Improvement on Load-Induced Cascading Failure in Asymmetrical Interdependent Networks : Modeling and Analysis

Many real-world systems can be depicted as interdependent networks and they usually show an obvious property of asymmetry. Furthermore, node or edge failure can trigger load redistribution which leads to a cascade of failure in the whole network. In order to deeply investigate the load-induced cascading failure, firstly, an asymmetrical model of interdependent network consisting of a hierarchical weighted network and a WS small-world network is constructed. Secondly, an improved “load-capacity” model is applied for node failure and edge failure, respectively, followed by a series of simulations of cascading failure over networks in both interdependent and isolated statuses. The simulation results prove that the robustness in isolated network changes more promptly than that in the interdependent one. Network robustness is positively related to “capacity,” but negatively related to “load.” The hierarchical weight structure in the subnetwork leads to a “plateau” phenomenon in the progress of cascading failure.


Introduction
Complex network theory has been a successful tool in modeling and analysis of modern systems [1].A variety of network models have been proposed to approximate realistic systems such as power grids [2], transportation systems [3], communication networks [4], and other systems.Most of them focus on single and isolated systems.Recently, interdependent network [5] provides a new insight in understanding the structure [6], percolation [7], spreading processes [8], evolution games [9], and robustness [10,11] of complex systems.This kind of structure usually consists of two or more subnetworks, with the subnetworks working dependently on each other [12,13].For example, power grids and computer systems must depend on each other because computers require power grids to supply electricity.Power grids, in turn, rely on computer systems to control power transmission process.
In 2010, Buldyrev et al. [14] found that interdependent networks have become significantly more vulnerable than their noninteracting counterparts under random attack.Gao et al. [5] reviewed the connectivity properties of "networks of networks" formed by interdependent random networks.
Hu et al. [15] constructed a partially coupled network with both interdependent and interconnecting links.The interconnections satisfy "one to one" condition.They found that the change of interconnecting links leads to the change of the phase transition from second order to first order through hybrid phase transition.Huang et al. [16] studied the robustness of interdependent networks under targeted attack on high or low degree nodes.It provided a routine method to study the degree-based targeted attack problems in both single networks with dependency links [17,18] and other general randomly connected and uncorrelated interdependent networks.Moreover, Parshani et al. [19] described the dynamic process of cascading failures on two partially interdependent networks.
Currently, researchers focus on the problem of modelling an interdependent network model for analyzing the progress of cascading failure.Based on classical network models such as ER [20], WS [21], and BA [22], researchers studied well some symmetrical and asymmetrical networks [19,23,24], such as ER-ER, WS-WS, BA-BA, ER-WS, BA-ER, and BA-WS; but these models differ greatly from real-world systems [25,26].Taking interdependent networks like ground transportation network and airline network as an example, ground transportation network displays a hierarchical property.That is, stations at provincial level have higher capacity and more importance than municipal stations.However, the airline network can be seen as a single level network with small world property [27].So, some other researchers [28][29][30] made their trials to construct framework of asymmetrical interdependent networks which are more authentic to reality.
Moreover, load such as cargoes transported in the transportation network and electric stream in power grids can trigger cascading failures [31].The load carried by the failed nodes or edges will not disappear but will flow to the remaining part of the network, which, in a possible way, will cause further failures.As far as we are concerned, not enough attention has been paid to the cascading failure induced by load redistribution in interdependent networks.
In order to investigate the load-induced cascading failure in interdependent networks, we propose asymmetrical interdependent networks model in Section 2 and the load-induced cascading failure model in Section 3. Section 4 simulates the cascading failure in the proposed model and compares it with a WS-WS symmetrical network when nodes and edges suffer from intentional attacks, respectively.Section 5 summarizes the contribution of this paper and identifies future research needs.nodes in the same level except for the top level, where all nodes are connected with each other.Internal edges connecting different levels have diverse weights representing the difference in importance of interlevel connections.The hierarchical structure is established as follows.Firstly, we establish a fully connected top level, the level marker is set to ℎ = 1, the number of nodes is  1 , and the edge weight is  1 .Then, each node in top level establishes relations with  subordinate nodes from the next level, where the level marker is set to ℎ = 2. Nodes in this level do not connect with each other.We repeat the former step until ℎ = ; then, Network  with hierarchical weight is completed.The weight of edges connecting level ℎ and level ℎ + 1 is set to

