Modeling and analysis of cascading node-link failures in multi-sink wireless sensor networks

Modeling and Analysis of Cascading Node-Link Failures in Multi-Sink Wireless Sensor Networks

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Modeling and Analysis of Cascading Node-Link Failures in Multi-Sink Wireless Sensor Networks Xiuwen Fu, Yongsheng Yang PII: DOI: Reference:

S0951-8320(19)30842-7 https://doi.org/10.1016/j.ress.2020.106815 RESS 106815

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Reliability Engineering and System Safety

Received date: Revised date: Accepted date:

28 June 2019 14 December 2019 19 January 2020

Please cite this article as: Xiuwen Fu, Yongsheng Yang , Modeling and Analysis of Cascading NodeLink Failures in Multi-Sink Wireless Sensor Networks, Reliability Engineering and System Safety (2020), doi: https://doi.org/10.1016/j.ress.2020.106815

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Highlights • We present a cascading node-link model for multi-sink WSNs. • We propose two sink-oriented load metrics to measure the load distribution on sensor nodes and wireless links respectively. • We evaluate the network robustness under link attacks and node attacks. • We explore the impacts of critical thresholds on the cascading robustness of the network.

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Modeling and Analysis of Cascading Node-Link Failures in Multi-Sink Wireless Sensor Networks Xiuwen Fu, Yongsheng Yang Institute of Logistics Science and Engineering, Shanghai Maritime University, Shanghai 201306, China [email protected]

Abstract Due to the prominent advantages in network lifetime and energy balance, the application of multi-sink wireless sensor networks (WSNs) is becoming more and more widespread. However, their cascading robustness is still rarely studied. Therefore, in this paper, a realistic cascading model for multi-sink WSNs is proposed. In this model, two load metrics are proposed to characterize the load distributions of sensor nodes and wireless links, and the cascading process of the network is jointly promoted by node overload events and link overload events, which can better reflect the cascading characteristics of multi-sink WSNs in practical scenarios. In addition, we focus on the cascading robustness of the network in the face of node attacks and link attacks. Through extensive experiments, we found that there are critical thresholds for both node capacity and link capacity, which can determine whether capacity expansion is helpful; there is a critical threshold for network load distribution, which can determine whether cascading failures occur; node attacks are more likely to trigger cascading failures than link attacks; increasing node capacity can more effectively reduce the damage of cascading failures to the network. The discovery of the above results can provide theoretical guidance for users to build a more robust multi-sink WSN against cascading failures. Keywords: cascading failures; multi-sink wireless sensor networks; node capacity; link capacity; load distribution; critical threshold; 1. Introduction Wireless sensor networks (WSNs) are an important part of the Internet of things (IoTs) system, which are widely used in industry, agriculture, military and other fields. WSNs are composed of general sensor nodes and sink nodes[1, 2, 3, 4]. General sensor nodes collect environmental data and forward the data to sink nodes, which then upload the data to the cloud. According to the number of deployed sink nodes, WSNs can be divided into single-sink networks and multi-sink networks. Compared with single-sink networks, multi-sink networks are more widely used due to their outstanding advantages in network lifetime and energy balance[5, 6, 7, 8]. Fig.1 shows examples of single-sink WSNs and multi-sink WSNs. Due to the low-cost hardware and constrained energy, WSNs usually adopt “narrowband communication mode[9, 4, 10]”. In this communication mode, the capacity of nodes and links is significantly limited. When the load of a few nodes or links in the network exceeds their capacity, cascading failures may be triggered and the global network may be paralyzed[11, 8, 12]. Because of the serious threat to the entire network, cascading failures in WSNs have been researched extensively. Different cascading models have been proposed to simulate the cascading process of WSNs. However, given the actual load distribution and hardware configurations of WSNs, existing models have several evident limitations: 1) in WSNs, all the messages collected by general sensor nodes will be aggregated at sink nodes and then uploaded to the cloud, Preprint submitted to Elsevier

February 3, 2020

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Figure 1: Examples of single-sink WSNs and multi-sink WSNs.

