Rethinking failure and attack tolerance assessment in complex networks

Physica A 390 (2011) 4684–4691

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Rethinking failure and attack tolerance assessment in complex networks Cinara G. Ghedini ∗ , Carlos H.C. Ribeiro Technological Institute of Aeronautics, Computer Science Division, São José dos Campos, SP, Brazil

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Article history: Received 30 April 2010 Received in revised form 1 May 2011 Available online 20 July 2011 Keywords: Vulnerability in complex networks Failure and attack tolerance Network connectivity

abstract Studies have revealed that real complex networks are inherently vulnerable to the loss of high centrality nodes. These nodes are crucial to maintaining the network connectivity and are identified by classical measures, such as degree and betweenness centralities. Despite its significance, an assessment based solely on this vulnerability premise is misleading for the interpretation of the real state of the network concerning connectivity. As a matter of fact, some networks may be in a state of imminent fragmentation before such a condition is fully characterized by an analysis targeted solely on the centrally positioned nodes. This work aims at showing that, in fact, it is basically the global network configuration that is responsible for network fragmentation, as it may allow many other lower centrality nodes to seriously damage the network connectivity. © 2011 Elsevier B.V. All rights reserved.

1. Introduction It is increasingly recognized that organizing principles operate in most real networks [1]. These networks form and evolve in an ad hoc manner, being naturally self-organizing and self-adaptive. Examples can be found in nature, such as in ecological food webs [2], Escherichia coli [3], and neuronal topologies in Caenorhabditis Elegans worms [4]. Such network structures may also serve as models for social [5,6] and technological networks, such as P2P, overlay, sensor, and communication networks [7–10]. Although real network formations are targeted to achieve specific goals in diverse contexts and applications, most networks in fact exhibit similar organizational principles [4,11,1,12]. In general, this implies similar topological properties. The best-known property is that, on average, pairs of nodes can be connected by short path lengths (the so-called smallworld phenomenon) [11]. Another property is a high clustering coefficient, which means that there is a high likelihood of any two nodes with a common neighbor being connected. Moreover, the degree distribution of many real networks follows a power-law tail, which, in a simplified way, means that a few nodes have many connections. However, networks exhibiting uniform and exponential degree distributions have also been reported [13]. As these networks do not rely on any fixed or predefined infrastructure and are not supported by any central management, nodes are usually autonomous to leave or join the network, causing frequent changes in the network topology. Besides, topological changes can also be induced by node failure (e.g., in sensor networks). Despite such variability, these networks are often able to maintain their main topological properties. However, this is not necessarily the case when such topological variability is biased to nodes of high centrality that either leave or fail. In fact, attacks on high centrality nodes are frequently considered (e.g., in communication networks) as a means to seriously harm the network operation, for instance by increasing the average path length. Several researchers have studied the impact that both failures and targeted attacks have on the network efficiency and connectivity [14,15]. This assessment has been made through simulations of node disconnections based on two main criteria: (a) at random, and (b) choosing the high centrality nodes, to mimic, respectively, failures and targeted attacks.

∗ Corresponding address: Instituto Tecnológico de Aeronáutica, Divisão de Ciência da Computação, Praça Mal. Eduardo Gomes, 50, 12228-900, São José dos Campos, SP, Brazil. Tel.: +55 12 3947 6887; fax: +55 12 3947 5989. E-mail addresses: [email protected], [email protected] (C.G. Ghedini), [email protected] (C.H.C. Ribeiro). 0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.07.006

