Abstract
In this chapter, we study the link-level network robustness to area attacks. We first derive the average probabilities of an arbitrary link being attacked for both the LCR random network and the traditional random network. Afterwards, we present the expected numbers of the destroyed links for both cases.
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In this chapter, we study the link-level network robustness to area attacks. We first derive the average probabilities of an arbitrary link being attacked for both the LCR random network and the traditional random network. Afterwards, we present the expected numbers of the destroyed links for both cases.
3.1 Average Probability of an Arbitrary Link being Attacked
In this section, we first present the probability of being attacked for a link with a given length. Then we derive the average loss probability for an arbitrary link for both the LCR random network case and the traditional random network case.
As shown in Fig. 3.1 with a given attack area \(\mathcal{A}(\mathbf{c},r)\), for a particular link L 1 with length l 1, it is destroyed if the center of the attack area is in the olivary area S. Therefore, the probability of being attacked for L 1 can be expressed as
where | ⋅ | denotes the volume.
Since the nodes and the attack area are uniformly distributed, the network geometry has a symmetric property, such that the probability of a link being killed is the same as other links of the same length (by neglecting the boundary effects).
3.1.1 The LCR Random Network Case
In a LCR random network, we assume the PBC on \(\mathcal{W}\). We denote f LCR (x) as the probability density function (PDF) of the link length, and we have
Since we assume the PBC along both the x and y directions, the network is homogeneous. Let us fix one end of a link; and P LCR (x) is equal to the probability of the other end being located in the ring area centered at the first end along the radius from x to x + dx [52], which could be expressed as
where 2πx ⋅dx is the area of the ring and πl 2 is the area of the disk with the radius of l.
Therefore, we have
Combining (3.1) and (3.6), the average probability of an arbitrary link being attacked in the LCR random network model is given by
3.1.2 The Traditional Random Network Case
In the traditional random network, there is no local limitation on the link length and the range of a link length is \((0,\sqrt{2})\). We denote f RN (x) as the PDF of the link length in the traditional random network. Similar to the LCR random network case, the average probability of an arbitrary link being attacked in the traditional random network model is given by
where f RN (x) is derived in Appendix and is given by
3.2 The Expected Number of Destroyed Links
In this section, we first derive the expected number of destroyed links in an arbitrary LCR random network. We then present the result in an arbitrary traditional random network. Finally, we investigate the case when the network topology is fixed.
3.2.1 The LCR Random Network Case
In the LCR random network, the expected total number of node pairs is given by
where N is a Poisson random variable with density λ.
The probability that a link exists between two randomly chosen nodes n i and n j can be expressed as
where | | n i − n j | | is the distance between n i and n j .
Therefore, the expected number of links in the network can be calculated as
Theorem 3.1.
Although the attacks over different links may be correlated, the expected number of destroyed links in the whole network is given as
which is the same as the case when each link gets attacked independently.
Proof.
First, we define some notations in Table 3.1.
We assume that L and A are independent, and N and A are independent. Let \({1}_{\{{l}_{j,N},A\}}\) denotes an indicator function given as:
The number of the destroyed links in N by an area attack A can be expressed as
Therefore the expected number of destroyed links in the network under the area attack is given as
which implies that, we count the number of the destroyed links for every possible network realization and every possible area attack, and then average the number over the all the possible network realizations and all the possible area attacks.
On the other hand, the expected number of links in the network can be expressed as E N ( | L N | ). Therefore, we have
where \({x}_{{l}_{j,N}}\) is the length of l j, N , \(0 = {a}_{0} \leq {t}_{1} \leq {a}_{1} \leq {t}_{2} \leq {a}_{2} \leq \cdots \leq {a}_{n-1} \leq {t}_{n} \leq {a}_{n} = l\) is a tagged partition over the range of the link length, \({\Delta }_{i} = {a}_{i} - {a}_{i-1}\) is the width of sub-interval i, \({\sum \nolimits }_{N\in {\Omega }_{N}}\vert {L}_{N}\vert \) is the total number of links in all network realizations, and f RN (t i )Δ i is the probability of the length of an arbitrary link being in (a i − 1, a i ). The Riemann sums in (3.24) can be written as
As such, we prove that \({N}_{d\_LCR} = {N}_{LCR}{P}_{LCR}\), i.e., the expected number of destroyed links under area attack is the same as the one under link-independent attacks.
From the above theorem it is clear that, in general, the attacks over different links are correlated in each area attack realization. However, the expected number of attacked links, which is averaged over all realizations, is only dependent on P LCR ; the attack correlation between the links is eliminated by the expectation operation.
3.2.2 The Traditional Random Network Case
Given the fact that the expected total number of links in the traditional random network is
the expected number of destroyed links by a randomly located area attack can be calculated as
The above simple relation follows a similar argument to that of Theorem 3.1.
3.2.3 The Fixed Network Case
In the previous analysis of the section, all the measures are expected values over all possible random realizations of the network topology. In the next, we fix a network realization, i.e., the geographic locations of all the links and the corresponding connectivity graph are fixed. We then derive the expected number of destroyed links only over the random realizations of the attack area. The following result is applicable to analyzing the performance of a given deterministic network under area attacks.
Let M be the number of destroyed links in a given network. The expectation of M over attack locations can be expressed as
where q i denotes the probability of a total of i links being attacked.
For a given link, we know that the link is attacked if the center of the attack area is within the corresponding olivary region around the link as shown in Fig. 3.1. Therefore, q i can be expressed as
where | S i | represents the sum area of all the overlapping parts intersected exactly by i olivary regions around all sets of i links. With the example shown in Fig. 3.2, for i = 1, | S 1 | denotes the sum area of the shadowed parts that each covers exactly one link, i.e., \(\vert {S}_{1}\vert = \vert {S}_{11}\vert + \vert {S}_{12}\vert + \vert {S}_{13}\vert + \vert {S}_{14}\vert \); for i = 2, \(\vert {S}_{2}\vert = \vert {S}_{21}\vert + \vert {S}_{22}\vert + \vert {S}_{23}\vert + \vert {S}_{24}\vert \); and for i = 3, | S 3 | = | S 31 | .
Substituting (3.31) into (3.30), we have
where N is the total number of links and T n denotes the area of the olivary region around link n.
For the example in Fig. 3.2, we now illustrate the details on how to calculate (3.32). Specifically, we have
Therefore, the expectation (over attack locations) of M in this given network realization can be expressed as
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Zhou, Q., Gao, L., Liu, R., Cui, S. (2013). Link-Level Network Robustness to Area Attacks. In: Network Robustness under Large-Scale Attacks. SpringerBriefs in Computer Science. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4860-0_3
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