The Robustness of Interdependent Networks With Traffic Loads and Dependency Groups

Existing researches on cascading failures of interdependent networks are mainly based on the percolation theory and do not consider the influence of dynamic load propagation and dependency groups. In this paper, we develop a novel interdependent system model to capture this phenomenon, also known as the hybrid cascading failure model. A degree-based targeted attack strategy on the cascading failure process of interdependent networks is studied. Combining dependency groups, interdependent relations, and traffic loads, small fraction of initial failed nodes may lead to the complete fragmentation of interdependent networks. The influence of two different dependency groups distributions on the robustness of interdependent networks under three coupling preferences is studied respectively. We provide a thorough analysis of the dynamics of cascading failures in interdependent networks initiated with a targeted attack. The system robustness is quantified as the surviving fraction of nodes in the giant connected component at the end of cascading failures. Our results highlight the need to consider loads, group effects and coupling preferences when designing the robust interdependent networks. And it is necessary to take steps in the early stage to reduce the losses caused by the large-scale cascading failures of infrastructure networks.


I. INTRODUCTION
In the past decades, with the continuous deepening of complex network theory and application research, many scholars have devoted themselves to studying complex phenomena in many infrastructures, such as cyber-physical networks, Internet networks, social networks, transportation systems and metabolic systems [1]- [4]. However, there is no purely isolated network, one network is more or less physically or logically related to other networks by some preferences, such as communications and power network, water and food supply network, transportation and logistics network, etc.. One of the most important researches in the complex network fields is that of the cascading failure phenomenon, which represents that even small portion of nodes in the network damaged would trigger an avalanche of the network. Due to the role of coupled connections, disturbances can not only The associate editor coordinating the review of this manuscript and approving it for publication was Fei Chen.
propagate along the connecting edges inside the network, but can also pass from one network to another along the coupling edges. Studies have shown that interdependent networks are quite vulnerable to malicious attacks, and small proportion of failed nodes are likely to cause catastrophic consequences to the networks.
Buldyrev et al. [5] firstly using percolation theory, established a theoretical model for studying the process of cascading failures in interdependent networks caused by the random initial failures of nodes, which acquire that interdependent networks are more fragile than a single network, and demonstrate a first order phase transition. Parshani et al. [6] considered two partial interdependent networks and found that reducing the coupling strength between networks lead to a change from a first order percolation phase transition to a second order percolation transition at a critical point. In the real world, however, the destruction of a system is often caused by deliberate attacks which means attacking important nodes in networks, Huang et al. [7], Dong et al. [8] developed a general technique that maps a targeted-attack problem in interdependent networks to a random-attack problem, discusses the robustness of completely and partially interdependent networks under target attack, respectively. In [9], the oneto-one random coupling model is extended to the multiple coupling model, that is, one node can be coupled with other nodes. Considering that actual networks are often distributed in a fixed geographical space, Bashan et al. [10] carried out detailed calculations and theoretical derivations for the cascading failure of spatially embedded coupled networks, and found that such networks are very fragile and extremely prone to first-order phase changes. Later, Gao et al. [11], [12] discussed the robustness of network of network under different structures and attack conditions based on the theoretical framework.
Recently, many systems are characterized by small dependency groups in which the entities belonging to a group strongly depend on each other. Under certain circumstances some nodes within one network tend to fail or survive together [13]. For example, transactions and sales relationships between companies form a financial network. Some companies are usually controlled by the same owner and there are dependencies links between these companies. If one companies goes out of business and the owner have not enough capacity to support other companies, other companies will go out of business too [14]. Another example is in the mobile Internet, an individual mobile device sometimes shares its internet connections with others if they have lost the connections. In this scenario, different devices also tend to keep the internet connections or not (fail) together, as a dependency group [13]. Relationships between nodes in the dependency group is called group-related dependency links. If one node in the dependency group fails, other nodes also fail at the same time, that is, nodes belonging to the same dependency group will fail together. It was firstly investigated by [14] to study cascading failures on a single network using the percolation theory to consider group-related dependency links and topological connection links between nodes. Reference [14] shows that group-related dependency links make the network more fragile than the one without dependency groups. In addition, the network with dependency groups undergoes a first-order phase transition