Network Model
where constant  controls the weightiness of internal edges which connect neighboring levels.Network  is evolved by the rule of WS small-world network because small-world network yields the shortest and most effective paths [32].The weight of each internal edge is equivalent.It turns from regular network to random network by adjusting the reconnecting probability  2 ∈ [0, 1], where  2 = 0 corresponds to the case of nearest-neighbor coupled network; the nodes are placed on a ring lattice with periodic boundary conditions and each node is initially connected to  nearest neighbors.When  2 ∈ (0, 1), the WS small-world network model is created by rewiring a small fraction of the links with probability  2 to nodes chosen at random.The average degree of node is still .When  2 = 1, it corresponds to a completely random network.
The coupling proportion is set to  1 = 0.7 between Network  and Network  .The sizes of the subnetworks are set to   =   = 340.In Network  ,  = 4,  = 4, and  1 = 4, so the numbers of subordinate nodes in the next three levels are 16, 64, and 256.Because  = 2 and  1 = 10, the weights of internal edges that connect different levels are  1,2 = 8,  2,3 = 6, and  3,4 = 4 according to (2).Network  is a WS small-world network with  2 = 0.1 and  = 4. Due to the randomly "one to one" relationship, we set the weight of coupling edges to be equal to the maximum weight of internal edges connected to the coupled node in Network  .

Cascading Failure Model
The coupling property makes the interdependent networks fragile when suffering from intentional attack [16,18].Here we focus on the cascading failure of interdependent network induced by load.We define the initial loads   (0) of node  as a function of the node intensity   : where load parameters  > 0 and  > 0 are adjustable to control the distribution of initial load.We can see that each node bears more or less some initial loads according to the node intensity and the loads on each node are a nonlinear function unless the parameter  is equal to 1. ML model [33] conjectures that the capacity of a node is proportional to its initial loads.There is a linear relationship between capacity   of node  and its initial load   (0).Consider where  > 0 is the tolerance parameter.It is well studied that big  leads to excellent robustness.However, considering other perspectives such as cost, the capacity is impossible to be infinite, so it tends to a finite .In addition, the linear relationship does not fit for many real-world networks, so a nonlinear "load-capacity" model is proposed as follows: where two tunable parameters  > 0 and  > 0 are introduced.If  = 1, it decays to the linear "load-capacity" model.According to a local nearest redistribution strategy, the proportions of load distribution Π  and the new added load Δ  →  where the failed node  passes to its neighbor  are as follows [34]: where  is the neighbor of the failed node  and Γ  is the set of neighbors which node  connects.If the initial load of node  plus the load that node  transfers to exceeds its capacity (  <   + Δ  →  ), node  fails and leads to a new round of load redistribution.The process repeats until there is no overloaded node or the entire network is paralyzed.The evolving procedure is illustrated in Figure 2.
At stage 1 in Figure 2(a), node  1 is attacked and fails; the load distributes to its neighbors { 2 ,  3 ,  4 ,  5 }.Node  2 fails because of overloading and it triggers the failure of  1 because the load of  1 exceeds its capacity after receiving some amount of load from  2 .Meanwhile, nodes  3 ,  4 , and  5 are still active because they are not overloaded.At stage 2 in Figure 2(b), due to the load redistribution of node  1 , node  6 is overloaded, but nodes  2 and  5 are still active. 6 distributes the load to its neighboring nodes which leads to the failure of nodes  7 and  8 .In this way, the cascading failure propagates in both Network  and Network  .At stage 3 in Figure 2(c), the failures of nodes  7 and  8 trigger the failures of  4 and  5 .There is no new failure in the interdependent networks in Figure 2(d) and the cascading progress ends at stage 4.
We use   to express the relative scale damage caused by node failures: where  (  ) is the sum of initial node intensity before (after) node failures.Similar to the case of node failure, the initial load of edge   (0) is related to the node intensities of  and , as defined in (8).The "load-capacity" model of edges can be expressed in (9). Figure 3 shows the load redistribution rule for edge failure.Consider where  > 0 is a tunable parameter to control initial load of edges.Consider If a single internal edge    ℎ  in Network  fails, the load on the broken edge will be redistributed bidirectionally to its neighboring internal and coupling edges.If the load on neighboring edges plus the extra load exceeds their capacity, this may trigger a recursive process of cascading failures, such as internal edge      and internal edge      shown in Figure 3  Network  in interdependent status.(c) A comparison of Network  in both isolated status and interdependent status.Capacity parameters  ∈ [0, 2) and  ∈ [0.2, 1.6].Load parameter  = 0.8.In (c), the critical value   is labeled with green dot.The solid lines denote the isolated status, while the dashed lines denote the interdependent status.Simulation results are averaged over 1000 independent trials.We finally adopt the relative size of removed edge as  to describe the network robustness under edge attack: where   is the number of failed edges due to cascading failure after   is cut off, while  is the total number of edges in the whole system before cascading failures.