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Despite high centrality nodes playing a crucial role regarding network connectivity, we argue that an assessment considering only this aspect misinterprets the role of the actual network configuration as far as the possibility of disconnection is concerned. In fact, some networks may reveal an imminent state of disconnection before it is characterized through standard centrality measurements. The imminent state of disconnection is related to the concept of vulnerability in several areas, from biology [16], sociology and ecology [17–19] to network engineering and computer science [20–22]. This paper aims at showing that, in general, the main feature that is responsible for network fragmentation is the global network configuration. In this sense, nodes of relatively low centrality can also seriously damage network connectivity. The rest of the paper is organized as follows. A brief theoretical background is presented in Section 2. Section 3 states the problem addressed herein, namely the impact of topological changes induced by failures and attacks on network efficiency and connectivity, and the usual approach adopted to assess it. In Section 4, we explore another point of view concerning this problem, and show the results of experiments conducted to support the argument of the existence of a global state of imminent disconnection. 2. Background Two major branches of research on complex networks are the development of methods for network analysis and for network modelling, as the combination of modelling and measurement tools provides a benchmark to simulate network dynamics and allows a targeted analysis. This section addresses the main analytical measures and network models which are of interest for the approach reported in this paper. 2.1. Measuring network properties Regarding the topological properties of complex networks, both global and local characteristics are relevant. Global assessment is often performed – especially in settings where information transmission is at stake – by computing the average of the shortest distance between any two nodes in the network, the so-called characteristic path length L [11]: L=

1 1 n 2

−

(n + 1)

dij ,

(1)

i≥j

where dij is the geodesic distance from vertex i to vertex j. The characteristic path length (L) indicates how far apart the nodes are from each other, in other words, how efficient a network is with respect to information dissemination through its elements. However, L is not an appropriate metric to deal with disconnected networks. Lattora et al. [14,23] introduced the efficiency E, which measures how efficiently the nodes exchange information in a local or global scope, independently of whether the network is disconnected, weighted, or topological. Consider a graph G where dij is the smallest sum of the physical distances through every possible path between nodes i and j. The efficiency Eij is inversely proportional to the shortest distance: Eij = d1 . If there is no path between them, the distance dij is +∞, and therefore Eij = 0. Thus, the global ij

efficiency of a graph G may be defined as

∑ Eglob (G) =

=

Eij

i̸=j∈G

n( n − 1 ) 1

− 1

n(n − 1) i= d ̸ j∈G ij

.

(2)

Notice that Eglob range is [0, ∞]. To normalize it, consider the ideal case Gideal where all the possible n(n − 1)/2 edges are in the graph; this is the case when Eglob assumes its maximum value. The normalized efficiency is then defined as

Eglob (G)

Eglob (Gideal )

.

The same idea can be extended to estimate the local efficiency (3). A local perspective provides mechanisms to quantify the existence of tightly linked subgraphs and may express the structure of the cluster a given node takes part in. The average over all network nodes represents the cohesion of the nodes (4): Eloc (G) =

1− n i∈G

E (Gi )

(3)

where E (Gi ) =

1

−

1

ki (ki − 1) l= d ̸ m∈Gi lm

(4)

and Gi is the subgraph containing all nodes directly connected to i (ki is its degree). If the nearest neighborhood of i was part of a clique, there would be ki (k1 − 1)/2 edges among them [1].

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2.2. Centrality measures Another important class of information inferred from the network topology is related to the relative importance of nodes in the network. There are different points of view with respect to this issue and, reflecting this, diverse measures have been defined in the literature to highlight the importance of any node in the network, such as degree centrality (DC ), closeness centrality (CC), betweenness centrality (BC ), and information centrality (IC) [24]. The choice of any one of them is tied to the considered application. The most widely used, however, are the degree centrality (DC ) and the betweenness centrality (BC ). DC (5) defines as the most important nodes the ones which present more ties to other nodes in the network [24]: CD (ni ) = d(ni ),

(5)

where d(ni ) is the degree of node i. BC (6) establishes higher scores for nodes which play an important role in the interactions among other nodes, that is, nodes which are contained in most of the shortest paths between every pair of nodes in the network have highest BC scores. The BC of node i is computed considering the number of geodesic paths linking all pairs of nodes (not including i) present in the network (gjk (ni )) in which i is included. As in such a case all paths are equally likely to be chosen, the probability of a link is calculated by gjk (ni )/gjk , where gjk is the number of geodesic paths between nodes j and k. Then, the BC of a node i is the sum of the estimated probability over all pairs of nodes not including i [24]: CB (ni ) =

− (gjk (ni )/gjk ).