process, which is different from a network consisting only of topological connectivity links. And [15], [16] considered a general cascading failures model that dependency groups have different sizes. Later, a cascading failures model was proposed that a dependency group fails only when more than a fraction of nodes of the group fail [17] and a percolation model on the network with assortative dependency groups is established. Furthermore, the effects of dependency groups on cascading failures was proposed to explore the robustness of interdependent networks. Bai et al. [18] studied the time-dependent of interdependent nodes and the recovery mechanism in reality to explore the network robustness. Wang et al. [13] investigated the effects of dependency groups on the cascading failures in the interdependent networks, they assumed that there are dependency groups in each network, which can improve the resilience of interdependent networks significantly. The above cascading failures studies on the single network and interdependent networks have considered the nodes exist direct and explicit coupling relationships, that is, the failures of nodes cause the subnetwork to fall off from the entire network, forming physical disconnections of the network, which induces percolation effects in the network. Only the giant connected component does not fail, and its relative size is used to measure the robustness of the network. The cascading failures caused by these coupling relationships are called structural coupling cascading failures.
However, the structural coupling model only considers the topological factors, and does not consider the distribution of the network flow. Many real-life networks such as communication networks and transportation networks could be modeled as the load-dependent networks [19]- [25]. For example, each device in a wireless ad hoc network handles a load of data traffic; airports and sectors in an air transportation network accommodate loads of air traffic. Compared with the previously mentioned researches about the cascading failures in the interdependent networks based on the percolation theory, considering the cascading propagations induced by overload failures on an node or edge in the interdependent networks is more realistic, which has inspired the attentions of many scholars [26]- [30]. Many cases have shown that the cascading failures caused by nodes overload has become one of the main threats to the safe operation of interdependent networks, and the global system collapse is more likely to occur under the coupling effect. The network model based on the load-capacity propagation generally assumes that all nodes in the network have a certain initial load L and capacity C. When the network is attacked, the load of failure node i will be transferred to other nodes of the network in a certain way. If the total loads of a node j is greater than its capacity limit (L j +L ij > C j , L ij represents loads obtained by the node j from failure nodes), then the node fails. In the process of seeking new dynamic balances, the network may further experience overloaded cascading failures. The cascading failure caused by this coupling relationship is called the functional coupling cascading failure. In the actual network, two types of faults based on the structural and functional coupling cascading failures interact and strengthen to form a new fault chain effect, which bring great challenges to the fault analysis.
Nevertheless, previous studies on the robustness of interdependent networks just consider the interdependent links between nodes in different networks and the topological connectivity links in one network, the effects of dynamic loads on cascading failures in interdependent networks with dependency groups still puzzle us. Therefore, it is still an important open question how these node grouping phenomena in real world systems impact the robustness of entire systems. Inspired by these researches on the robustness of interdependent networks, in this work, we for the first time develop a general model of ''group effect'' in the interdependent networks with traffic loads, which provides a new concept in complex network studies. To better understand the catastrophic impacts on networks caused by load-failures, dependency groups and interdependent links, it is natural and important to construct a hybrid cascading load model to explore the propagation of cascading failures in the interdependent networks. The general case of an interdependent networks with dependency groups is depicted in Fig.1. Here the networks A and B are fully interdependent with the same number of nodes. Connectivity links between nodes in one network are shown as black straight lines both in network A and network B. Interdependent links between nodes in different layers are depicted as dashed lines. Nodes in the same dependency group are surrounded by the blue dashed circle. Our work goes beyond the existing literature in the following three aspects. Firstly, in spite of the tremendous recent interest in robustness of complex networks and in cascading failures of interdependent networks, the idea combined traffic loads and dependency groups has not appeared in any previous work. Secondly, our idea of hybrid cascading failures exploits the effects of dependency groups and dynamic load redistributions on the robustness of interdependent networks, which has not been studied in the literature. Thirdly, our hybrid cascading failures strategy is physically meaningful and realizable.
The overall framework of this paper is organized as follows. In section II, a hybrid cascading failure model is proposed to explore the robustness of interdependent networks experienced cascading failures. Then we built the interdependent networks to analyze the proposed model by simulation results in section III. Finally, in section IV relevant conclusions and future researches are summarized.

