Enhancing the robustness of interdependent networks by positively correlating a portion of nodes

Cascading failures caused by interdependencies make modern coupled systems extremely fragile to failures. In existing network robustness enhancing methods, maximizing interlayer degree-degree correlations has been proven to be an effective way to improve the robustness of interdependent networks under random failures. Here, we propose a portion of nodes positively correlated strategy (PNC) to improve network robustness by positively correlating a portion of nodes, in which the nodes that are positively correlated are selected in descending order of degree, starting at a cutoff value. Based on percolation theory, we verify the effectiveness of PNC on different networks. And find that, when the nodes with the highest degree are preferentially correlated, i.e. the cutoff value takes the maximum degree, this strategy achieves a state-of-the-art optimization effect. In particular, for interdependent scale-free networks with power-law exponent γ that satisfies 2<γ<3 , we theoretically demonstrate that the highest degree preferentially correlated mode can maximize network robustness by changing the coupling state of the near 0 proportion of nodes. For γ = 3, such a mode can make the network turn into a second-order phase transition at the collapse point. Finally, we discuss the relationship between the robustness of optimized networks and common links in real-world networks.


Introduction
Network percolation theory has been widely used to analyze the robustness of networks by theoretically portraying the changes in the network's giant connected component during the node removal process [1][2][3][4][5][6][7][8].Modern systems are always coupled with each other for performing functions [9][10][11][12][13][14][15].For example, communication networks require energy from the power grid to function properly.Such relationships can be effectively modeled using interdependent networks in which the failures of nodes in one layer can cause the failures of nodes in other layers, leading to cascading failure processes throughout the network.Buldyrev et al [16] first constructed a percolation framework to analyze the robustness of interdependent networks and found that the network undergoes a first-order phase transition at the critical point.In subsequent studies, this framework was extended to different attack patterns, such as targeted attacks [17], localized attacks [18], combined attacks [19], and attacks with limited knowledge [20,21].Similar first-order phase transitions were observed in these studies, indicating that cascading failures caused by interdependencies can abruptly fragment the system.
The most straightforward way to improve the robustness of interdependent networks is to reduce the dependency ratio.Parshani et al [22] discovered that when the dependence ratio is below a threshold, the network transforms from a first-order phase transition to a second-order phase transition at the critical point.Dong et al [23] generalized this framework from a random failure to a targeted attack using equivalent substitution.Schneider et al [24] proposed a more effective network robustness optimization strategy by preferentially removing the dependencies of hub nodes.Zhou et al [25] analyzed the robustness of partially interdependent scale-free (SF) networks to random failures and observed three types of phase transitions in the process of decreasing the dependence ratio, i.e. first-order, second-order, and hybrid phase transitions.
The methods used in these studies revealed feasible ways of optimizing the robustness of interdependent networks and have been widely applied to more complex coupled systems [1,26].In addition, adjusting the proportion of coupled nodes can also be used to optimize the robustness of other types of networks.For example, there exists an optimal fraction of interconnected nodes in modular interacting networks [27].
Changing the interlayer correlation of the network is another effective method for enhancing network robustness.The first correlation to receive attention from researchers was the interlayer degree-degree correlation (IDDC), which describes the correlation between the degree values of nodes in different layers [28].Numerous studies have shown that networks with positive correlations are more robust to random failures, while networks with negative correlations are more robust to target attacks [3,[29][30][31][32][33][34].Another indicator used to quantify the correlation between different layers is intersimilarity, which portrays the overlapping levels of edges of different layers [32,35].Since the interdependent network is close to a single-layer network, when the overlap of edges across layers is high, networks with higher intersimilarity are more robust to random failures.Kleineberg et al [36] observed a hidden correlation in real-world interdependent networks known as interlayer geometric correlations.Later, they constructed this type of interlayer correlation by mapping the synthetic network into hyperbolic space and found that it could mitigate the network's vulnerability to target attacks [37].Additionally, numerous studies have improved the robustness of networks by designing interlayer structure optimization algorithms [38][39][40].
As mentioned above, maximizing the IDDC has been demonstrated to be effective in improving the robustness of the network under random failures.However, in real-world scenarios, only a portion of nodes' coupling state can be adjusted.For example, the interlayer coupling state of the aviation network can be adjusted by changing the routes of different airports [3,41]; the coupling state between the power grid and the internet can be adjusted by changing the dependencies between the nodes [42][43][44].
In this context, we propose a general model to study network robustness enhancement by positively correlating a portion of nodes.The principle of our portion of nodes positively correlated strategy (PNC) is to select the nodes that need to be positively correlated in descending order of degree, starting at a cutoff value.Accordingly, we construct a new percolation framework to analyze the percolation behavior of optimized networks.Based on this framework, the effectiveness of PNC is verified in different types of networks.We demonstrate that the highest degree preferentially correlated mode (DPC), a special case of PNC, is the most effective mode.In particular, we provide exact theoretical proof of the effectiveness of DPC in SF networks.Furthermore, in real-world networks, we investigate the relationship between the robustness of PNC-optimized networks and the proportion of common links, where common links are defined as overlapping edges between the layers.
The rest of this paper is organized as follows.In section 2, we introduce the percolation framework for correlating a portion of nodes in interdependent networks.The effectiveness of PNC in synthetic networks is theoretically analyzed in section 3.In section 4, we investigate the impact of PNC in real-world networks.The paper is summarized and discussed in section 5.