Analysis of Network Robustness under Node Attack.
A similar phenomenon of the isolated status and interdependent status is that   has positive relations with the capacity parameters  and .This means that high capacity benefits the robustness of Network  .Additionally, it is notable that the curve of   abruptly emerges when  approaches a critical value in Figure 4(a), and then it rapidly grows to 1; but, in Figure 4(b), after its abrupt emergence, the robustness indicator then shows a slow increase to 1.In order to find the detailed difference between the two statuses, we fix  at 0.2, 0.4, and 0.6 separately to compare the curves of   in isolated status and interdependent status in Figure 4(c).It can be seen that the solid lines are above the corresponding dashed ones because the interdependent relationship leads to a weaker robustness of Network  .Furthermore, the robustness indicator   in isolated status undergoes a short "plateau" state before it grows to 1.In comparison, in the interdependent status, the robustness undergoes a hybrid phase transition.The robustness of Network  has a positive relation with the capacity parameter  in Figure 5.However, it has a negative relationship with load parameter  as shown in Figures 5(a) and 5(b).We fix  at 1.2, 1.4, and 1.6 separately to compare the changes of robustness in both statuses.We also find that the robustness of Network  in isolated status is better than that in interdependent status.In addition, the trend of each curve in Figure 5(c) is similar to that in Figure 4(c).Network  in interdependent status.(c) A comparison of Network  in both isolated status and interdependent status.Capacity parameters  ∈ [0, 2) and  = 0.8.The critical value   is labeled with green dot.The solid lines denote the robustness of Network  in isolated status, while the dashed lines denote the robustness of Network  in interdependent status.Simulation results are averaged over 1000 independent trials.
Because of the hierarchical weight structure in Network  , nodes from the same level have equal initial load and capacity according to (1) and ( 3).If a node fails, the proportion of load redistribution is divided by the level to which the neighboring nodes belong.The failure probabilities of neighboring nodes from the same level are equivalent, while those of the nodes from different levels are not.That finally leads to the sharp change of   when most neighbors fail or the short plateau without large scale of failures in the neighborhood.The hierarchical weight structure in Network  also leads to a cascade of failures in the isolated status unlike a regular first-order phase transition.Meanwhile, the interdependent relationship leads to the failures from a first-order phase transition to a second-order phase transition.

