*\Delta \nu \end{aligned}$$
(17)
Table 2 Physical parameters for simulation.
Full size table

We consider a set of modulation formats8 as shown in Table 3 , including their corresponding spectrum efficiencies and required GSNR thresholds. Considering achievable network capacity, identify the highest-order modulation format corresponding to the GSNR threshold that the flow GSNR can meet, to maximize the spectral efficiency. The achievable network capacity is the sum of the information rate of all flows.

Table 3 Modulation formats.
Full size table

We compare three different cases as illustrated in Fig. 4:

  1. 1.

    Case 1: Assign the launch power without optimization for the SRS effect. Each flow is assigned the same launch power \({P_0}\). This case requires one variable \({P_0}\).

  2. 2.

    Case 2: Optimize the launch power with frequency information. We set the offset and tilt for the launch power in each band. Referring to54, the launch power is obtained by Formula 18. This case requires two variables for each band: OFFSET and TILT. \({f_S},{f_C},{f_L}\) correspond to the lowest frequency in the S-, C-, and L- bands, respectively.

  3. 3.

    Case 3: Optimize the launch power with flow coupling strength. We adjust the launch power using the PAA. The launch power is obtained by Formula 13 and Formula 14. This case requires two variables: \({P_{origin}}\) and \(\Delta {P_{\max }}\).

$$\begin{aligned} {P_i} = \left\{ {\begin{array}{*{20}{c}} {OFFSE{T_S} + TIL{T_S}*({f_i} - {f_S}),\quad for \quad S-band}\\ {OFFSE{T_C} + TIL{T_C}*({f_i} - {f_C}),\quad for \quad C-band}\\ {OFFSE{T_L} + TIL{T_L}*({f_i} - {f_L}),\quad for \quad L-band} \end{array}} \right. \end{aligned}$$
(18)
Fig. 4
figure 4

Three cases for comparison. The case 1 does not set tilt and optimize targeting the SRS effect; The case 2 sets tilt based on frequency as a straight line; The case 3 sets tilt based on the coupling effect as an irregular curve.

Full size image

To compare the performance of three cases, it is necessary to optimize the parameters of each case to achieve their best performance. According to current research, the genetic algorithm is a commonly used and effective method for optical network optimization20,21. We implement it to find the best parameters for each case, and the searching object is the network capacity.

Figure 5a–d respectively present the achievable network capacity for USNET, NSF, GERMAN and COST239 in C+L and S+C+L systems.

First, in S+C+L systems, case 2 and case 3 yield a greater enhancement in achievable network capacity than in the C+L systems. Moreover, in topologies with longer link lengths (USNET and NSF), the capacity enhancement is greater compared to topologies with short link lengths (GERMAN and COST239). Because stronger network coupling strength occurs in topologies with long link lengths and more flows. Consequently, power optimization targeting the SRS effect leads to more improvement. In contrast, in datasets where the network coupling strength is weak (GERMAN and COST239 in C+L systems), the network capacity enhancement is negligible. Those phenomena are consistent with the discussion in the next section.

In addition, comparing case 2 and case 3, case 3 consistently outperforms case 2 in achievable network capacity. This not only indicates that the network capacity enhancement is greater in PAA but also suggests that, in the network layer, observing the influence of the SRS effect through the coupling effect is more effective than relying solely on frequency.

Fig. 5
figure 5

Achievable network capacity for C+L and S+C+L systems in four topologies comparing three schemes: (a) USNET, (b) NSF, (c) GERMAN, (d) COST239.

Full size image

Influence of network coupling strength

In this section, we investigate the impact of the network coupling strength on the network capacity.

For the generalizability of the experimental outcomes, we conduct the experiment in a randomly generated network with 20 nodes. Then, we increase the number of flows in the network and record the network coupling strength every time an additional 200 flows are added until the BP reaches \({10^{ - 1}}\). For the optimized scenario, we assign power with PAA and utilize the genetic algorithm for parameter optimization. For non-optimized scenarios, all flows are assigned 0 dBm. Under the optimized parameters, we compared the total Shannon network capacity55 between optimized and non-optimized scenarios. The Shannon network capacity is given by Formula 19.

$$\begin{aligned} C = \sum \limits _{i = 1}^N {[2*{B_i}*{{\log }_2}(1 + GSN{R_i})]} \end{aligned}$$
(19)
Fig. 6
figure 6

The impact of network coupling strength on the network capacity.