II. INTERDEPENDENT HYBRID CASCADING FAILURE MODEL A. INTERDEPENDENT NETWORKS MODEL
Here, without loss of generality, the interdependent networks is composed of two networks, labeled A and B. The links in network A or network B are called intra-links, the number of intra-links that a node possess is called intra-degree. Two networks are considered with the same size (i.e., the number of nodes N = N A = N B ) and the same average degree k A = k B = k . Correspondingly, the links between network A and network B are called interdependent links or coupling links. We assume that for each i = 1, . . . , N , nodes a i and b i are interdependent with each other meaning that if one fails, the other will fail at the same time. Although oversimplified, the one-to-one interdependence model is considered to be a good starting point and has already provided great insights [5]. Assuming that nodes in network A are with dependency groups (i.e., all nodes in the same group are functional dependent on each other) and network B has no dependency groups which means that it only exists topological connectivity links. A similar argument can be made for the case that there are dependency groups in every network because the mechanism that causes cascading failures of interdependent networks is similar. The specific steps to build a two-layered interdependent networks with groups are as follows. (a) two networks are coupled by interdependent links. Interdependent links only reflect interdependent relationships like in [5]. When a node stops functioning owing to attack or failure, its interdependent counterpart also stops functioning. It is considered that each node has at most one interdependent link [5]. (b) inducing dependency groups in layer A and randomly dividing all nodes in network A into non-overlapping (without shared nodes) groups according to a given group size distribution P(s). We only focus on the influence of dependencies within the dependency group on the cascade. This is a reasonable assumption that considers in previous research [13], [14].

1) CAPACITIES AND LOADS OF NODES
In this paper, we use the data-packet transport model where the traffic load of node i is relying on the information of node's betweenness centrality. In layer X ∈ {A, B}, the initial load L Xi (t) of the node i in one network at time t is defined as the betweenness of the node [19], which is calculated by where θ i st is the the number of shortest paths between nodes s and t that run through node i, θ st denotes the total number of available shortest paths from s to t, N X represents the number of nodes in the network A or network B. The betweenness of one node is defined as the proportion of the number of paths passing through the node in all shortest paths in the network to the total number of shortest paths. Because in most real-world systems, the information flow is transmitted through the shortest path between nodes, betweenness centrality can better describe the function of nodes when the information flow propagates from one component to another. For example, data packets travel from the source node to the destination node along the shortest path in Internet.
The capacity of the node signifies the ability of each node to process its load which usually is influenced by both the technology used and the limited costs. We adopt the nonlinear load-capacity [31] as follows where C Xi stands for the capacity of node i in each network; and L Xi (0) is the load of node i at initial time t = 0, α and β are the tunable coefficients and α, β > 0. The effects of different values of parameters α and β are discussed in paper [31]. The formula (2) is used to describe the nonlinear characteristics of nodes' capacities and loads. A node fails when its load exceeds its capacity.

2) FAILED NODES
In cascading failure studies, an attack on a fraction of nodes or links may cause catastrophic damage to the network. Common attack strategies include the random attack strategy and the targeted attack strategy. In the paper, we consider the mechanism of cascading failure for interdependent networks that are triggered by an initial attack. We studied a degree-based targeted attack strategy on the cascading failure process of interdependent networks. The intra-degree of nodes in one subnetwork is arranged in descending order and the top-ranked N del nodes are selected as initial failure nodes. The value of N del indicates the intensity of one initial attack. When the number of initially failed nodes is equal to or larger than a certain value, the initiated cascading failure will disintegrate the interdependent networks very quickly. Furthermore, we will explain that the removal of some nodes will change the topology of the networks so that the traffic loads of nodes may change, which would trigger a new round of loads redistribution process. After a fraction, 1 − p, of nodes are removed, A p (k) represents the number of nodes with degree k. P p (k) is the new degree distribution of the remaining fraction p of nodes in the network, When the new node is removed, A p (k) changes as where k(p) β ≡ P p (k)k t . Based on the above analyses, removal of some nodes will change the nodes's degree distribution that cause the traffic loads of nodes to change.

B. COUPLING MODES
Previous studies [30], [32], [33] have shown that the interdependent networks with various coupling preferences exhibit different robustness under targeted or random attacks and the robustness of interdependent networks under targeted attacks changes more obviously. Therefore the effects of coupling preferences on the interdependent networks draw more attentions. To investigate the impact of coupling preferences, we propose three cases as follows.
Assortative link pattern: The betweenness of nodes in network A and network B is arranged in descending order respectively. The interlinking of nodes by their intra-layer betweenness in the monotonic order, which means that the ith highest nodes in network A linked to the ith highest nodes in network B. (If some nodes happen to have the same betweenness, we randomly choose one of them).
Disassortative link pattern: This is called by antimonotonic matching, which indicates that the ith highest betweenness nodes in network A linked to the ith lowest betweenness nodes in network B.
Random link pattern: Randomly select two nodes in network A and B are connected. Repeat this process until N interconnected links between networks A and B are added.

