2. SYSTEM RELIABLE PROBABILITY (SRP)

Invulnerability research on topological structures is important in reliability research on multi-AUV cooperative system networks (Albert et al., Reference Albert, Jeong and Barabasi2000). Attack (see Albert et al., Reference Albert, Jeong and Barabasi2000) information can be divided into three modes (Seyyedmohsen, Reference Seyyedmohsen2010; Chen and Zhang, Reference Chen and Zhang2006; Deng et al., Reference Deng, Wu, Lv, Li and Tan2008): zero information, complete information and incomplete information. Zero information attack occurs when the information of the network structure is not contained in the attack information and nodes are randomly targeted by the attack. Complete information attack refers to the situation in which the attack information contains all the network structure information, and the key nodes are preferentially attacked under a certain degree of importance criteria. In reality, zero and complete information attacks are two ideal extreme cases. In most cases, especially under complex marine environments and enemy interference attacks, the attack is neither a zero-information attack nor a complete information attack. Most attacks are incomplete information attacks. In other words, one part of the information is known, and the other part of the information is unknown.

A multi-AUV cooperative system network can be abstracted to a structural graph G=(V, E) (Schneider et al., Reference Schneider, Rainwater, Pohl, Hernandez and R-Marquez2013; Zuev et al., Reference Zuev, Wu and Beck2015), where $V=\{v_{1},v_{2},v_{3},\ldots,v_{N}\}$ represents a collection of individual AUVs that comprise the system. $E=\{e_{1},e_{2},e_{3},\ldots,e_{W}\}$ shows a collection of communication links between two individual AUVs. N=|V| is the number of individual AUVs in system G. W=|E| is the number of the communication links in G. d _i is defined as the communication degree of v _i, that is, the number of communication links of v _i, and 0<d _i<N. $\bar{k}=\frac{\sum_{i} d_{i}}{N}$ is defined as average communication degree of G, and $\bar{k}=\frac{2W}{N}$.

The individual AUVs of a multi-AUV cooperative system are rearranged by descending order according to their communication degree (Carlesi et al., Reference Carlesi, Michel, Jouvencel and Ferber2011). When the communication degree is the same, the arrangement is random. After rearrangement, the sequence number of v _i is r _i. Then, $\{d_{1}^{\prime},d_{2}^{\prime},\ldots,d_{r}^{\prime},\ldots,d_{N}^{\prime},\}$ is the communication degree sequence of G, where $d_{1}^{\prime} \ge d_{2}^{\prime} \ge \cdots \ge d_{N}^{\prime}$.

Under an incomplete information attack, a measurement of attack information is an important foundation and prerequisite of invulnerability modelling and analysis on multi-AUV cooperative systems. This study supposes a number of S AUVs are attacked. S=Nα, where α indicates the proportion of individual AUVs attacked, namely, the parameter of attack information range. Accordingly, a large value of α indicates that the information is extensive. Under a zero information attack (α=0, random failure), the attack information range is zero. Under a complete information attack (α=1, intended failure), the attack information range is the whole system.

For G, $\varepsilon_{i}$ is defined as the state of an individual AUV (v _i) with information obtained by the attacker. If the communication degree sequence number r _i of the individual AUV (v _i) has been acquired by the attacker, then $\varepsilon_{i}=0$; otherwise, $\varepsilon_{i}=1$. Depending on the state of an individual AUV with information that has been obtained (the value of $\varepsilon_{i}$), a collection of individual AUVs V of G can be divided into two parts: individual AUVs with information that has been acquired V _s (when $\varepsilon_{i}=1$) and individual AUVs with information that has not been obtained V _U (when $\varepsilon_{i}=0$).