Full size image

Figure 6a exhibits the increase in \({C_P}\) and network capacity gain under different network loads. The network capacity gain is defined as the growth rate in Shannon network capacity consequent to power optimization in the presence of SRS, given by \(({C_{optimized}} - {C_{origin}})/{C_{origin}}\). The \({C_{optimize}}\) is the Shannon network capacity implementing the PAA optimization, and the \({C_{origin}}\) is the Shannon network capacity without power optimization. Although network capacity gain and \({C_P}\) differ in magnitude, there is only a minor difference in their growth patterns. Moreover, we have verified the linear correlation between them, as depicted in Fig. 6b, and they exhibit almost linear correlations. This phenomenon can be reasonable. The QoT in optical networks is influenced by a multitude of distinct factors, including the nonlinear noise, the amplifier spontaneous emission noise, and the SRS effect. In scenarios where the network coupling strength is not strong enough, the impact of the SRS effect on network capacity will not be the primary factor limiting the network capacity. Instead, the amplifier spontaneous emission noise and the nonlinear noise may be the main contributors to the decline in network capacity. In this scenario, power optimization targeting the SRS effect will not yield an enhancement in network capacity. As the number of flows increases, the randomness in routing and spectrum allocation leads to the formation of complex coupling effect relationships. Consequently, the network coupling strength continuously intensifies, and the impact of the SRS effect on network capacity also grows. The linear relationship between network capacity and \({C_P}\) indicates that the \({C_P}\) can effectively reflect the network coupling strength. The greater the \({C_P}\) is, the greater the enhancement in network capacity that can be achieved through power optimization targeting the SRS effect.

Table 4 Statistical tests for the linear correlation.
Full size table

We repeated the above experiment in four topologies (GERMAN, USNET, COST239, NSF) as Fig. 3 to verify the universality of this conclusion. Under the same experimental settings, we use the Pearson coefficient to measure the linear correlation and the p-value to provide statistical significance tests. We report the results of a two-tailed t-test in Table 4. The same conclusion as that of Fig. 6 can be obtained.

Thus, we can conclude the influence of the coupling effect on the network. When the network coupling strength continues to increase, a few flows experience significant power gain or losses, and the received signal power distribution becomes highly uneven. Power optimization targeting the SRS effect yields greater gains in network capacity. Moreover, the \({C_P}\) can be an effective metric for observing the network coupling strength.

Network coupling strength in different networks

Next, we investigate the impact of network topological features on the network coupling strength.

Previous researches show that flows have exhibited different features in different networks56. In this section, we consider three different networks, BA scale-free networks, ER random networks, and WS small-world networks, to study how network structures affect the coupling effect between flows.

In the BA network, the distribution of node degree follows a power-law distribution, where a few nodes have high connectivity (referred to as “hub” nodes), while the majority have fewer connections. We consider a parameter m when constructing a BA network. This parameter represents the number of connections that a new node establishes with the existing network. It directly influences the average node degree of the network. A smaller m leads to a low average node degree in the network. We examine the network behavior for a range of m, m = 1, 3, 5, 7, 9.

The WS network captures the small-world phenomenon characterized by short path lengths and high clustering coefficients. This model is constructed by starting with a regular ring lattice, where each node is connected to its k nearest neighbors. It then introduces randomness by re-wiring edges with a probability p, thus creating shortcuts that significantly reduce the average path length. We set k to 6 and examine the network behavior for a range of p, specifically p = 0.1, 0.3, 0.5, 0.7, 0.9. Higher p brings the network closer to a random graph, yet the average node degree remains unchanged.

In the ER network, we consider a parameter p to control the probability of creating edges. The value of p determines the average node degree under a fixed n, which is approximately \(n(n - 1)*p/2\), where n is the number of nodes in the network. We examine the network behavior for a range of p, p = 0.1, 0.3, 0.5, 0.7, 0.9.