this sink-oriented delivery paradigm distinguishes WSNs from general peer-to-peer networks. Existing cascading models for WSNs are developed based on general degree-load metrics or betweenness-load metrics, which cannot properly reflect the impacts of sink nodes on network load distribution; 2) existing cascading models for WSNs are typical cascading node models, in which the cascading process can only be triggered by the failures of sensor nodes. However, in actual WSNs, wireless links are highly likely to fail due to external interference and are also subject to capacity constraints. The failures of links or nodes will result in an update of message routing paths, further renewing network load distribution. In this process, some nodes or links may be overloaded and a new round of load redistribution may occur. In order to explore the actual cascading process of WSNs, the impacts of link failures should not be ignored; 3) due to the outstanding advantages in terms of network lifetime and energy conservation, multi-sink WSNs have become the dominant network architecture in practical applications. Compared with the traditional single-sink WSNs, the load distribution characteristics of multi-sink WSNs are more complicated, which makes their cascading process significantly different from that of general networks. Therefore, in this work, we develop a realistic cascading model for multi-sink WSNs. The major contributions of this work are summarized as follows: 1) To properly characterize the cascading process of multi-sink WSNs, a cascading node-link model is proposed, in which cascading failures are caused by overload of nodes and links. 2) Two robustness metrics are proposed to measure the cascading robustness of multi-sink WSNs in the face of node-failure attacks and link-failure attacks respectively. 3) Through extensive experiments, the soundness and effectiveness of the proposed cascading model have been verified. The impacts of modeling parameters on network robustness have also been explored. The reminders of the paper is organized as follows. A brief survey of related work is presented in Section 2. The details of the proposed cascading model for multi-sink WSNs are presented in Section 3. Experimental results are given in Section 4. Finally, conclusions and the future work are shown in Section 5. 2. Overview and Discussion of Literature In this section, we first present an overview of literature regarding cascading failures in WSNs. Then, we mainly discuss the limitations of existing models by presenting some examples. 2.1. Literature overview Liu et al.[13] presented a betweenness-based cascading model for WSNs. In this model, the load of sensor nodes is represented by their betweenness values and the overload function 3

is defined as a monotone non-decreasing function with the load of sensor nodes as the only input. The cascading robustness of three topological structures (scale-free topology, plane k connected topology and k -connected dominating topology) was studied. Yin et al.[14] proposed a cascading model based on exponential degree for scale-free WSNs. In this model, the load of sensor nodes is exponentially determined by their own degrees. They mainly explored the relationship between the load and the largest connected component after cascading failures on scale-free WSNs. Based on the exponential-degree cascading model, Ye et al.[15] used the probability generating function method to analyze the cascading robustness of scale-free WSNs under a single random node failure. They found that the scale of cascading failures is positively related to the power law exponent of scale-free networks; Hu et al.[16] researched the cascading robustness of three typical topologies of WSNs (i.e., small-world topology, scale-free topology and random topology) through the general betweenness-based cascading model. In our previous work[17], we developed a degree-based cascading model for hierarchical WSNs. In this model, the connections between nodes are divided into inter-cluster connections and intra-cluster connections. We assumed that the relay load of a sensor node is only related to the number of inter-cluster connections it has, and the sensing load of a sensor node is only related to the number of intra-cluster connections it has. In [18], we researched the cascading robustness of WSNs based on the coupled map lattice (CML) method under three attack schemes (i.e., random attack, max-degree attack, and max-status attack). Although the above researches have made some progress, the cascading models they established fail to consider the decisive impacts of sink nodes on network load distribution, making them unable to properly characterize the load and capacity in multi-sink WSNs. Moreover, existing models mainly consider the impacts of node overload events on the cascading process of WSNs, but ignore the impacts of link overload events. In practical scenarios, wireless links often fail due to interference. As with node failures, link failures can also renew the network load distribution and trigger the cascading process of WSNs. 2.2. Problem statement In this part, we discuss the limitations of existing cascading models in terms of multi-sink orientation and cascading node-link process. 2.2.1. Multi-sink orientation 200 180 160 140

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Fig.2 demonstrates a general network topology of multi-sink WSNs referencing [18]. In this topology, 300 sensor nodes and 3 sink nodes are randomly deployed. Fig.3 shows the 4

network load distributions generated by two most-widely used multi-sink routing protocols (i.e., MUSTER[19] and SGF[20]) respectively. We can clearly observe that the closer the sensor node is to the sink node, the greater the load it will carry. This phenomenon has been confirmed by many researchers and is called “sink hole” effect[21, 22, 23]. The reason for the ”sink hole” effect is that all messages will be gathered at sink nodes, so sensor nodes near sink nodes need to take more load than other nodes in outer regions. 200

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The network load distributions created by degree-based and betweenness-based cascading models are shown in Fig.4. In the network load distribution obtained by degree-based cascading model, the load distribution is only positively related to the density of wireless links. In the network load distribution obtained by betweenness-based cascading model, sensor nodes that are closer to the center of the network can get a higher load. We can easily observe that in these two models the load of the sensor nodes near sink nodes is not higher than that of sensor nodes in other areas. Apparently, compared with the load distributions shown in Fig.3, the load distributions obtained by degree-based and betweenness-based cascading models fail to reflect the multi-sink orientation characteristics in actual networks. 200

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Figure 4: Network load distributions generated by degree-based model and betweenness-based model.