C. DISTRIBUTION OF DEPENDENCY GROUPS
In this paper, two types of dependency group distributions, Poisson distribution and normal Gaussian distribution [15], [34], are discussed respectively.

1) POISSON DISTRIBUTION OF DEPENDENCY GROUPS
The case considered is that the size of dependency groups follows a shifted or scale adjusted Poisson distribution with s 1. That is, the probability P(s) that one node belongs to a dependency group of size s is given by where λ = D size − 1, D size represents the mean size of dependency groups. D size is regarded as a major parameter to measure the effect on the robustness of interdependent networks against cascading failures.

2) GAUSSIAN DISTRIBUTION OF DEPENDENCY GROUPS
The other case considered is that the size of dependency group follows the normal Gaussian distribution. The probability of a random node which belong to a dependency group of size s is given by where A is a normalization constant. Note that the purpose of P(s) > 0 only for the range that 1 < s < 2D size − 1 is to have a symmetrical distribution around D size . In the following simulation section, we will further explore the effects of different group distributions on the robustness of interdependent networks combining with dynamic loads redistribution and interdependent links.

D. HYBRID CASCADING FAILURE PROCESS OF THE INTERDEPENDENT NETWORKS
The following statement will discuss the failure propagation process of the interdependent networks with dependency groups. A hybrid cascading failure model is formed by the interaction of three types of relationships. Cascading failure in the interdependent network can be divided into two processes within subnetworks and one process between subnetworks. First, the dynamic load redistribution within subnetworks is one of the reasons for cascading failure. The failure nodes in network A(B) and their intra-links are 98452 VOLUME 8, 2020 removed from the network and some shortest paths between pairs of nodes in the network are changed. Loads are then redistributed along with the updated shortest paths may arouse extra load to some nodes, leading to the failure of nodes in A(B) due to overload. The network may further trigger the overloading stage in the process of seeking a new dynamic balance. Once the load exceeds the node capacity, these nodes will be considered overloaded and crashed. They are removed from the network, and the shortest path between nodes in the network is changed again. The new round of load redistribution is iteratively triggered in this way. This process is called the structure-induced cascading failure. Second, some nodes within one network belong to the same dependency group, so one node fail will result in the failure of all nodes in the same dependency group. This process is called the dependency group cascading failure. Third, during the process between subnetworks, interdependent nodes interact with each other. The failed nodes in A(B) will cause the failure of their interdependent counterparts in B(A), interdependent nodes and their connected interdependent links are removed from the network. This process is called the function-induced cascading failure. The isolated nodes that are not in the largest connected component failed and the corresponding interdependent nodes failed as well. The cascading failure processes end until there is no further failure from one network to the other. Based on the hybrid cascading failure model, after a fraction of nodes failed, the failures caused by dynamic load redistribution, dependency groups and interdependent links happen recursively until no more damage takes place. Fig. 2 illustrates the hybrid cascading process.
To trigger the cascading failure simulation on the interdependent networks with dependency groups, we select the top N del nodes with maximum intra-degree in one subnetwork as initial failure nodes. The main steps of the simulation, including three iterative failure processes, are depicted in Table 1.
In order to ensure network connectivity, it is assumed that only the nodes in the giant component remain functional, the relative size of the giant connected component G is denoted as an evaluation index to describe the robustness of interdependent networks, which is defined as follows where N f is the final number of nodes in the giant component which include nodes from interdependent networks composed of networks A and B. At the end of cascading VOLUME 8, 2020 process, the final G is calculated to measure the robustness of interdependent networks.

III. NUMERICAL SIMULATIONS AND RESULTS
In this section, we apply our model to interdependent networks, consisting of ER-ER networks and SF-SF networks, to verify the effect of the dynamic loads, dependency groups and interdependent links on the robustness of interdependent networks. The node quantities of two sub-networks of interdependent networks satisfied N A = N B = 1000. Each point in figures of simulations corresponds to the average value obtained by triggering the same initial failure on 20 randomly generated interdependent networks with the same size. The fluctuations of data in some curves are due to the randomness of simulation. The effects of different dependency group distributions and tunable parameters α, β in formula (2) are analyzed respectively.