Percolation theory of interdependent network
In this study, we consider the interdependent networks arranged in two layers and assume that the nodes of the network are one-to-one interdependent.This condition can be expressed as follows: if node i of layer A depends on node j of layer B and node j of layer B depends on node l of layer A, then l = i.The node degree, which is the number of neighbors of a given node, is denoted as k.We assume that each layer α is generated by a configuration model with the same degree distribution P α (k) = P(k) (α = A, B).The coupling relationship between the two layers of the network can be captured by the joint degree distribution P(k A , k B ), which represents the proportion of coupled nodes with degree k A in layer A and degree k B in layer B.
For interdependent networks, it is generally assumed that only the nodes in the giant mutually connected component (GMCC), which is defined as the largest set of coupled nodes connected through each layer, are functional [16].On this basis, initial node failures can trigger a dynamic cascading failure process until the whole system reaches a steady state.Random failures correspond to natural failures and are widespread in different types of systems.Thus, we focus on random failures where a fraction 1 − p of randomly selected nodes in layer A are initially removed.Then, the relative size of the GMCC S, after cascading failures, can be calculated theoretically through the self-consistent probabilities method [45,46].
A randomly chosen node with degree k A in layer A, belonging to the GMCC, must satisfy the two conditions simultaneously.First, this node must not be removed, and at least one of its edges can lead to the GMCC.Second, its dependent nodes in another layer must have at least one edge that can lead to the GMCC.
Assume the degree of its dependent nodes in layer B is k B .And define the probability that a randomly selected edge in layer A (B) cannot lead to the GMCC as x A (x B ).The probability of this node belonging to the GMCC is p(1 − x kA A )(1 − x kB B ). Thus, the relative size of the GMCC S satisfies where m is the minimum degree and M is the maximum degree.Then, we consider the equation for x A .Suppose a randomly selected edge in layer A leads to a node u with degree k A and u depends on a node with degree k B .Analogously, this edge can lead to GMCC equivalent to u is not removed and the remaining k A − 1 edges of u and the edges of its dependent node both have at least one edge that can lead to the GMCC.The probability of this event occurring is p(1 ).Thus, we can obtain the equation for x A as where ⟨k⟩ = ∑ M k=m kP(k) is the average degree, and k A P (k A , k B ) / ⟨k⟩ is the probability of a randomly selected edge in layer A can lead to a pair of coupled nodes with degrees (k A , k B ). Similarly, we can obtain the equation for x B as The critical point at which the network undergoes a first-order phase transition corresponds to the tangency of the curves represented by equations ( 2) and (3) [46].If these equations are transformed into x A = F A (x B , p) and x B = F B (x A , p), this condition can be represented as By combining equations ( 2)-(4), we can obtain the critical removal fraction 1 − p c of the network.