2.2.2. cascading node-link process In this part, we still use the network topology shown in Fig.2 and observe the changes of load on each node and each link before and after deleting first 10 wireless links in the descending order of the load. According to Fig.5, we can clearly observe that the load distributions on nodes and links under MUSTER have changed significantly after deleting highest-load links. If we take the highest-load values of nodes and links in the network before removal as the capacity, we can find that there will be 9 nodes and 25 links exceeding the capacity after deletion. Similarly, 5

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Figure 6: Normalized load on each node and each link under SGF.

according to Fig.6, we can find that under SGF, 12 nodes and 29 links will be overloaded after the removal. These results clearly tell us that: 1) in addition to node failures, link failures can also renew the network load distribution, which may trigger cascading failures; 2) similar to node overload, wireless links are also at risk of overload during the cascading process; 3) the actual cascading process is driven by overload failures of nodes and links. Apparently, since most of the existing models for WSNs are developed based on the cascading node models, they cannot properly reflect the cascading node-link process in multi-sink WSNs. 3. Cascading Node-Link Model Without loss of generality, a multi-sink WSN can be depicted by an undirected graph G=(V, E), where V ={1, 2, ..., N } is the set consisting by general sensor nodes and sink nodes. E={ei,j |i, j ∈ V } is the set of wireless links between sensor nodes. The N ×N adjacency matrix [aij ] has aij =1 if there is a wireless link between node i and node j. We use the node sets Vg and Vs to represent the set of general sensor nodes and the set of sink nodes respectively. We can easily get that V =Vg ∪Vs . Ng and Ns represent the number of general sensor nodes and the number of sink nodes in the network. Ne is the number of links in the network. 3.1. Multi-oriented betweenness To properly reflect the load distribution of multi-sink WSNs, we propose two load metrics: multi-oriented node betweenness (MNB) and multi-oriented link betweenness (MLB), which are to measure the load on sensor nodes and wireless links respectively. MNB is defined as follow: Dj (t) =

X X wi,m,j (t)/wi,m (t) , Ng NS(i) i∈V g

(1)

m∈S(i)

S(i) = {k|d(i, k) = min [d(i, j), j ∈ Vs ]} , 6

(2)

where S(i) is the set of sink nodes that are closest to node i and d(i, j) is the minimum number of hops between node i and node j. wi,m,j (t) is the number of the shortest paths from node i to its nearest sink node m which pass through node j at time t. wi,m (t) is the number of the shortest paths from node i to its nearest sink node m. Ng is the number of general sensor nodes in the network. NS(i) is the size of the sink set S(i), which can also be considered as the number of the nearest sink nodes of node i. In the case that each of the shortest path from any sensor nodes to their nearest sink nodes passes through node j, Dj (t) takes the maximum value of 1. In the case that no sensor nodes need to go through node j to reach their nearest sink nodes, Dj (t) takes the minimum value of 0. MLB is defined as follow:  X X wi,m,ej,k (t) wi,m (t) , (3) Dej,k (t) = Ng NS(i) i∈V g

m∈S(i)

where wi,m,ej,k (t) is the number of the shortest paths from node i to its nearest sink node m which pass through link ej,k at time t. wi,S(i) (t) is the number of the shortest paths from node i to its nearest sink nodes m. In the case that each of the shortest path from any sensor nodes to their nearest sink nodes pass through link ej,k , Dej,k (t) takes the maximum value of 1. In the case that no sensor nodes need to go through link ej,k to reach their nearest sink nodes, Dej,k (t) takes the minimum value of 0. In order to verify the soundness and effectiveness of the proposed metrics MNB and MLB, we propose two hierarchical topologies (i.e., single-sink topology and multi-sink topology). Fig.7 shows an example of a single-sink hierarchical topology. In this topology, node 1 is the sink node and all other nodes are required to transmit data to the sink node 1. By observing the network traffic distribution, we can easily obtain that all the traffic in the network needs to reach the sink node 1 through node 2; nodes 3 and 4 need to relay the traffic from three nodes (including their own traffic) respectively; nodes 5∼8 only need to relay the traffic generated by themselves. Table 1 gives the MNB of each node in this topology. By comparing the values of MNB with the actual traffic distribution, we can easily conclude that the proposed load metric MNB perfectly characterizes the load distribution in single-sink WSNs. Next, we observe the traffic distribution on each link in Fig.7. We can easily get that link e1,2 needs to relay the traffic of the entire network; links e2,3 and e2,4 need to relay 3/7 of the traffic in the network; links e3,5 , e3,6 , e4,7 and e4,8 only need to relay the traffic from one sensor node. By comparing the values of MLB in Table 1 with the actual traffic distribution on each link, the rationality of MLB in single-sink WSNs can be verified. 5

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Fig.8 shows an example of a multi-sink hierarchical topology. In this topology, nodes 1 and 2 are sink nodes and all other sensor nodes need to transmit data to the sink node closest to 7

Table 1: MNB of each node in the single-sink hierarchical WSN

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Figure 8: An example of a multi-sink hierarchical topology.