(6)

j
2.3. Complex network models Several studies have pointed out that it is possible to find similar patterns in the formation of real networks, such as a small characteristic path length and a high clustering coefficient [25,1,11]. Many authors have thus proposed models to generate networks with such specific topological properties as convenient simulations of real networks. The most widely used models are the Watts and Strogatz β -model [26], the Barabasi and Albert (BA) model [1], the Klemm–Euguluz (KE) model [27], and the Erdős and Rényi (ER) model [28]. The β -model builds networks exhibiting a high clustering coefficient and a small characteristic path length from a construction process that starts with a regular lattice with n nodes and ⟨k⟩ average degree, where each node has precisely k/2 neighbors on either side. Then, edges are randomly rewired with probability β . When β = 0, the regular lattice is completely preserved, and with β = 1 a random graph is generated. The small-world property is achieved with a small β -value. In this paper, we refer to β -model network implementations as WS networks. The BA model was designed based on the statement that many large networks exhibit a small characteristic path length and a power-law degree distribution. Then, the resulting network shows these two main properties, with P (k) ∼ k−3 , but a very small clustering coefficient. Klemm and Euguluz [27] claimed, however, that most real networks exhibit a small characteristic path length, a power-law degree distribution, and a high clustering coefficient. Based on the BA model, they proposed a new model in which initially a fully connected network with m active nodes is created. At each time t a new node i is inserted in the network with m connections. For each connection, it is decided with probability 1 − µ if a node will be connected to an active or a random node. The random node is chosen by preferential attachment—the probability of any node j in the network being connected to new node i depends on the degree kj of node j [1]. Then, an active node is chosen to ∑ −1 1 be deactivated with probability k− i / l kl . The new node i is set as active. The parameter µ adjusts the network topology. If µ = 1 the BA model is reproduced; otherwise, if µ = 0, a highly clustered network is generated. The networks with the expected properties are those where µ is set to a small value (10−3 < µ <= 10−1 ). The Erdős and Rényi (ER) model generates networks with a small characteristic path length, a small clustering coefficient, and a Poisson degree distribution for large n. Its application is mainly for validation purposes, acting as a benchmark for complex networks applications. 3. Failures and attacks in complex networks Assessing the impact of topological changes induced by failures and attacks on network efficiency and connectivity requires a protocol that is briefly sketched in this section. This protocol is based on approaches already reported in the literature [29–31,15,32]. As the intention is to measure performance in unknown topologies, for the sake of generalizing results three different topological models were considered, namely KE, WS, and ER networks. All of them show a small characteristic path length, differing, however, in the degree distribution and in the clustering coefficient values. KE networks exhibit a power-law tail degree distribution; random networks have a Poisson degree distribution, and WS topologies have a uniform distribution. Regarding the clustering coefficient, KE and WS networks exhibit a high C -value, but ER networks exhibit a very low clustering coefficient. For simulating failures, we assume nodes as autonomous agents that can leave the network at random with a uniform dropping probability distribution. The result of attacks, on the other hand, may be simulated through removal of high

1

1

0.9

0.9

0.8

0.8

0.7

ER WS KE

0.6

Local efficiency

Global efficiency

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0.5 0.4 0.3

0.6 0.5 0.4 0.3 0.2

0.1

0.1 0

0.1 0.2 0.3 0.4 Fraction of Nodes Removed

0

0.5

ER WS KE

0.7

0.2

0

4687

0

0.1 0.2 0.3 0.4 Fraction of Nodes Removed

0.5

Fig. 1. Values of global Eglob (G) and local Eloc (G) efficiencies in case of failures for ER, WS, and KE network models.