Analysis of Network Robustness under Edge Attack.
When the edge with maximum initial load max(  (0)) in Network  suffers from intentional attack, the relationship among load, capacity, and robustness is analyzed based on the results in Figures 6 and 7.Under edge attack, the robustness indicator  also shows positive correlations with the capacity parameters  and  but a negative correlation with load parameter .The results shown in Figures 6 and 7 also prove that the robustness can be enhanced by high capacity as well as low initial load.As shown in Figures 6(a) and 6(b) and Figures 7(a) and 7(b), the enhancement of robustness is more prompt in isolated status than in interdependent status.In Figures 6(c) and 7(c), the solid line denoting each value of  and  is above the corresponding dashed ones which also indicates a weaker robustness of Network  in interdependent status.The hierarchical weight structure in Network  also leads to a plateau on the curves of  in Figure 7(c).Because the initial load and capacity of internal edge   which connect the neighboring levels are equal, according to (1), (2), and Load parameter  = 0.8.The solid lines denote the isolated status, while the dashed lines denote the interdependent status.Simulation results are averaged over 1000 independent trials.(8).Edges connecting the neighboring levels have equal initial load and capacity.If one internal edge fails, the failure probabilities of edges connecting the same neighboring levels are equivalent, while those of the edges connecting different levels are not.So it leads to the sharp change of  when most edges fail or the plateau without large scale of edge failures.

Comparison of Asymmetrical and Symmetrical Models.
We compare between robustness of the asymmetrical network model we propose and that of a symmetrical network model (WS-WS).The configuration of WS subnets in the symmetrical network model is the same as that of the WS subnet in the asymmetrical network model.The robustness of network under different initial conditions is shown in Figures 8 and 9.
We test intentional node attacks on the studied models under two initial conditions in Figure 8.It can be seen that the robustness of the symmetrical model is stronger than that of the asymmetric model.Additionally, no matter in the symmetrical or the asymmetrical case, the robust levels of Network  and Network  under the same conditions are very close to one another.
Additionally, we test intentional edge attacks on the two models under different initial conditions in Figure 9. Similar to the cases on node attack, the robust levels in the symmetrical model are better than those in the asymmetric model.However, what is notable is that  −  curves of each subnetwork in the asymmetrical model deviate from each other, while those of the symmetrical model remain resemblant.Network  is less vulnerable than Network  .

Conclusion
Interdependent network is a representative abstraction of realistic systems, where failures in one system may trigger a cascade of failures in the other system through interactions among systems.For instance, load redistribution caused by node or edge failures in one subnetwork may trigger an iterative progress of cascading failure in interdependent networks.In this paper, firstly, we propose an interdependent network model considering asymmetrical property of the two subnetworks.Secondly, a nonlinear "load-capacity" model is also proposed for nodes and edges, respectively, to model the cascading failures.Thirdly, we conduct a systematic series of experiments to analyze the robustness of interdependent network under attack and make comparisons with the situation of well-studied symmetrical WS-WS network model.
(1) Interdependent networks are more fragile compared with isolated ones.(2) The hierarchical weight structure leads to a short "plateau" phenomenon in isolated status unlike the regular first-order phase transition in nonhierarchical isolated networks.Under node attack, only if the capacity reaches a critical value   can the robustness abruptly emerge and rapidly approach 1.Furthermore, the interconnections lead to the robustness curve from a first-order phase transition to a second-order phase transition in interdependent networks.
(3) Under edge attack, the hierarchical weight structure also causes a plateau in an iterative process of cascading failures in isolated status.The change of robustness in isolated network is more prompt than that in interdependent network.(4) Compared with symmetrical WS-WS network model, the robustness of the asymmetrical model is weaker.Under node attack, no matter in the symmetrical or the asymmetrical case, the two subnetworks have similar robustness.However, under edge attack, subnetworks in the asymmetrical model perform differently, while those of the symmetrical model remain resemblant.