Interlayer degree-degree correlation
The interlayer structure of interdependent networks can be characterized by the interlayer degree-degree correlation (IDDC) [28], which is calculated as where which is reached when the two layers are maximally positively correlated (MP) (figure 1(d)).Meanwhile, when the two layers are randomly correlated (RC) (figure 1 ) and thus r = 0.It has been shown that networks are more robust to random failures when the two layers are positively correlated (r > 0) [33].Especially, the MP case is the most stable structure of the network (figure 1(d)), and the joint degree distribution can be calculated as

Portion of nodes positively correlated strategy
The maximally positively correlated case is an ideal situation.In real-world scenarios, the IDDC of man-made or naturally formed systems is usually low [3,16,28].Thus, we assume that the initial state of the network is RC.On this basis, maximizing IDDC requires changing the coupling state of all nodes, which is difficult to achieve (figures 1(a) and (d)).To this end, we propose a portion of nodes positively correlated strategy (PNC) to improve network robustness by positively correlating the degrees of a portion of nodes.Specifically, we select the nodes that are positively correlated in descending order of degree, starting at a cutoff value.The nodes that are not selected remain RC.The steps of this strategy are given by • Step1: Sort the degree values of the two layers in descending order.
• Step2: Select q proportions of nodes in descending order starting at cutoff degree K 2 of two layers and make them positively correlated.Let the remaining 1 − q proportion of nodes be RC.
A simple example of this strategy is shown in figure 1(b).In this example, we set K 2 = 2 and q = 0.25.Under this assumption, nodes 4 and 7 in two layers are positively correlated, and the remaining nodes are RC.When K 2 = M, the coupling state of the PNC-optimized network is equivalent to the coupling state studied in literature [31].
Previous research has shown that the internal mechanism by which IDDC enhances network robustness is that the network's connectivity is provided by hub nodes and the mutual coupling of them reduces their failure risk during the removal and cascading process [1,34].Therefore, in this paper, we also focus on the case K 2 = M in which the nodes with the largest degree are protected.Figure 1(e) displays a simple example of such a highest degree preferentially correlated mode (DPC) .Under the assumption that q = 0.25, nodes 3 and 6 in the two layers are positively correlated, and the remaining nodes are RC.Furthermore, as shown in figures 1(d)-(f), the MP case equivalent to the DPC taking q = 1.
To theoretically analyze the effect of PNC, we next derive the joint degree distribution of the network after the optimization of the above two steps.For a given degree distribution P(k), we assume that the minimum degree value of the selected q proportion nodes is K 1 , i.e.K 1 = inf{k| ∑ K2 i =k+1 P(i) ⩽ q}, where inf is the infimum.As shown in figure 1(c), nodes with degree values k that satisfy k ∈ (K 1 , K 2 ] can only be coupled to nodes with the same degree value in the other layer.Therefore, For k A = k B = K 1 , nodes in two layers can be classified into two categories.The first category is the nodes that are chosen to be in the q proportion nodes.The proportion of such nodes to all nodes is q − ∑ K2 k=K1+1 P(k) (figure 1(c)).These nodes will be coupled with the nodes with degree K 1 in another layer.The second category is the unchosen nodes, whose proportion to all nodes is ∑ K2 k=K1 P(k) − q.Thus, in the process of random coupling, the probability that such nodes in the two layers can be coupled is By multiplying by the proportion of randomly coupled nodes 1 − q, we can get the proportion of coupled nodes with degrees (K 1 , K 1 ) due to random coupling as Using the same method, we can obtain that, when For Combining the above discussion, we can obtain the joint degree distribution of PNC-optimized networks as By substituting K 2 = M into equation ( 11) (figure 1(f)), we can obtain the joint degree distribution of the DPC-optimized network as When q = 1, equation ( 12) is equivalent to equation (6).