In this experiment, we consider the network load at BP = \({10^{ - 1}}\) as the fully-loaded state. The experimental data are derived from the average of 50 random experiments.

Figure 7a–c illustrate the increase in network coupling strength resulting from the flow number in three different networks. We increase the number of flows in three kinds of networks with varying parameters until the BP = \({10^{ - 1}}\) with a step of 500. The scale of all networks is 20 nodes, and the link length is 50 km. Comparing WS networks with the other two networks, parameters do not change the average node degree. The alterations in network structure solely change the capacity to carry flows. At the full-loaded state, the differences in network coupling strength are not significant. In the BA and ER networks, as the value of m and p increase, the average node degree increases, and the network coupling strength becomes weak at the fully loaded state. Compared with BA networks, the network coupling strength is much weaker in ER networks. This phenomenon can be attributed to the power-law degree distribution of nodes in BA networks, which leads to the emergence of key nodes. Key nodes carry a significant amount of flows, leading to a greater coupling effect strength among the flows. Although the increase in the average node degree results in the sparsity of flows, key nodes still bear a substantial amount of flows. This network characteristic becomes an important source of the coupling effect strength in BA networks. Thus, we can conclude that networks with higher average node degree exhibit lower network coupling strength. However, key nodes can enhance the network coupling strength.

Fig. 7
figure 7

(a) The network coupling strength in the WS network with different loads. (b) The network coupling strength in the BA network with different loads. (c) The network coupling strength in the ER network with different loads. (d) The network coupling strength in the WS network with different link lengths. (e) The network coupling strength in the BA network with different link lengths. (f) The network coupling strength in the ER network with different link lengths. (g) The network coupling strength in the WS network with different nodes. (h) The network coupling strength in the BA network with different nodes. (i) The network coupling strength in the ER network with different nodes.

Full size image

Figure 7d–f exhibit the impact of link length on the network coupling strength across three different networks. We conducted experiments in three different networks, with a fixed number of nodes at 20, by increasing the link length from 50 km to 500 km with a step of 50 km, and recording the \({C_P}\) at the full-loaded state. It can be observed that there is a linear correlation between link length and the network coupling strength, which can be understood from the Formula 10. Notably, the rate at which the network coupling strength increases with link length varies across different networks. In BA and ER networks, networks with higher average node degree exhibit a more rapid increase of \({C_P}\) as the link length increases. In contrast, in WS networks, despite changes in network structure, the average node degree remains constant. Consequently, the \({C_P}\) increases with link length at essentially a consistent rate across different p.

Figure 7g–i exhibit the impact of network scale on the network coupling strength across three different networks. We fixed the link length at 50 km and increased the node number from 20 to 100 with a step of 10 in three different networks and recorded the changes in \({C_P}\) at the full-loaded state. In BA networks, it is still the case that networks with lower node degree exhibit greater network coupling strength. However, changes in network size have a minimal impact on the coupling strength of the optical network at the full-load state. In ER networks, there is a tendency for the network coupling strength to decrease in larger networks. This can be understood that the network coupling strength is related to the node degree, which is determined by m and p. At the BA network with a fixed m, the average node degree remains constant, and consequently, the coupling effect stays unchanged. At ER networks with a fixed p, the average node degree becomes smaller as the network scale increases, and consequently, the network coupling strength becomes weak. As network scale increases, so does the network diameter, flows distributed throughout the network, and only couple with other local flows. In comparison, because of the characteristic of short path in WS networks, the increase in network scale results in a dense coupling effect among flows across the overall network, not only local flows. Thus, the coupling strength of the optical network in WS networks at the full-loaded state continuously strengthens as the network scale increases. We further extend the network scale to 450 nodes with a step of 50 and conduct comparative experiments among WS (p= 0.5), BA (m= 1), ER (p= 0.3), as depicted in Figure 8. The network coupling strength in WS increases intensively in large-scale networks compared with BA and ER networks.