them. We can clearly obtain that nodes 3 and 4 relay the traffic from three nodes (including their own traffic) respectively; nodes 5∼8 only relay their self-generated traffic; links e1,3 and e2,4 take the traffic from 3 sensor nodes; links e3,5 , e3,6 , e4,7 and e4,8 only take the traffic from one sensor node; there is no traffic exchange between node 3 and node 4, so link e3,4 takes no load. Table 2 shows the MNB and MLB of each node and each link in this topology. By comparing the values in Table 2 with the actual traffic distribution, the rationality of MLB and MNB in multi-sink WSNs is verified. Table 2: MNB of each node in the multi-sink hierarchical WSN

Nodes MNB 3,4 0.5 5,6,7,8 0.167

Links e1,3 , e2,4 e3,4 e3,5 ,e3,6 ,e4,7 ,e4,8

MLB 0.5 0 0.167

In WSNs, according to the forwarding direction, message delivery can be divided into two modes: application communication and infrastructure communication[24, 25]. The application communication mode mainly relates to the sensing data transferred from general sensor nodes to sink nodes. The application communication model mainly relates to the configuration and maintenance data transferred from sink nodes to general sensor nodes (i.e., routing set-up query). Since our proposed load metrics evaluate the load of nodes and links from the perspective of network topology, they are applicable to both of the above messaging modes. For example, in Fig.7 and Fig.8, we have verified the soundness and effectiveness of the proposed two load metrics in the application model. If we observe the load distribution of the topology shown in Fig.8 based on the infrastructure communication mode, we can obtain that: the message delivery from the sink node 1 to sensor nodes 5, 6 needs to go through sensor node 3 and link e13 ; the message delivery from sink node 2 to sensor nodes 7, 8 needs to go through node 4 and link e24 ; nodes 5, 6, 7 and 8 only need to take their own load. According to the results in Table 2, MNB and MLB perfectly characterize the above load distribution in the infrastructure mode. 3.2. Load and capacity Since the proposed load metrics MNB and MLB can perfectly reflect the network load distribution on each sensor node and each wireless link respectively, it is reasonable to use 8

MNB and MLB as inputs to define the load functions of nodes and links respectively. For this consideration, we define the load functions of node i and link ej,k at time t as Lj (t) = Dj (t)α ,

(4)

Lej,k (t) = Dej,k (t)α ,

(5)

where α≥0 is the load-exponential coefficient to adjust the network load distribution. In most of the existing cascading models for WSNs, the capacity of a sensor node is positively correlated with its initial load, but this assumption does not match the real situation. In actual WSNs, due to the large-scale deployment, the capacity of each sensor node cannot be customized according to their initial load. Therefore, in this model we define the capacity of each sensor node as P Lj (0) j∈Vs WN = (1 + β)LN (0) = (1 + β) , (6) Ng where β is the node-tolerance coefficient to indicate the capacity resources owned by each sensor node; LN (0) is the average of the node load in the initial network; Ng is the number of nodes in the network. Since in actual WSNs the transmission modules of sensor nodes are always the same, each node has the same capacity. WN can be regarded as the maximum load that can be transmitted by each node. When the current load of a sensor node is greater than its capacity, it will fail at the next moment. We also define the capacity of each wireless link as P Lej,k (0) ej,k ∈E

, (7) Ne where γ is the link-tolerance coefficient to indicate the capacity resources owned by each link; LE (0) is the average of the link load in the initial network; Ne is the number of links in the network. WE is the maximum load that each link can take. When the current load of a link is greater than its capacity, it will fail at the next moment. WE = (1 + γ)LE (0) = (1 + γ)

3.3. Cascading mechanism The initial load on all nodes and links in the network is less than their initial capacity. When a certain proportion of nodes or links are attacked, the network load distribution will be renewed, which may cause some nodes and links to fail due to overload. This cascading process will continue until no new nodes or links fail. In multi-sink WSNs, if a sensor node loses all the paths to sink nodes, it will fail from a functional point of view because it cannot continue to deliver messages to sink nodes. In this case, we consider this kind of nodes as isolated nodes. Fig.9 demonstrates an example of the cascading process in a multi-sink WSN. In this network, nodes 1 and 7 are sink nodes, and other nodes are general sensor nodes. We assume that the load-exponential coefficient α=1, the capacity of both nodes and links is 0.8. In Fig.9(a), all the nodes and links in the network are not overloaded. As is shown in Fig.9(b), when the link between node 6 and node 7 is attacked, the network load distribution will be renewed, where the load on node 6 drops and the load on nodes 2, 3, 4, 9, 10 rises. Since all the paths to the sink node 7 are destroyed, all traffic in the network has to go through node 2 and link e1,2 to reach the remaining sink node 1. At this point, the load on node 2 and link e1,2 will be updated to 1, which is greater than their capacity. Then, they will be overloaded, which means that the paths to the sink node 1 are also destroyed. In Fig.9(c), there will be a huge isolated component where nodes fail because they cannot establish a valid path to sink nodes. At this point, the entire network are paralyzed due to cascading failures. 9