A. EFFECTS OF POISSON DISTRIBUTION
We first investigate the case that the distribution of dependency group size in one subnetwork following Poisson distribution. Previous study [34] illustrates that the changes of D size and k have significantly influences on the network robustness. Here D size means the average size of dependency groups in the network. Therefore, in the following discussion, we discuss the influence of these changing parameters on the robustness of interdependent networks. We set the capacity redundancy parameters α = 1.05 and β = 0.5.

1) DIFFERENT MEAN SIZES OF D size
Here we compare the robustness of ER-ER interdependent networks that have different average size D size and coupling modes, as depicted in Fig. 3. D size = 1 means that there are no dependency groups in the interdependent networks which is regarded as a comparison with other cases that have dependency groups. First, we analyze Fig. 3 (a) to show that the increase of N del leads to the decrease of G. The interdependent networks experiences a first-order phase transition in different coupling modes, which indicates that the network can easily crash when attacked. Compared with ER-ER interdependent networks without dependency groups, the downward trend of G is more obvious in the interdependent networks with dependency groups. And this downward trend is particularly significant as D size increases which provides a hint that the mean size of D size has an impact on the robustness of interdependent networks. When the number of initial failed nodes exceeds a certain value, the interdependent networks may be fatally damaged. From the three subgraphs of Fig. 3, it can be seen that the ER-ER interdependent networks exhibit similar robustness in the three coupling modes, which means that the effect of coupling mode on the robustness of ER-ER interdependent networks is weaker than that of the hybrid cascading failures. Because dependency groups represent tight relationships between nodes in one network that can cause disruptive damages to interdependent networks. The results about the robustness of SF-SF interdependent networks with dependency groups and non-dependency groups in the assortative coupling mode is depicted in Fig. 4, which includes two cases of k = 3 and k = 4. As the increase of D size , the downward trend of G with the increase of N del is more obvious which also suggests that D size has an impact on the robustness of SF-SF interdependent networks.

2) EFFECTS OF DIFFERENT AVERAGE DEGREES k
Paper [11] revealed that the network average degree k of the subnetwork can bring influences on the robustness of interdependent networks. Fig. 5 shows the robustness results of ER-ER interdependent networks with dependency groups satisfying Poisson distributions. D size affects the critical threshold of Nc max with different k , k = 6, 8, and 10, respectively. The critical threshold of Nc max means that when the number of initial failed nodes exceeds it for the first time, the value of G will be less than 0.2. When the value of k is fixed, Nc max decreases as the increase of D size in all coupling links. Particularly, the change trend between Nc max and D size in assortative coupling mode is more obvious than the trends of other two coupling modes. When D size takes a certain value, Nc max increases as the increase of k .   But the overall change trend of Nc max is not very significant, which indicates that the average degree k of the subnetwork has little impacts on the robustness of interdependent networks with dependency groups. Fig. 6 shows the results of dependency group satisfying Poisson distribution in SF-SF interdependent networks where D size = 1, 2, 3. When k = 2, the small number of failed nodes can bring catastrophic damage to the networks. But the change of G shows almost the same slow downward trend in both cases k = 3 and 4, respectively, when increasing N del . These results indicate that the SF-SF interdependent networks become more stable as k increases.

B. GAUSSIAN DISTRIBUTION OF DEPENDENCY GROUPS
Similar to the method discussed above, we further consider the change trend of robustness of interdependent networks when the distribution of group size follows Gaussian distribution. As Eq. (6) shows the possibility that a node belongs to one dependency group depends on two different parameters, different sizes of D size and variance of group size σ 2 . The following simulations show the variation trend of the robustness of interdependent networks when these two parameters take different values. Fig. 7 illustrates that the impact of G on the robustness of ER-ER interdependent networks under three different coupling preferences when the variance σ 2 is fixed. Similar to what we discussed earlier when the dependency group follows Poisson distributions, the interdependent networks experiences a first-order phase transition when the hybrid cascading failure process is triggered. It shows that the  dependency group is a crucial factor that cause the cascading failures of interdependent networks. When the value of D size becomes larger, the interdependent networks is more unstable, which means that only a small fraction of initial failed nodes may trigger the whole system collapsed. And the downward trend of G shows almost the same under three coupling modes when D size is fixed. The effects of coupling modes on the robustness of interdependent networks is weaker than that caused by dependency groups For the SF-SF interdependent networks, we consider the impact of D size on the robustness of networks in assortative coupling mode. Fig. 8 illustrates the change trend of G when D size = 1, 2, 3, 4. D size = 1 means that there are no dependency groups in the SF-SF interdependent networks which is regarded as a comparison with other cases that have dependency groups. When D size becomes larger, the downward trend of G is more urgent, which means the interdependent networks is more unstable.