The precision of attack information, combined with the sequence number r of individual AUV v _i, is considered. Then, an auxiliary variable is constructed as:

(1)$$\varphi_i=r_i^{-\delta}$$

where $\delta \in [0,\infty)$ is the parameter of the attack information accuracy. In a single sample, the probability of the individual AUV v _i to be drawn is:

(2)$$\eta_i=\frac{\varphi_i}{\sum_{t=1}^N\varphi_t}=\frac{r_i^{-\delta}}{\sum_{t=1}^N r_t^{-\delta}}$$

Evidently, a large value of δ indicates a high probability of obtaining the information of an individual AUV with a large communication degree, that is, the accuracy of attack information is high.

When δ=0, the probability of the individual AUV v _i to be drawn is $\eta_{i} =\frac{\varphi_{i}}{\sum_{t=1}^{N} \varphi_{t}}=\frac{r_{i}^{-\delta}}{\sum_{t=1}^{N} r_{t}^{-\delta}}=\frac{1}{N}$. In other words, the information of each individual AUV is acquired with equal probability. This kind of attack information is called random information.

When $\delta=\infty$, $\sum_{t=1}^{N} r_{t}^{-\infty}=\sum_{t=1}^{N} t^{-\infty}=1$. If r _j=1, then the probability of the individualAUV v _i to be drawn is $\eta_{i}=\frac{1}{\sum_{t=1}^{N} r_{t}^{-\infty}}=1$ when i=j; the probability of the individual AUV v _i to be drawn is $\eta_{i}=\frac{r_{i}^{-\infty}}{\sum_{t=1}^{N} r_{t}^{-\infty}}\frac{r_{i}^{-\infty}}{1^{-\infty}+2^{-\infty}+3^{-\infty}+ 4^{-\infty}+\cdots +r_{i}^{-\infty}}=\frac{0}{1+0+0+0+\cdots+0}=0$ when i ≠ j. Therefore, the information of an individual AUV with the largest communication degree is always acquired first. This kind of attack information is called priority information.

To avoid drawing out individual AUVs with a large communication degree repeatedly, that is, repeated attack, the acquisition process of the attack information is abstracted into an unequal probability sampling schedule without replacement (Ma and Lu, Reference Ma and Lu2009; Schillewaert et al., Reference Schillewaert, Langerak and Duhamel1998; Robson, Reference Robson2002).

First, an individual AUV sample is drawn-out in accordance with the probability of $\eta_{i}=\frac{\varphi_{i}}{\sum_{t=1}^{N} \varphi_{t}}=\frac{r_{i}^{-\delta}}{\sum_{t=1}^{N} r_{t}^{-\delta}}$. Second, the remaining AUVs are rearranged from a large to a small communication degree. The auxiliary variable φ_i and the probability of being drawn-out η_i are recalculated. Third, a sample is drawn-out on the basis of the newly calculated η_i. Finally, the second and third processes are repeated until S samples are drawn-out.

A measure for determining the network collapse is (Sun et al., Reference Sun, Liu and Ding2013b; Sun and Wang, Reference Sun and Wang2013; Gupta et al., Reference Gupta, Behnam, Lian, Estrada, Pop and Kumar2013):

(3)$$z=\frac{\sum_{k=m}^M k(k-1)p(k)q(k)}{\sum_{k=m}^M kp(k)}$$

where k is the communication degree of an individual AUV, p(k) is the communication degree distribution, q(k) is the unaffected probability of an individual AUV with communication degree denoted as k, m is the minimum communication degree in the system and M is the maximum communication degree. If z<1, then the multi-AUV cooperative system is in a state of collapse; if z>1, then the system is not in a state of collapse; if z=1, then the system is in a state of critical collapse.

When the portion of known information V _s is determined, the unaffected probability q(k) of an individual AUV with communication degree denoted as k can be determined. As long as the communication degree distribution p(k) is given, whether the multi-AUV cooperative system has collapsed or not can be determined.

Definition: System Reliable Probability (SRP) is defined as:

(4)$$\beta=\lim_{n\to \infty} \frac{b(n)}{n}\approx \frac{b(n)}{n}$$

where n is the number of random experiments under certain attack condition and information. b(n) is the number of experiments that will not collapse in n random experiments. If n is sufficiently large, then $\frac{b(n)}{n}$ will approach β.