LL33(0)=0.1875 (0)=0.1875 LL44(0)=0.125 (0)=0.125 LL5(0)=0.1875 5(0)=0.125 L3(0)=0.1875 L4(0)=0.125 L5(0)=0.125 3

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4 5 LLee3,434(0)=0.0625 LeL5,6e56(0)=0.1875 (0)=0.0625 (0)=0.0625 LLee4,545(0)=0.0625 (0)=0.125 Le34(0)=0.0625 Le45(0)=0.0625 Le56(0)=0.125 LL22(0)=0.5 L6(0)=0.5 (0)=0.5 LLee2,323(0)=0.25 L6(0)=0.5 (0)=0.25 L2(0)=0.5 Le23(0)=0.25 L6(0)=0.5 66 77 11 22 6 7 2 L e 6,7(0)=0.5 Le67(0)=0.5 ee2,10 (0)=0.25 LLee1,212(0)=0.5 2,10(0)=0.25 (0)=0.5 Le2,10LL(0)=0.25 Le67(0)=0.5 Le12(0)=0.5 LLee9,109,10(0)=0.0625 (0)=0.0625 LLee8,989(0)=0.0625 (0)=0.0625 LeL6,8e68(0)=0.1875 (0)=0.125 Le9,1010(0)=0.0625 Le989(0)=0.0625 8 Le68(0)=0.125 10

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(a) LL33(0)=0.4375 (0)=0.4375 LL44(0)=0.3125 (0)=0.3125 LL5(0)=0.1875 5(0)=0.1875 L3(0)=0.4375 L4(0)=0.3125 L5(0)=0.1875 3 3

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4. Results and Analysis In this section, we first present two metrics to measure the cascading robustness of multi-sink WSNs. After that, we explore the impacts of modeling parameters on the cascading robustness of multi-sink WSNs. The simulation experiments are based on Matlab. We use the GUROBI optimization tool[26] to speed up the simulation process. 4.1. Performance metrics In most existing works[27, 28, 29], they used the relative size of the giant component after removing a certain number of nodes to measure the cascading robustness of the network. Although this metric is reasonable enough in peer-to-peer network systems, it is not suitable for WSNs. In WSNs, we are more concerned about the number of sensor nodes that can still communicate with sink nodes after attacks. Therefore, we present a new concept “effective component” consisting of nodes that can communicate with sink nodes. An example of the giant component and the effective component is shown in Fig.10. In this example, the giant component is made up of 12 sensor nodes that cannot communicate with sink nodes, while the effective component is made up of 7 sensor nodes that can have at least one path to sink nodes. Another disadvantage of the giant-component based robustness metric is that it can only reflect the network robustness under specific attack modes.

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Figure 10: An example of effective component and giant component in WSNs.

Therefore, in order to correctly reflect the cascading robustness of multi-sink WSNs, we design two new robustness metrics based on the effective component. One metric is to evaluate the cascading robustness of the network in the face of node failures, and the other metric is to evaluate the cascading robustness of the network in the face of link failures. In the cascading metric against node failures (CMNF), we first remove a node j from the initial network randomly and observe the size of the effective component Cj (i.e., the number of nodes in the effective component). In a similar way, we can get the size of the effective component after removing remaining nodes one by one. To quantify the network robustness of the whole network against node failures, we use the normalized size of the effective component, i.e., P Ck CN =

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(8)

which is the sum of the normalized effective components by removing each sensor node. It is easy to understand that CN = 0 indicates that any node failure can trigger the cascading process and paralyze the entire network. CN = 1 indicates that any node failure in the network will not trigger cascading failures. In the cascading metric against link failures (CMLF), we first remove a link ei,j from the initial network randomly and observe the size of the effective component Cei,j . In a similar 11

way, we can get the size of the effective component after removing remaining links one by one. To quantify the network robustness of the whole network against link failures, we adopt the normalized size of the effective component, i.e., P Cei,j CL =

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(9)

which is the sum of the normalized effective components by removing each wireless link. CL =0 means that any link failure can trigger the cascading process and paralyze the whole network, while CL =0 means that any link failure will not trigger cascading failures. 4.2. Multi-sink oriented characteristics In this part, we evaluate the multi-sink oriented characteristic of the proposed cascading model. As discussed in many literature works[30, 31], due to the sink-oriented characteristic of WSNs, sensor nodes near sink nodes will take heavier traffic load, resulting in faster energy deletion. This phenomenon is known as the “sink hole” effect. Therefore, the sink-oriented characteristic can be considered as one of the most important criteria for judging the soundness and reasonability of a cascading model of WSNs. In this part, we use the network topology referencing [18]. This topology consists of 300 randomly deployed nodes. The transmission radius of sensor nodes is set to 20m. A certain number of sink nodes will be deployed in the network. The network topology is shown in Fig.11. 200 180 160 140

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Figure 11: Network topology consisting by 300 general sensor nodes.