2) EFFECTS OF VARIANCE σ 2 OF GROUP SIZE
Figs. 9 (a) and 9 (b) show the effects of different σ 2 on the robustness of ER-ER and SF-SF interdependent networks with fixed D size and k , respectively. σ is equal to 0.8, 1.6, 2.4 and 3.2. NOTE: In the figures, var = σ . Fig. 9 (a) displays the relation between G and N del on the robustness of ER-ER interdependent networks under the assortative coupling mode, where D size = 4 and k = 10. As σ changes from 0.8 to 1.6 or 2.4 to 3.2, the trend of G almost has no change. And when the values of σ grows from 0.8(1.6) to 2.4(3.2), the downward trend of G becomes slow. In cases of σ = 0.8 or 1.6, the value of G is less than 0.2 when the number of initial failed nodes N del exceeds 5. And in cases of σ = 2.4 or 3.2, when the number of initial failed nodes N del exceeds 10, the value of G is less than 0.2 for the first time. Although the value of N del increases from 5 to 10 in these cases, it shows that a small number of initial failed nodes can cause deadly damage to the interdependent networks. It indicates that the increase of σ changes the robustness of interdependent networks, but the overall change is not large. The ER-ER interdependent networks exhibits fragile characteristics when deliberately attacked.  Meanwhile, the effect of σ on the robustness of SF-SF interdependent networks with D size = 3 and k = 3 is considered in Fig. 9 (b). The change trend of G shows similar downward trends when the value of σ changes from 0.8 to 3.2. It indicates that the growth of σ brings weak impacts on the robustness of SF-SF interdependent networks. From the above analysis, we can conclude that σ has different effects on different interdependent networks.

3) EFFECTS OF DIFFERENT AVERAGE DEGREES k
The effect of different k on the robustness of ER-ER interdependent networks under three coupling modes is presented in Fig. 10. From the plotted graphs, the trend of Nc max fluctuates slightly with the change of k , which indicates that k has little impact on the robustness of ER-ER interdependent networks. As for the effect of different k on the robustness of SF-SF interdependent networks in assortative coupling mode, we show the results in Fig. 11. Increasing k from 2 to 3 significantly changes the trend of G. And this trend has almost no change as k increases from 3 to 4. The above results show that k has an impact on the robustness of SF-SF interdependent networks, but this effect becomes weaker as k increases.

C. THE PARAMETERS α AND β IN THE CAPACITY
This section will discuss the influences of α and β on the robustness of interdependent networks, taking the ER-ER  Fig. 12 (a), β is equal to 0.5. The downward trend of G slows down as α changes from 1 to 1.10, which indicates that the increase of α enhances the robustness of interdependent networks. This result is because the increase of α leads to the increase of the node's capacity C i , which can be obtained from formula (2). The increase of node's capacity will improve the ability of node to resist overload failures, and thus further improve the robustness of interdependent networks against the hybrid cascading failures. The trends of the curves in Fig. 12 (a) coincide with the theoretical derivation. And from Fig. 12 (b), the change of β FIGURE 12. Effects of α and β on the robustness of ER-ER interdependent networks when the dependency group follows Poisson distribution. VOLUME 8, 2020 can actually affect the robustness of interdependent networks. But the increase trend of G and β are not consistent because there is not a linear relationship between the capacity C i and β based on the definition of the node's capacity(formula (2)). As discussed above, it can be obtained that analyzing the effects of dependency groups and relevant tunable parameters on the robustness of interdependent networks can provide insightful ideas for the study of complex systems.

IV. CONCLUSIONS
Combining with load redistributions, dependency groups and interdependent relations, in this paper, a hybrid cascading failures model is proposed to explore the robustness of the interdependent networks. In this model, two types of interdependent networks, ER-ER and SF-SF interdependent networks, are built. When the size of the dependency group obeys Poisson distribution and Gaussian distribution, respectively, the influence of related parameters on the robustness of interdependent networks is discussed separately. The results illustrate that the mean size of dependency groups is a major factor which effects the robustness of interdependent networks. The SF-SF interdependent networks with dependency groups shows better robust than the ER-ER interdependent networks with dependency groups when attacked. Further researches can explore the corresponding protection measures to improve the robustness of interdependent networks. Her research interests include intelligent information processing and computing, modeling and analysis of network information systems, and data processing. VOLUME 8, 2020