Results in synthetic networks
In this section, we verify the effectiveness of the PNC in two typical synthetic networks: interdependent Erdõs-Rényi (ER) random networks and interdependent SF networks.For interdependent ER networks with an average degree ⟨k⟩, the degree distribution satisfies the Poisson distribution; that is, where e is the natural constant, and k! is factorial of k.Based on the aforementioned percolation framework, we show in figure 2 the theoretical and simulation results of interdependent ER networks, demonstrating a strong agreement between them.First, we investigate the effectiveness of the DPC highlighted in the previous section.As shown in figure 2(a), the robustness of the DPC-optimized network gradually increases as the proportion of nodes that are positively correlated q increases.Especially, the DPC only needs to optimize a proportion q = 0.5 of nodes to make the network robustness close to that of the MP case.Note that the MP case corresponds to the DPC taking q = 1.This indicates that coupling the nodes with high degrees is the key factor in improving the robustness of the network.In real scenarios, positively correlating the nodes with highest degrees is usually difficult.Thus, we further test the optimization effect of PNC with different cutoff degrees K 2 .The results in figure 2  heat maps of the joint degree distributions with different cutoff degrees K 2 .It can be seen that as K 2 increases, the joint distribution of the network becomes more heterogeneous, which in turn reduces the failure probability of the nodes with large degrees.Furthermore, the heat maps in figures 2(c)-(e) show that the joint distribution of the PNC-optimized network gradually converges to the DPC-optimized network as the cutoff degree K 2 increases.A worthy concern is whether network robustness increases monotonically as K 2 increases, i.e. whether the optimization effectiveness of PNCs with higher cutoff degrees is certainly better than those with lower values.To analyze this issue, we show in figure 3(a) the critical removal fraction 1 − p c as a function of q with different K 2 .It can be seen that as q increases, the 1 − p c of networks with lower K 2 exceeds that of networks with higher K 2 .This occurred because, when q is large, the small K 2 makes a large number of small-degree nodes in the two layers positively correlated, which can indirectly reduce the coupling probability of large-degree nodes with small-degree nodes.On this basis, as shown in figure 3(b), network robustness does not monotonically increase with K 2 .
To explain this phenomenon more intuitively, we show in figures 3(c) and (d) the heat maps of the joint degree distribution of the network when such a phenomenon occurs.Here, we set q = 0.6, and compare the cases of K 2 = 6 and K 2 = 4.When ⟨k⟩ = 4, we have ∑ 4 k=0 P(k) = 0.6288 and ∑ 6 k=0 P(k) = 0.8893.Thus, in the case where K 2 = 4, almost all nodes with degrees less than or equal to 4 in the two layers are positively correlated.This makes nodes with degrees greater than 4 more likely to be coupled to nodes with degrees greater than 4 in another layer (figure 3(c)).In the case where K 2 = 6, there are still a large number of nodes with small degrees that are not positively correlated.These nodes will couple to nodes with large degrees in another layer by random coupling (figure 3(d)).For example, at this point, there is a large percentage of nodes with degrees (2, 7) and (7,2).Especially, this phenomenon may make the PNC have a negative optimization effect, i.e. the 1 − p c of the network is lower than that of RC (subplot of figure 3(a)).Therefore, the appropriate K 2 and q should be selected when using PNC to optimize the network structure.
Next, we focus on interdependent SF networks in which where γ is the power-law exponent, m is the minimum degree, and M is the maximum degree.The theoretical and simulation results for interdependent scale-free networks are shown in figure 4(a), where m = 2, M = 100, and q = 0.1.It can be seen that PNC only needs to change the coupling state of a few nodes to significantly increase the network's collapse point compared to the RC case.In particular, the DPC can make the network robustness close to the MP case.This occurs because, in scale-free networks, a small percentage of hub nodes can maintain network connectivity, and DPC can significantly reduce their failure probability.
To further analyze the effectiveness of DPC, we next investigate the properties of the network optimized by DPC under the assumption that the system is infinite, i.e.M → ∞.Since the degree distributions of the two layers are identical, we have x A = x B = u.Thus, equations ( 2) and (3) degenerate into one equation, which can be written as New J. Phys.26 (2024) 063030 Substitute equation ( 12) into the equation ( 15) yields Let and Equation ( 16) can be transformed into Let It is easy to obtain F(1) = 1 and F(0) > 0. Thus, equation ( 19) has a trivial solution u = 1.Next, we focus on the nontrivial solution of equation (20), which determines the value and type of the network's phase transition points.Consider the case of 2 < γ < 3, in which G(u) 1) is infinite.Thus, the curve represented by y = F(u) near u = 1 is on the lower side of the straight line y = u.Combined with F(0) > 0, for any 0 < 1 − p < 1, there is always an intersection between curves y = F(u) and y = u (figure 4(d)), which means that equation ( 20) has a nontrivial solution for any 0 < 1 − p < 1.Thus, the critical removal fraction 1 − p c of the network optimized by DPC is 1.This indicates that, for any q > 0, the robustness of the network optimized by DPC is maximized.To verify this conclusion, we show in figure 4(b) the relative size of the GMCC S with q = 0.01.It can be seen that as the maximum degree M increases, the critical removal fraction 1 − p c of the network gradually approaches 1.Furthermore, when q = 1, DPC is equivalent to MP (figures 1(d) and (e)).Therefore, under the assumption that the system is infinite, for any q > 0, the DPC-optimized network has the same critical point as the MP case.This phenomenon theoretically guarantees that the optimization effectiveness of DPC is close to that of the MP case (subplot of figure 4(a)).Especially, as shown in figure 4(c), the gap between them becomes smaller when the system size increases.
For the case of γ = 3, G ′ (u) diverges as −ln(1 − u) when u → 1.Thus, G(u) ) has a finite value at u = 1.Accordingly, F ′ (1) is finite.On this basis, when the removal fraction reaches the threshold 0 < 1 − p c < 1, the curves y = F(u) and y = u will be tangent at u = 1 (figure 4(e)).This indicates that the DPC-optimized network undergoes a second-order phase at the critical point 1 − p c .For the case of γ > 3, it is easy to obtain that F ′ (1) = 0.As shown in figure 4(f), curves y = F(u) and y = u are tangent at u < 1 when 1 − p reaches the critical removal fraction 1 − p c .At this point, the DPC-optimized network undergoes a first-order phase transition.Similarly, the DPC is equivalent to the MP, when q = 1 (figures 1(d) and (e)).Thus, the MP case has the same phase transition behavior as the DPC-optimized network.
In summary, when the system size is infinite (M → ∞), the DPC has the same critical behavior as the MP for any q > 0. This result provides a theoretical assurance of the effectiveness of DPC.Meanwhile, this result reconfirms the importance of hub nodes in the connectivity of SF networks [1,34].