Fig. 8
figure 8

The comparison of the network coupling strength between WS network (p = 0.5, nodes = 20, length = 50 km), BA network (m = 1, nodes = 20, length = 50 km) ,and ER network (p = 0.3, nodes = 20, length = 50 km).

Full size image

Comparing Fig. 7b,e,h, we can observe that the curves corresponding to larger m in the BA network consistently lie below those with smaller m. In the BA network, the larger the m, the greater the average node degree. Therefore, we believe that the larger the average node degree of the topology, the greater the network coupling strength.

Comparing Fig. 7c,f,i, we can observe that the curves corresponding to larger p in the ER network consistently lie below those with smaller p. In the ER network, the larger the p, the greater the average node degree. This phenomenon further illustrates that the greater the average node degree, the smaller the network coupling strength.

This judgment is reasonable and intuitive. Because the smaller the average node degree of the topology, the fewer the number of links in the topology, and the less communication resources available for flows. As a result, flows will be distributed more closely in the topology, the number of edges in the FCN will increase, and the network coupling strength will also increase.

Thus, we can conclude that networks with low average node degree and long distance exhibit stronger network coupling strength, and the “key node” in the network can enhance the network coupling strength. What’s more, the network features can influence the network coupling strength, and small-world networks exhibit stronger network coupling strength in large-scale networks than BA and ER networks.

Conclusions and discussions

In this chapter, we analyze the practical deployment of our method and summarize the research.

We analyze the time complexity of FCN. When there are N flows, it is only necessary to traverse N flow nodes in FCN and calculate flow coupling strength for each node once, so as to evaluate the power variation of all flows. Therefore, the time complexity is O(N). When calculating with Formula 1, since there is no analytical solution but only numerical solution, piecewise calculation must be adopted. Assuming that the propagation distance of the flow is L meters and the calculation is performed every 50 meters, the number of iterations \(K=L/50\). M is the number of channels, V is the number of links in the optical network. In a fiber link, each wavelength is affected by all other wavelength channels, and all links in the optical network need to be calculated to evaluate all flows. Then, the time complexity of estimating power variation suffered by all flows is \(O(K*V*M^2)\). It is easy to derive that \(N < V*M\). Our method can bring about a significant improvement in computational complexity. Moreover, as the network scale expands, the advantages of our approach become increasingly evident.

We conduct an analysis of the space complexity of FCN. Since FCN is a directed weighted network, when there are N flows, calculating according to the adjacency matrix requires \(O(N^2)\) space for storage. This poses requirements on the storage capacity of hardware.

What’s more remarkable is that the PAA only needs two parameters for power optimization, while traditional heuristic methods for optimizing power tilt involve extremely high parameter-tuning complexity. Suppose there are R transmitters in the optical network and B bands are used (\(B=2\) for C+L systems, \(B=3\) for S+C+L systems). Conventional approaches would require tuning \(R*B\) parameters to optimize the entire network. In contrast, the PAA algorithm can achieve power optimization for all flows in the network by adjusting only two parameters (\({P_{origin}}\) and \(\Delta {P_i^{ad}}\)), regardless of network scale or the number of bands used. Moreover, it outperforms traditional methods in terms of performance. Therefore, our approach is suitable for large-scale networks.

In the current work, our main contribution is modeling the coupling effect and constructing the FCN. We introduce the complex network into the interactions between flows in the optical network. We propose the flow coupling strength and the network coupling strength.

In response to the flow coupling strength. We validate the model rationality based on the linear correlations and optimize the launch power based on the flow coupling strength. We conclude that optimizing power based on flow coupling strength outperforms optimization based on frequency.

In response to the network coupling strength. We find that stronger network coupling strength leads to greater network capacity gain from power optimization. Furthermore, we investigate how the optical network structure influences the network coupling strength. Networks with low average node degree and long link length show stronger network coupling strength. In large-scale networks, WS networks show stronger network coupling strength than BA and ER networks because the network coupling strength in WS networks increases as the network scale.

Based on the FCN, we can directly analyze the influence of the SRS effect in the network layer. In the future study, based on the FCN, we can introduce further complex network theory to investigate the uncovered phenomena in the multi-band optical network, such as cascade failures, emergencies, link prediction, etc.