1

1 0.9 0.7 0.6 0.5 0.4 0.3

0.7 0.6 0.5 0.4 0.3

0.2

0.2

0.1

0.1

0

0

0.1 0.2 0.3 0.4 Fraction of Nodes Removed

ER BC WS BC KE BC ER DC WS DC KE DC

0.8 Local efficiency

0.8 Global efficiency

0.9

ER BC WS BC KE BC ER DC WS DC KE DC

0.5

0

0

0.1 0.2 0.3 0.4 Fraction of Nodes Removed

0.5

Fig. 2. Values of global Eglob (G) and local Eloc (G) efficiencies for BC /DC attacks in ER, WS, and KE network models.

centrality nodes. Two centrality metrics were adopted to classify the nodes: the degree centrality (DC ) (Eq. (5)) and the betweeness centrality (BC ) (Eq. (6)). To assess the impact of failures and attacks, global (Eq. (2)) and local (Eq. (3)) efficiencies, as well as the size of the giant component (S ) as a function of the fraction of nodes dropped, were used. At each iteration, a node was dropped from the network according to the priorities previously defined: highest DC , highest BC , and at random (RN ). Then, the values of global and local efficiencies and the size of the giant component were computed. All the results were averaged over 30 networks containing 100 nodes with average degree of 6. Fig. 1 illustrates the scores for global and local efficiencies when different network topologies are exposed to failures. Global and local efficiencies were only slightly affected for small fractions of node removal: for a 20% node removal fraction, we note a maximum average efficiency decay of 8% in the KE network model. This happens because the choice for node removal is at random. In practice, this means that only occasionally high centrality nodes fail, but this is not a frequent event. ER networks are also little affected by attacks. On the other hand, attacks (either on high BC or DC nodes) affect WS and KE networks heavily. Fig. 2 shows the variations for global and local efficiencies considering attacks. Notice that WS topologies inherit characteristics of a regular lattice, but with just a few nodes playing the role of connecting nodes far apart, thus supporting shortcuts in network communication. This implies that there is a fairly uniform degree centrality distribution, making DC -targeted attacks less harmful, as the few nodes acting as shortcuts being as likely for attacks as many nodes with the same degree which do not play an important part in network communication. In contrast, BC scores are more heterogeneous in WS topologies. Nodes playing the role of shortcuts show higher BC and, when attacks are directed to these nodes, the network loses important channels of communication, with a resulting decrease in global efficiency. Besides, these nodes connect clusters far apart, and therefore, by attacking them, the local efficiency is increased. Notice that for a high node removal fraction (more than 40%), the WS properties of the topology do not hold anymore, thus explaining the observed decrease in local efficiency. KE networks are also considerably affected by attacks. This is a consequence of their degree distribution characteristics: there are few nodes responsible for creating shortcuts on the networks (nodes with high BC ) and a few nodes with a higher degree (nodes with high DC ) are linked to nodes with a low degree. These also implies that the global efficiency decreases a little faster for BC than DC attacks, and the local efficiency decreases faster for DC attacks.

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1

1

0.9

0.9

0.8

0.8

S – Giant Component

S – Giant Component

4688

0.7 0.6 0.5 0.4

ER BC WS BC KE BC ER DC WS DC KE DC

0.3 0.2 0.1 0

0

0.7 0.6 0.5 0.4 0.3 0.2

ER WS KE

0.1

0.1 0.2 0.3 0.4 Fraction of Nodes Removed

0.5

0

0

0.1 0.2 0.3 0.4 Fraction of nodes removed

0.5

Fig. 3. Size of the giant component (S ) versus fraction of nodes removed when networks are exposed to failures and BC /DC attacks for ER, WS, and KE network models. Table 1 Pearson’s correlation Eglob (G). Network