Results in real-world interdependent networks
In this section, we study the impact of PNC in three real-world networks: the US Air Transportation network [5], the European Air Transportation network [47], and the Homo Sapiens network [5,48,49].In both the US air transportation network and European Air Transportation network, each layer represents the routes of a different commercial airline.The US air transportation network contains three layers of networks, which can form three different interdependent networks, as shown in table 1.In the European air transportation network, there are 37 layers of networks.Here, we choose three interdependent networks combined from four layers of networks as test subjects; that is, Ryanair-Easyjet, Easyjet-Lufthansa, and Lufthansa-Air Berlin (see table 1 for details).In the Homo Sapiens network, the nodes represent proteins and different layers represent different interactions between them, i.e. direct interaction, physical association, suppressing genetic interaction.Similar to the US Air Transportation network, these layers can form three different interdependent networks (table 1).In simulating 1 − p c , we assume that the network collapses when S is smaller than √ N/N.Due to the different degree distributions in the two layers, we select the q proportion of nodes that are positively correlated, starting from the nodes at the percentile of the degree value ranking.Under this assumption, DPC corresponds to selecting nodes at the 100th percentile.Without loss of generality, for PNCs that are not the highest degree preferentially correlated, we select the nodes at the 90th percentile.
We show in figures 5(a)-(c) the 1 − p c of these interdependent networks.It can be seen that PNC can effectively increase the critical removal fraction 1 − p c of the network compared to the RC case.And, DPC only needs to optimize a small percentage of nodes to make different types of networks as robust as MP.However, in some optimization ratios, PNCs that are not the highest degree preferentially correlated can increase the coupling probability of large-and small-degree nodes, which, in turn, makes the network less robust than the RC case.Such negative optimization phenomena are similar to what we observe in the synthetic networks (figure 3(a)).To more accurately analyze the effectiveness of different methods, we show in figures 5(d)-(f) the 1 − p c of different networks with q = 0.1.It can be seen that the optimization effect of DPC is close to that of MP.However, it is difficult to positively correlate the nodes with highest degrees.Therefore, a PNC with suitable parameters should be selected to optimize the structure of the network.
Meanwhile, the results in figure 5 show that the original version of the Homo Sapiens network exhibits a high level of robustness.The reason behind this is that the interdependent networks of Homo Sapiens have a high proportion of common links f e , which is defined as the proportion of overlapping edges between the layers over all edges.When the proportion of common links f e is high, the nature of the network is close to that of a single-layer network and, therefore, it will perform more robustly.To verify this conclusion, we show in figure 6 the f e of three interdependent networks.It can be seen that the f e of the original Direct-Physical network is much higher than that of the PNC-optimized networks (figure 6   The coupling between nodes with large degrees can increase the probability of generating common links.As a result, a larger IDDC can produce more common links, and the MP case can maximize this effect.To analyze the effectiveness of the PNC in generating common links, we show in figures 7(a)-(c) the f e as a function of q.It can be seen that the PNCs with different parameters can all effectively generate common links.And the DPC only needs positively correlated proportion q = 0.1 of nodes to make f e close to that of MP.As mentioned above, the MP case corresponds to the DPC taking q = 1.This again demonstrates that coupling nodes with large degrees is the key factor in generating common links.Furthermore, in the US Air Transportation network and European Air Transportation network, the number of common links of the optimized network exceeds that of the original network (figures 7(a) and (b)); thus, an excellent optimization result is achieved (figures 5(a) and (b)).In figures 6(d)-(f), we also visualize the state of the common links of the American-Delta interdependent network with q = 0.1.It can be seen that the common links' structure of the DPC-optimized network is similar to that of MP.  ) fall in the first quadrant of the coordinate area.This proves that another reason why DPC is effective is its advantage in generating common edges.
In summary, two of the three studied networks, the US air transportation network and the European air transportation network, are infrastructure networks.Such networks are modifiable and form the backbone of society's functioning.The results in figures 5(a), (b) and 7(a), (b) show that our PNC strategy can improve the robustness of these infrastructure networks by adjusting the coupling state of a small proportion of nodes, which demonstrates that this strategy has high practical significance.