Figure 2 :
Figure 2: The cascading failure under node attack on asymmetrical interdependent network.The initial topology of the asymmetrical network, shown in (a), has a set of nodes in Network  and Network  , labeled as { 1 ,  2 , . . .,  8 } and { 1 ,  2 , . . .,  7 }, respectively.The internal edges are represented as solid black lines and the coupling edges are represented as dashed blue lines.Red circles represent the failed nodes and green circles represent the nodes that suffered from the load redistribution but are still active.

Figure 3 :
Figure 3: The load redistribution triggered by an edge failure.The internal edges are represented as solid black lines, while the coupling edges are represented as dashed blue lines.The black arrowed lines represent the directions of load redistribution from the failed edge and the red solid line represents the edge which is overloaded.

Figure 4 :
Figure 4: The relationship between capacity and robustness of Network  under node intentional attack.(a) Network  in isolated status.(b)Network  in interdependent status.(c) A comparison of Network  in both isolated status and interdependent status.Capacity parameters  ∈ [0, 2) and  ∈ [0.2, 1.6].Load parameter  = 0.8.In (c), the critical value   is labeled with green dot.The solid lines denote the isolated status, while the dashed lines denote the interdependent status.Simulation results are averaged over 1000 independent trials.

Figure 5 :
Figure 5: The relationship between load and robustness of Network  under node intentional attack.(a) Network  in isolated status.(b)Network  in interdependent status.(c) A comparison of Network  in both isolated status and interdependent status.Capacity parameters  ∈ [0, 2) and  = 0.8.The critical value   is labeled with green dot.The solid lines denote the robustness of Network  in isolated status, while the dashed lines denote the robustness of Network  in interdependent status.Simulation results are averaged over 1000 independent trials.

Figure 6 :
Figure 6: The relationship between capacity and robustness of Network  under edge intentional attack.(a) Network  in isolated status.(b) Network  in interdependent status.(c) A comparison of Network  in both isolated status and interdependent status.Capacity parameters  ∈ [0, 2) and  ∈ [0.2, 1.6].Load parameter  = 0.8.The solid lines denote the isolated status, while the dashed lines denote the interdependent status.Simulation results are averaged over 1000 independent trials.

Figure 7 : 1 A 1 AFigure 8 :Figure 9 :
Figure 7: The relationship between load and robustness under edge intentional attack in isolated status (a) and in interdependent status (b).(c) A comparison of Network  in both isolated status and interdependent status.Capacity parameters  ∈ [0, 2) and  ∈ [0.2, 1.6].Capacity parameter  = 0.8.The solid lines denote the robustness of Network  in isolated status, while the dashed lines denote the robustness of Network  in interdependent status.Simulation results are averaged over 1000 independent trials.
Figure1shows an asymmetrical interdependent network model.Network  has a hierarchical and weighted structure, where nodes are assigned with weights and distributed into different levels, where Network  is a single level network.The numbers of nodes in Network  and Network  are equal.The coupling proportion is set to  1 .Nodes in different subnetworks randomly construct "one to one" connections; for example, node   in Network  merely couples with   in Network  .The model can be described mathematically by (, , ), where  =   +  is the set of nodes from Network  and Network  .=   +   +   denotes the set of edges, where   (  ) denotes the internal edges of Network  (Network  ) and   represents the coupling edges between the two subnetworks.Matrix  = [  is the neighborhood of node , including neighboring nodes in both Network  and Network  .As mentioned before, in Network  , nodes are divided into several levels.There are  levels in Network  and each node from a certain level has  subordinate nodes from the next level.There are no connections between subordinate . After coupling edge      fails, its load proportionally is reassigned to neighboring edges connected to node   in Network  and node   in Network  .Consider Δ   →   =      ∑ ∈Γ       + ∑ ∈Γ       − 2          , Δ   →   =      ∑ ∈Γ       + ∑ ∈Γ       − 2          , where Γ   (Γ   ) is the set of neighbors of node   (  ) and      is the initial load of edge      .Δ   →   is the additional load distributed to edge      .Δ   →   is the additional load distributed to edge      .Only if   >   + Δ  →  for any edge will there be no new edge failure in the system.