BC

DC

Random

ER WS KE

0.94 0.97 0.96

0.93 0.90 0.92

0.92 0.96 0.95

Let us now consider the size of the giant component in all the topologies considered above (Fig. 3). Notice that for BC /DC attacks, mainly in highly clustered topologies (KE and WS), networks can disconnect much earlier than for failures. For instance, for KE and WS networks, with 10% of the highest BC nodes dropped, the giant component had on average only 70% of the nodes in the network. For failures, networks become disconnected, on average, with more than 30% of nodes dropped. As the main point is to evaluate the maintenance of the network connectivity, the question of concern is: Is the measure adopted for global efficiency a reliable estimator for such condition? Table 1 shows the Pearson correlation between the global efficiency value (Eglob (G)) and the size of the giant component (S ). Despite the high correlation, there are no clues about an imminent state of disconnection. The results demonstrate that, regardless of the attack strategy (either directed to high BC or high DC nodes), topologies more similar to those found in real networks (i.e. WS and KE models) are more affected by attacks (regarding connectivity) than ER networks. Trying to figure out how an imminent disconnection state could be evaluated, we realize that the methodology above, characterized by an analysis targeted solely on the centrally positioned nodes, which is widely adopted, is misleading, as mentioned above and discussed in the next section. 4. Reassessing failure and attack tolerance Analysis of Fig. 3 points to the fact that the loss of high centrality nodes is potentially harmful to network connectivity, as network fragmentation happens after successive node losses. The question however is how the network is changing, and what is the real influence of the high centrality nodes in this process. Indeed, attacks on high centrality nodes affect not only the network connectivity, but also its efficiency (see Fig. 2). If we turn our attention to the overall network configuration during this process, from a certain point on the network is so severely affected that not necessarily the highest centrality node will disconnect it, but other nodes may do so. In other words, the factor that is responsible for network fragmentation is a global state of network vulnerability. In such a state, two situations may occur. The first is the network becoming disconnected by the removal of the highest centrality node, which probably implies that others nodes may do that. The second is when the highest centrality node does not do that, but some of the other nodes are able to do so. In both cases, the impact caused by nodes other than the highest centrality one may be not just similar, but worse. To demonstrate this, we reapply the protocol of Section 3, but, before the node with the highest BC /DC is permanently removed from the network, the impact of the removal of each one of its nodes is evaluated. For evaluation purposes, we define four classes of network configuration: (1) neither the highest BC /DC node nor any of the other nodes disconnects the network (0–0), (2) the highest BC /DC node does not disconnect the network, but some of the other nodes do (0–1), (3) the highest BC /DC node disconnects the network and none of the others do (1–0), and (4) the highest BC /DC node and some of the other nodes disconnect the network (1–1). Besides the size of the giant component (S ), we also computed the average of the proportion of nodes which produced disconnection (Nd ), the BC /DC node scores (i.e. a ranking of node centralities), and the shortest distance between nodes and the highest BC /DC one (distv ), normalized by the network diameter.

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1 0.9

0–0 0–1 1–0 1–1

Fraction of Iterations

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

ER–BC ER–DC ER–RN KE–BC KE–DC KE–RN WS–BC WS–DC WS–RN

Fig. 4. Fraction of network configurations after successive failures or attacks.

0.45

Nd – Proportion of Nodes

0.4 0.35 0.3 0.25 0.2

Only Non–highest BC/DC (0–1) Both (1–1)

0.15 0.1 0.05 0 KE–BC KE–DC KE–RN WS–BC WS–DC WS–RN

Fig. 5. Proportion of nodes that disconnect the network for non-highest BC /DC attacks: 0–1 (only non-highest BC /DC attacks) versus 1–1 (both) configurations for WS and KE network models.

Fig. 4 shows the fraction of iterations in which each network configuration occurred. The score for the 0–0 configuration is high because, at initial stages, networks are kept connected despite successive losses of nodes. For the remaining cases, the proportion of iterations where the highest BC /DC node did not disconnect the network but some of the other nodes did (0–1) is significant when compared to the opposite (1–0). Notice that in just a few iterations was only the highest BC /DC responsible for the network disconnection. Fig. 4 suggests that there may be a state of network vulnerability, in which several nodes can be harmful to the network connectivity. Fig. 5 shows, for WS and KE network models, the proportion of non-highest BC /DC nodes (Nd ) which disconnect the network in two configurations: 0–1 (only non-highest BC /DC attacks disconnect the network) and 1–1 (both non-highest BC /DC and highest BC /DC attacks disconnect the network). On average, these nodes were more distant from the highest BC /DC one and exhibit a higher BC /DC -value in the 0–1 configuration than those in the 1–1 configuration. The most important aspect to be considered, however, is the high proportion of nodes other than the highest centrality ones producing network disconnection when the highest centrality node does not disconnect the network, suggesting an overall state of network vulnerability. Fig. 6 shows the size of the giant component produced by the removal of the node which most impacts the network connectivity in each network configuration. For the 1–1 configuration, the values produced for both highest and non-highest centrality nodes are shown. The values for KE networks demonstrate that the size of the giant component generated by the removal of the nonhighest BC node is very close to those generated by the removal of the highest BC node (KE–BC ). Notice that, for the 1–0 configuration, S = 0.83, almost the same size when only a non-highest BC node disconnects the network (S = 0.86). When both did (1–1 configuration), there was a similar impact: the size of the giant component is 0.72 for the disconnection of the highest BC node and 0.73 for the non-highest BC node.