Fig.12 demonstrates the load distributions under different number of sink nodes. As can be seen, the load in the area near sink nodes is significantly higher than that in other areas. The multi-sink oriented characteristic has been verified. Moreover, we can also easily observe that with the increase of the number of sink nodes in the network, the network load will become more balanced. The advantages of multi-sink WSNs are further verified. 4.3. Impacts of modeling parameters on cascading robustness As is shown in Fig.13, no matter when the network faces node attacks or link attacks, there is a critical threshold α∗ that can determine whether the network will have cascading failures. When α is less than or equal to α∗ , the network will not have cascading failures, and when α is greater than α∗ , the cascading failures can be triggered. Moreover, in the case that α is greater than α∗ , the cascading robustness of the network tends to decrease with the increase of α. This is because, as the only parameter to determine the exponential distribution of load, the 12

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Figure 13: CMNF and CMLF with varying α, β and γ.

13

increase of α will make the network load distribution more uneven, and then increase the risk of cascading failures. In addition, we can also observe that the increase of [β,γ] can significantly improve the cascading robustness of the network. This improvement is mainly reflected in two aspects: (1) under the same setting of α, the increase of [β,γ] can significantly improve CMNF and CMLF, indicating that the damage of cascading failures to the network will also decline; (2) the critical threshold α∗ will increase with the increase of [β,γ], which means that the risk of cascading failures will be reduced.

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Figure 14: CMNF and CMLF with varying β and γ (α=2).

Fig.14 demonstrates the cascading robustness with varying β and γ. As can be seen, as a key parameter for adjusting the node capacity, the rise of β can significantly improve the cascading robustness of the network. Taking CMNF as an example, in the case of β=0.5 and γ=2, the CMNF is 0.21, which means that there is a high possibility of cascading failures. When γ stays the same and β rises to 2, the CMNF is 0.64, which means that the risk and damage of cascading failures are greatly reduced. In addition, we also observe that there is a critical threshold β ∗ . When β is less than β ∗ , the increase of β is helpful to the cascading robustness of the network. When β is greater than or equal to β ∗ , the lifting effect of node capacity expansion is saturated. This result tells us an important fact: excessive capacity expansion for sensor nodes will not bring about an improvement in the cascading robustness of the network, but it will lead to an increase in network costs. Fig.15 shows the comparison results of CMNF and CMLF with varying β and γ. We can clearly find that compared with node attacks, the network demonstrates stronger cascading robustness against link attacks. For example, in the case of β=0.5 and γ=0.5, the CMNF is 0.21 and the CMLF is 0.59. These results clearly tell us that in this case, the network is prone to large-scale cascading failures when facing node attacks, while more than half of the nodes in the network can function normally when facing link attacks. This is because, compared with deleting links, the disturbance caused by deleting nodes to the network load will be more severe, which in turn will be more likely to cause cascading failures. As is shown in Fig.16, as the only parameter to determine the link capacity, the rise of γ can help the network obtain better cascading robustness. Taking CMNF as an example, in the case of β=0.5 and γ=0.5, the CMNF is 0.25. When β=0.5 remains unchanged and γ rises to 2, the CMNF is 0.46. Through the above results, the suppression effect of link capacity expansion on cascading failures is verified. Moreover, we can also observe a critical threshold γ ∗ . When γ is greater than or equal to γ ∗ , the lifting effect of link capacity expansion on cascading robustness is saturated. Fig.17 shows the comparison results of CMNF and CMLF with varying γ. Under the same setting of γ, the obtained CMLF is always greater than the CMNF. From the perspective of 14

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Figure 17: Comparison results of CMNF and CMLF with varying γ and β (α=2).

cascading failures, the conclusion that link attacks are less harmful to the network has been further verified. 4.4. Critical thresholds β ∗ and γ ∗ According to the obtained results from Section 4.3, there is a critical threshold β ∗ . When β is greater than or equal to β ∗ , the lifting effect of node capacity expansion on cascading robustness will be saturated. Similarly, we can also find the critical threshold γ ∗ for link capacity expansion. It is easy to understand that the critical thresholds β ∗ and γ ∗ are important references to guide users to optimize network capacity, so we will focus on them in this subsection. Fig.18 presents the heatmaps of CMNF and CMLF within the parameter space [β,γ]. In both heatmaps, we can easily observe that the critical thresholds β ∗ and γ ∗ divide the heapmap into four zones: sensitive zone, link tolerance zone, node tolerance zone and safety zone. When the capacity parameters [β,γ] fall into the sensitive zone, the cascading robustness of the network (i.e., CMNF and CMLF) varies according to the settings of [β,γ]. In the link tolerance zone, the cascading robustness is only sensitive to the changes of β, which means that adjusting link capacity has no effect. In the node tolerance zone, the cascading robustness can only respond to the changes of γ, indicating that the expansion of node capacity cannot help the network resist cascading failures. In the case that [β,γ] is within the safety zone, the cascading robustness reaches the maximum value of 1, which means that cascading failures will not occur. In the study of cascading failures, we not only focus on how to reduce the damage caused by cascading failures, but also on how to configure the network so that cascading failures do not occur. Apparently, the safety zone is exactly what we need to focus on. The larger the safety zone, the lower the costs of capacity expansion to prevent cascading failures on the network. By comparing the size of the safety zones of CMNF and CMLF, we can clearly find that the network needs more capacity to ensure no cascading failures when facing node attacks. 16

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Figure 18: Heatmaps of CMNF and CMLF within the parameter space [β,γ] (α=2).