3. OCEAN CURRENT INFLUENCE

An individual AUV of a multi-AUV cooperative system will inevitably be affected by ocean currents during sailing. For a relatively slow AUV (or one with no rapid speed requirement), ocean currents will considerably affect its navigation, positioning and communication with other AUVs. For example, an individual AUV executing a task within a system is usually positioned intermittently. Between two instances of positioning, the AUV usually uses inertial navigation to sail toward the target location. In the inertial navigation process, which is influenced by ocean currents, the AUV may deviate from the established track position prior to its next positioning, and the tracks of certain AUVs may exhibit large drift. Ultimately, the entire system will suffer from unexpected effects.

In oceanography studies, the speed of ocean currents in the east–west direction is usually expressed by the symbol u, whereas the speed of ocean currents in the north–south direction is usually expressed by the symbol v (Liang, Reference Liang2013); this rule is followed here. The relationship between a multi-AUV cooperative system and ocean currents is shown in Figure 1.

Figure 1. Relationship between Multi-AUV cooperative systems and ocean currents.

In Figure 1, S ₁ is the maximum operating area of the multi-AUV cooperative system, the shaded part S ₂ is the area where sea currents exist, u represents the velocity of currents in the east–west direction, v represents the velocity of currents in the north–south direction, and S ₃ is the minimum rectangular area that contains the multi-AUV cooperative system.

When the interfering factor of ocean current is considered alone, the attack proportion is:

(5)$$f=\frac{S_2 \cap S_3}{S_3}$$

The attack information range is:

(6)$$\alpha =\frac{S_2}{S_1}$$

When the interference of simple currents belongs to unconscious attacks, the attack information accuracy is δ=0.

For an AUV, the attack strength of the multi-AUV cooperative system subject to interference of ocean currents is mainly reflected by the noise interference flow. According to Blokhin Zaitsev's theory (Liu and Lei, Reference Liu and Lei2010), the attack strength can be obtained by:

(7)$$c=k\bar {v}^n$$

where k is a constant, n is an amount comprising AUV underwater linearity and other factors, and $\bar{v}$ represents the speed of ocean currents. Combined with the velocity u of the west–east direction and v of the north–south direction, $\bar{v}$ can be expressed as:

(8)$$\bar {v}=\sqrt {u^2+v^2}$$

Definition: $\rho=\rho_{0} {\cdot} e^{-C}$ is the probability that the multi-AUV collaborative system is still available when a single AUV is attacked, where C is the attack strength and indicates the degree of interference from the harsh marine environment or the strength of the hostile attack. The value range of C is (0,+∞). A large value of attack strength C indicates a high interference degree of the harsh marine environment or high intensity of the attack by the hostile forces. ρ₀ is the benchmark reliability of a single AUV, that is, the probability that a single AUV is available in the ideal case of no harsh marine environment interference or hostile attack.

4. EXAMPLE

A multi-AUV cooperative system of 13 AUVs is analysed as an example. Figure 2 shows the distribution of the 13 AUVs and ocean current. Figure 3 shows a multi-AUV cooperative system topology structure operating in a sea area of 16, 000×8, 000 m. The system comprises 13 AUVs, which will jointly perform a task. The coordinates of the 13 AUVs are presented as follows:

$$\begin{matrix} v_1:(12{,}300,750)\hfill &v_2:(1{,}950,1{,}950)&v_3:(2{,}400,1{,}050)&v_4:(9{,}600,1{,}200) \hfill \\ v_5:(14{,}400,4{,}200)&v_6:(14{,}700,6{,}150)&v_7:(14{,}550,6{,}300)&v_8:(7{,}350,4{,}050) \hfill \\ v_9:(13{,}650,2{,}700)&v_{10}:(12{,}300,750)&v_{11}:(4{,}200,7{,}500)&v_{12}:(13{,}800,4{,}950) \hfill \\ v_{13}:(8{,}250,7{,}050)&&& \end{matrix}$$

Figure 2. Distribution of AUVs and ocean current.