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1 0.9

S – Giant Component

0.8 0.7 0.6 0.5 0.4

Only Highest BC/DC (1–0) Both Highest BC/DC (1–1) Both Non–highest BC/DC (1–1) Only Non–Highest BC/DC (0–1)

0.3 0.2 0.1 0 KE–BC

KE–DC

WS–BC

WS–DC

1

1

0.9

0.9

0.8

0.8

S – Giant Component

S – Giant Component

Fig. 6. Size of the giant component produced by the removal of the node which most impacts the network connectivity. For the 1–1 configuration, the values for both highest and non-highest centrality nodes are shown.

0.7 0.6

WS BC WS DC WS BC–α WS DC–α

0.5 0.4 0.3 0.2 0.1 0

0.7 0.6

KE BC KE DC KE BC–α KE DC–α

0.5 0.4 0.3 0.2 0.1

0

0.1 0.2 0.3 0.4 Fraction of Nodes Removed

0.5

0

0

0.05 0.1 0.15 Fraction of Nodes Removed

0.2

Fig. 7. Comparative values of the giant component generated by BC /DC attacks and for removal of node which most impact the network connectivity (apha) for WS and KE network models.

For DC attacks (KE–DC ), the connectivity impact for non-highest nodes is greater than for the highest one for both configurations: 0.98 and 0.92 for the highest DC node, and 0.78 and 0.81 for the non-highest DC node. The pattern of results is also valid for DC attacks in WS networks (WS–DC ), but for BC attacks (WS–BC ) the size of the giant component produced by non-highest nodes in the 0–1 configuration was significantly larger than for the cases in which the highest BC centrality node also disconnected the network. In summary, the results show that, when disconnection is imminent, in most cases the network configuration is such that attacks to nodes other than the highest centrality one can affect the network in the same way, sometimes even more heavily. We therefore argue that it is not the removal of the most centralized node which will fragment or impact mostly the network connectivity, but a global state of vulnerability of the network. Fig. 7 presents the size of the giant component (S ) considering the highest and the non-highest BC /DC node (α ) attacks which most impacted the network connectivity. Significantly, in both topologies considered (WS and KE) the removal of the most harmful non-highest node generates a network with a smaller giant component than the produced by the highest BC /DC node removal. 5. Conclusions Vulnerability of complex networks concerning network connectivity is still an open issue. New tools, measurements, and views are required to support its evaluation and management. The discussion presented here is a step towards rethinking the assessment of vulnerability as far as failures and attack tolerance are concerned. The results demonstrate the difference between thinking in localized terms – nodes with specific configurations – and looking at the whole network configuration. The main point showed is that the network fragments into clusters because it is in a previous state of vulnerability and often, in such a state, it is not the removal of the most centralized node which will do the harm, even though these nodes might be the main culprits for making the network achieve such a state of vulnerability.

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The discussion also reinforces that, despite local interactions playing a crucial role in the emergence of global properties, it is probably a combination of global properties that is responsible for giving networks certain singular configurations which allow intriguing events and phenomena to occur, among these a vulnerable state regarding network connectivity. For future work, we consider looking for tools to support vulnerability assessment considering not only particular cases, but a whole set of network configurations. The focus must be on understanding this process and how it is shaped. This will allow the development of tools not only for detecting vulnerability states, but also to adaptively minimize its probability of occurrence. Acknowledgments The authors thank CNPq–Brazilian National Research Council and FAPESP–São Paulo State Research Foundation for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

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