Fig.19 shows the critical thresholds β ∗ and γ ∗ with varying α. We can easily observe that β ∗ and γ ∗ will increase as α rises. This result tells us an important fact that the increase of α will not only increase the risk of cascading failures, but also increase the costs of capacity expansion to prevent cascading failures. In addition, we can also obtain that β ∗ is always greater than γ ∗ . This result shows that the node capacity is the key to determine whether the cascading failures occur. Therefore, when configuring the capacity resources, more resources should be provide for sensor nodes.

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Fig.20, Fig.21 and Fig.22 present the heatmaps of CMNF and CMLF with varying α. By comparison, we can clearly find that with the increase of α, the size of safety zones become smaller. The impacts of α on the costs of capacity expansion is further verified. By comparing the size of the safety zones of CMNF and CMLF under the same parameter setting, the conclusion that node attacks are more likely to cause cascading failures and require higher network maintenance costs is further verified. 4.5. Number of sink nodes In this subsection, we evaluate the number of sink nodes on the cascading robustness of the network. To give a comprehensive evaluation, we design four cases of parameter settings, as shown in Table.3. Fig.23 shows the CMNF and CMLF with varying number of sink nodes under different cases. We can easily obtain that the increase in the number of sink nodes in the network can significantly improve the cascading robustness of the network. In the case of CMNF, when only 17

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Cases

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18

one sink node is deployed, the CMNF is 0.16, which means that the network is very easy to be completely paralyzed due to cascading failures. In contrast, when the number of sink nodes increases to 5, the CMNF rises to 0.96, indicating that the cascading robustness of the network has been greatly improved and most of cascading failures can be resisted. This is because the increase of the number of sink nodes can contribute to the network load balancing and further reduces the risk and damage of cascading failures. 1

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Fig.24 depicts the critical thresholds β ∗ and γ ∗ of CMNF with varying number of sink nodes. As can be seen, β ∗ and γ ∗ tend to decrease as more sink nodes are introduced into the network. As discussed in Section 4.4, the critical thresholds β ∗ and γ ∗ can indicate the costs of capacity expansion required for preventing cascading failures. Therefore, we can easily conclude that the increase in the number of sink nodes can reduce the risk of cascading failures, and can also reduce the expansion costs to resist cascading failures. From Fig.25, we can come to a similar conclusion. Although more sink nodes will lead to an increase in the network hardware costs, we consider it worthwhile because of its significant effects on cascading robustness. 2 1.5

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4.6. Cascading process under multi-node attacks and multi-link attacks According to the proposed CMNF and CMLF, we can only evaluate the cascading robustness of the network when facing single-node attack or single-link attack. In practical scenarios, in addition to the above two attack schemes, multi-node attacks and multi-link attacks are also very common. In this subsection, we evaluate the cascading robustness of the network when 19

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facing these two attacks. Consistent with most of the existing literature[32, 33, 34], we choose the intentional attack mode (i.e., select a fixed proportion of nodes from high to low importance to attack). In this work, the attack proportion is set to 5% and the importance of nodes and links are determined by their multi-oriented betweenness values (i.e., MNB and MLB). We use the survival ratio Ps (t) (i.e., the proportion of nodes in the network that can still work normally at time t) to observe the cascading process. We use Ps (∞) to denote the final survival ratio (i.e., the proportion of surviving nodes after the cascading process stops). Moreover, we also explore the impacts of network size (i.e., number of sensor nodes in the network) on cascading robustness. 1

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Fig.26 depicts the cascading process under multi-node attacks and multi-link attacks. As can be seen, in Case 1, the network has the strongest cascading robustness, and its cascading process stops at t=3. In contrast, the cascading robustness of Case 4 is the worst and its cascading process does not stop until t=7. We can easily conclude that the duration of cascading process is positively related to the scale of cascading failures. This result reminds us that the larger the cascading failure scale is, the longer the failure-spreading process will take, which also provides us with more time to adjust the network to further prevent cascading failures. Moreover, in the cascading process, although the changes in network load will affect the energy consumption of sensor nodes, the impacts of energy consumption on the network can be ignored when compared with the immediate failures of nodes or links caused by overload. Fig.27 shows the final survival ratio with varying network size under multi-node attacks and multi-link attacks. We can easily obtain that with the expansion of network size, the 20

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Figure 27: Survival ratio with varying network size under multi-node attacks and multi-link attacks.