Figure 3. Two topological structure schemes.

The speeds of ocean current in the west–east and north–south directions are u=5 cm/s and v=2 cm/s, respectively.

Scheme A:

The interaction of the SRP with environmental factors centres on the attack proportion f, attack information range α, attack information accuracy δ and attack strength c. These parameters are in turn closely related to the effects of ocean currents.

The shaded area S ₂ in Figure 2 is the region with ocean current. The entire area of S ₁ $(16{,}000\times 8{,}000\,{\rm m})$ is the region where the multi-AUV cooperative system navigates. S ₃, which is surrounded by a dotted line, is the smallest rectangle that contains all the AUVs. From the data in Figure 2, the following data can be calculated:

$$\begin{align}& S_1=1{\cdot}04\times 10^8\,m^2,\quad S_2 =4{\cdot}864\times 10^7\,m^2,\\ & S_3=8{\cdot}118\times 10^7\,m^2,\quad S_2 \cap S_3 =2{\cdot}4735\times 10^7\,m^2.\end{align}$$

The attack proportion is $f=\frac{S_{2} \cap S_{3}}{S_{3}}=0{\cdot}30$. The attack information range is $\alpha =\frac{S_{2}}{S_{1}}=0{\cdot}47$. The current attacks are unconscious ones. The attack information accuracy is δ=0. The current speed is $\bar {v}=\sqrt{u^{2}+v^{2}} =0{\cdot}054\,\mbox{m/s}$. The attack strength is $c=k\bar{v}^{n}$, where $k=1{\cdot}2\times 10^{5}$ and n=2; thus, $c=k\bar {v}^{n}=349$.

For this multi-AUV cooperative system, f=0·3; thus, Nf=3·9. When rounded down, the nearest whole number of actual attacked AUVs is three. $N\alpha=6{\cdot}11$ when $\alpha=0{\cdot}47$. Nα can be rounded to six, as the number of AUVs within V _s. The standard reliability of an individual AUV is taken as $\rho_{0}=0{\cdot}5$. Thus, the reliability of an individual AUV is $\rho =\rho_{0} {\cdot} e^{-c}\approx 0$. Therefore, an individual AUV under attack is nearly completely unreliable.

Data after a large number of randomised simulations can be obtained, as shown in Table 1, in accordance with the standards of disintegration and collapse.

Table 1. Randomised simulation data of reliable and collapsed systems.

The System Reliable Probability (SRP) of Scheme A is 0·8675.

Scheme B:

Similar to Scheme A, λ, α, δ, and c are defined.

Under the same ocean environment of Scheme A, the following data can be the same:

$$\begin{align}& S_1,\quad S_2, \quad S_3, \quad S_2 \cap S_3,\quad \textit{f},\quad \alpha=\frac{S_2}{S_1}, \quad \delta, \quad \bar {v}=\sqrt {u^2+v^2},\quad c=k\bar {v}^n,\\ &\textit{Nf}, \quad N\alpha, \quad \rho =\rho_0 \cdot e^{-c}.\end{align}$$

Data after a large number of randomised trials can be obtained, as shown in Table 2, in accordance with the standards of disintegration and collapse.

Table 2. Randomised simulation data of reliable and collapsed systems.

The System Reliable Probability (SRP) of Scheme B is 0·8921.

The SRP of Scheme A is lower than that of Scheme B. In other words, the invulnerability of Scheme A is low when nodes are influenced by the ocean current environment, as shown in Figure 2. Under the same current environment as in Figure 2, the structure of Scheme B is more reliable than that of Scheme A.

The influence of the same current environment on different structures of the multi-AUV cooperative systems differs. This result provides a reference for the structure selection of multi-AUV systems.