Conclusion and discussion
In this paper, we propose to use a portion of ndes positively crrelated strategy (PNC) to enhance the robustness of interdependent networks.Accordingly, a new percolation framework is designed for analyzing the percolation behavior of optimized networks.The results in both the synthetic and real-world networks show that preferentially coupling the nodes with the highest degree is the most efficient mode of PNC.Moreover, for networks with high heterogeneity, the advantages of this mode are especially prominent.That is, for interdependent scale-free networks, DPC can optimize network robustness with a cost near 0 when the power-law exponent γ satisfies 2 < γ < 3. And, for γ = 3, DPC can change the network from a first-order phase transition to a second-order phase transition.These results provide a theoretical guarantee of the effectiveness of our strategy.
In the PNC, we positively couple nodes according to the ordering of degree values.However, specifying the degree values of the selected nodes is an ideal assumption.In many real-world scenarios, it is typically only possible to select nodes that are positively correlated with some degree preferences.The impact of such degree preferences on the effectiveness of optimization strategies is a question worth investigating.Furthermore, many networks are partially interdependent, and how to optimize the coupling state of such networks is also a worthwhile research area.Meanwhile, our results also show that the proportion of common links f e is also a key factor that affects the robustness of the network.Therefore, in different scenarios, designing optimization strategies that can simultaneously improve the IDDC and f e is significant for improving the robustness of real-world systems.

Figure 1 .
Figure 1.Illustrations of different coupling modes.((a), (b) and (d), (e)) Simple examples of RC (a), PNC (b), MP (d), and DPC (e).In ((a), (b) and (d), (e)), the black dotted and solid red line connect the nodes that are randomly correlated and positively correlated, respectively.The yellow dashed line in (a) is the edge that needs to be adjusted for the PNC shown in (b).(c), (f) Calculation methods for the joint degree distributions of networks optimized by PNC (c) and DPC (f).
(b) show that PNC with K 2 < M can also effectively improve the robustness of the network.To demonstrate the mechanism of the effectiveness of PNC, we show in figures 2(c)-(e) the

Figure 2 .
Figure 2. Results in interdependent ER networks.(a) Relative size of the giant mutually connected component S of the DPC-optimized network with different q.(b) Relative size of the giant mutually connected component S of PNC-optimized network with different cutoff degrees K2 when q = 0.5.The subplot of (a), (b) is the critical removal fraction 1 − pc as a function of ⟨k⟩.The symbols in (a), (b) represent the average simulation results for over 100 independent realizations of synthetic networks with 10 5 nodes, and the lines represent the theoretical results.(c)-(e) Heat maps of joint degree distributions of networks optimized by PNC with different cutoff degrees K2.The top side of them shows the average simulation results for over 100 independent realizations of synthetic networks with 10 5 nodes, and the bottom side shows the theoretical results.Here, we show the heat maps at log-scale.