final survival ratio Ps (∞) has a slight decline. Our analysis is that although the expansion of network size will increase the network load, the network connectivity will also be improved in this process and contribute to the network load balancing. Through the interaction of the positive and negative aspects above, the cascading robustness of the network will not decline dramatically. 5. Conclusions and Future Work Cascading failures are one of the main factors affecting the actual performance of multi-sink WSNs, which makes cascading robustness become a hot research topic in current research. Existing cascading models fail to properly reflect the cascading process of WSNs. On the one hand, these models cannot properly characterize the load distribution of multi-sink WSNs. On the other hand, they only consider the network cascading process as a load redistribution process caused by node overload, which ignore the impacts of link overload. Therefore, this paper proposes a more realistic cascading model for multi-sink WSNs. In our model, the cascading process of the network is jointly promoted by node overload events and link overload events, which can better reflect the cascading characteristics of multi-sink WSNs in practical scenarios. In addition, we focus on the cascading robustness of the network in the face of node attacks and link attacks. We have obtained some meaningful results, which can provide theoretical guidance for the establishment of networks with higher cascading robustness: 1) There are critical thresholds β ∗ and γ ∗ for node capacity and link capacity. Excessive capacity expansion will not lead to the improvement of network cascading robustness, but will lead to the increase of network costs; 2) There is a critical threshold γ ∗ for network load distribution. To reduce the risk of cascading failures, the network load should be balanced as much as possible; 3) Node attacks are more likely to trigger cascading failures than link attacks. To prevent cascading failures, node capacity expansion should be given more priority. In the future, we will focus on the optimization of cascading robustness. We believe that there are two aspects worth trying: 1) we can optimize the layout of multiple sink nodes to make the network more load balanced and reduce the risk of cascading failures. How to determine a reasonable number of sink nodes and how to evaluate the load balance of the network will be the key to this solution; 2) we can also introduce mobile sink nodes to help multi-sink WSNs resist cascading failures. If there is an overload risk in an area of the network, the mobile sink nodes can move to that area to reduce the risk by alleviating the load. The key to this solution is how to determine the high-risk area and how to schedule the trajectory of mobile sink nodes reasonably. 21

Acknowledgement This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No.61902238, and Science and the Technology Commission of Shanghai Municipality under Grant No.18510745100. Declaration of Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Author Statement Xiuwen Fu Conceptualization, Methodology, Software , Data curation, Writing Original draft preparation , Visualization, Investigation. Yongsheng Yang : Writing Reviewing and Editing , Supervision References [1] S. Chakraborty, N. Goyal, S. Mahapatra, and S. Soh, “A Monte-Carlo Markov chain approach for coverage-area reliability of mobile wireless sensor networks with multistate nodes,” Reliability Engineering & System Safety, vol. 193, p. 106662, 2020. [2] S. Xiang and J. Yang, “Performance reliability evaluation for mobile ad hoc networks,” Reliability Engineering & System Safety, vol. 169, pp. 32–39, 2018. [3] Y. Mo, L. Xing, W. Guo, S. Cai, Z. Zhang, and J. H. Jiang, “Reliability analysis of IoT networks with community structures,” IEEE Transactions on Network Science and Engineering, pp. 1–1, 2018. [4] X. Fu, G. Fortino, W. Li, P. Pace, and Y. Yang, “Wsns-assisted opportunistic network for low-latency message forwarding in sparse settings,” Future Generation Computer Systems, vol. 91, pp. 223–237, 2019. [5] H. Barani, Y. Jaradat, H. Huang, Z. Li, and S. Misra, “Effect of sink location and redundancy on multi-sink wireless sensor networks: a capacity and delay analysis,” IET Communications, vol. 12, no. 8, pp. 941–947, 2018. [6] X. Fu, G. Fortino, P. Pace, G. Aloi, and W. Li, “Environment-fusion multipath routing protocol for wireless sensor networks,” Information Fusion, vol. 53, pp. 4–19, 2020. [7] C. Wang, L. Xing, A. E. Zonouz, V. M. Vokkarane, and Y. Sun, “Communication reliability analysis of wireless sensor networks using phased-mission model,” Quality and Reliability Engineering International, vol. 33, no. 4, pp. 823–837, 2017. [8] X. Fu, H. Yao, and Y. Yang, “Modeling and analyzing cascading dynamics of the clustered wireless sensor network,” Reliability Engineering & System Safety, vol. 186, pp. 1–10, 2019. [9] Y. D. Beyene, R. Jantti, O. Tirkkonen, K. Ruttik, S. Iraji, A. Larmo, T. Tirronen, and J. Torsner, “NB-IoT technology overview and experience from cloud-RAN implementation,” IEEE Wireless Communications, vol. 24, no. 3, pp. 26–32, 2017. [10] L. Feltrin, G. Tsoukaneri, M. Condoluci, C. Buratti, T. Mahmoodi, M. Dohler, and R. Verdone, “Narrowband IoT: A survey on downlink and uplink perspectives,” IEEE Wireless Communications, vol. 26, no. 1, pp. 78–86, 2019. 22

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