Figure 3 .
Figure 3. Effectiveness of PNC with different cutoff degrees K2 in interdependent ER networks.(a), (b) Critical removal fraction 1 − pc as a function of q (a) and K2 (b).(c), (d) Heat maps of joint degree distributions of networks optimized by PNC with K2 = 4 (c) and K2 = 6 (d).In (c), (d), we set q = 0.6.The heat maps are shown at log-scale.

Figure 4 .
Figure 4. Results in interdependent SF networks.(a) Relative size of the giant mutually connected component S a function of 1 − p with M = 100 and q = 0.1.The symbols represent the average simulation results for over 100 independent realizations of synthetic networks with 10 5 nodes, and the lines represent the theoretical results.(b) Relative size of the giant mutually connected component S as a function of 1 − p with different M. (c) Critical removal fraction 1 − pc as a function of M. The dashed and solid lines represent the results for MP and DPC, respectively.(d)-(f) Graphical solution of equation (19) with 2 < γ < 3 (d), γ = 3 (e), and γ > 3 (f).In (b)-(f), we set q = 0.01.
(c)), and thus the robustness of the original version is close to that of the MP case (figure 5(c)).In American-Delta and Ryanair-Easyjet, the f e of the original version is low (figures 6(a), (b) and (d)-(f)), so the above phenomenon is not evident (figures 5(a) and (b)).

Figure 5 .
Figure 5. Robustness of real-world networks after optimization by PNC.(a)-(c) Critical removal fraction 1 − pc of the US Air Transportation network (a), European Air Transportation network (b), and Homo Sapiens network (c) as a function of q.The green line is the result of the PNCs that are not the highest degree preferentially correlated.(d)-(f) Critical removal fraction 1 − pc of the US Air Transportation network (d), European Air Transportation network (e), and Homo Sapiens network (f) with q = 0.1.For each network, we show the results of three types of interdependent networks.All of the results are the average simulation results for over 100 independent realizations.

Figure 6 .
Figure 6.Common links of American-Delta, Easyjet-Lufthansa, and Direct-Physical.(a)-(c) Proportion of common links fe of American-Delta (a), Easyjet-Lufthansa (b), and Direct-Physical (c).The original network is represented by OR.For DPC (red column) and PNCs that are not the highest degree preferentially correlated (green column), we set q = 0.1.All of the results are the average simulation results for over 100 independent realizations.(d)-(f) Illustrations of common links of American-Delta.(d) Common links of the original network.(e) Common links of the DPC-optimized network with q = 0.1.(f) Common links of the MP case.The orange edges and nodes are common links and nodes connected by common links, respectively.The black nodes are nodes that are not connected by common links.

Figure 7 .
Figure 7. Relationship between robustness and common links in optimized networks.(a)-(c) Proportion of common links fe of the US Air Transportation network (a), European Air Transportation network (b), and Homo Sapiens network (c).For each network, the results of the three types of interdependent networks are shown.The green line represents the result for the PNCs that are not the highest degree preferentially correlated.(d)-(f) Correlation between p O c − p P c and f P e − f O e in the US Air Transportation network (d), European Air Transportation network (e), and Homo Sapiens network (f).Different symbols represent interdependent networks composed of different layers.Here, we set q = 0.1.The blue and red symbols are the results after optimization using DPC and PNCs that are not the highest degree preferentially correlated, respectively.The blue and red regions in the coordinate plane are the regions where p O c − p P c > 0 and p O c − p P c < 0, respectively.The black dashed line is f P e − f O e = 0.All of the results are the average simulation results for over 100 independent realizations.

Table 1 .
Characteristics of the US Air Transportation network, European Air Transportation network, and Homo Sapiens network.The interdependent networks combined by different layers are shown in the second column.The number of nodes N, the minimum degree of layer A mA, the minimum degree of layer B mB, the maximum degree of layer A MA, and the maximum degree of layer B MB are also shown.