Evaluation of Connectivity Reliability in MANETs Considering Link Communication Quality and Channel Capacity

Abstract

Mobile Ad Hoc Networks (MANETs) exhibit diverse deployment forms, such as unmanned swarms, mobile wireless sensor networks (MWSNs), and Vehicular Ad Hoc Networks (VANETs). While providing significant social application value, MANETs also face the challenge of accurately and efficiently evaluating connectivity reliability. Building on existing studies—which mostly rely on the assumptions of imperfect nodes and perfect links—this paper comprehensively considers link communication quality and channel capacity, and extends the imperfect link assumption to analyze and evaluate the connectivity reliability of MANETs. The Couzin-leader model is used to characterize the ordered swarm movement of MANETs, while various probability models are employed to depict the multiple actual failure modes of network nodes. Additionally, the Free-Space-Two-Ray Ground (FS-TRG) model is introduced to quantify link quality and reliability, and the probability of successful routing path information transmission is derived under the condition that channel capacity follows a truncated normal distribution. Finally, a simulation-based algorithm for solving the connectivity reliability of MANETs is proposed, which comprehensively considers node characteristics and link states. Simulation experiments are conducted using MATLAB R2023b to verify the effectiveness and validity of the proposed algorithm. Furthermore, the distinct impacts of link communication quality and channel capacity on the connectivity reliability of MANETs are identified, particularly in terms of transmission quality and network lifetime.

1. Introduction

With reference to the theory of system reliability for multi-state components and multi-state systems, only components in intact or intermediate states that meet specific requirements or thresholds can perform their designated functions. Meanwhile, the state of these components must not deteriorate to a level below these requirements or thresholds during the specified task period [1,2,3]. This also applies to MANETs such as unmanned swarms and mobile wireless sensor networks: network links, as the medium for information transmission, have multiple intermediate states (in addition to the connected state $\left(e=1\right)$ and disconnected state $\left(e=0\right)$ [4]), each of which corresponds to a distinct communication quality. Not all established link states that are connected meet the requirements for information transmission. For links that are connected but fail to meet transmission quality requirements, they should still be classified as “non-working” or “failed” from the perspective of network reliability. For communication links that meet the transmission requirements, it is necessary to ensure that even if degraded operation occurs during the entire information transmission process from the source node to the sink node, the transmission quality will not fall below the requirements. It can be seen that the quality of links in MANETs is crucial for connectivity reliability evaluation and requires further analysis.

In the binary model or deterministic model of communication links [5,6,7,8], a link with a reliability of 1 is considered established when the Euclidean distance between any two nodes is less than the communication distance threshold, thereby achieving connectivity between the nodes. However, in practical applications, even if two MANET nodes are within each other’s transmission range or radio coverage, the signal of information transmission over the communication link will still be attenuated by factors such as noise, fading, and interference, leading to a decrease in link communication quality and the probability of successful communication. In severe cases, the communication function may be deemed unachievable [6], rendering the link “unreliable”. It can be seen that the assumption of the binary model of communication links will lead to an overestimation of the analysis and evaluation results of the connectivity reliability of MANETs, further affecting the deployment and optimization decisions of MANETs. In addition, the mobility of nodes within MANETs leads to frequent switching between link disconnection $\left(e=0\right)$ and establishment $\left(e=1\right)$, which also greatly impacts information transmission. Communication links established between nodes $N_i$ and $N_j$ at a given moment $t_1$ may disappear at another moment $t_2$ due to node movement, at which point the Euclidean distance $d_{ij}$ between them exceeds the communication distance threshold. If information transmission and reception between nodes $N_i$ and $N_j$ remain uncompleted within the time interval $\left[t_1,\hspace{0.33em}{t}_2\right]$, the link established at time $t_1$ cannot be deemed “reliable”, since it fails to perform its intended communication function. Based on the above analysis, in the actual deployment and application of MANETs, mobility characteristics will change the communication coverage relationship between network nodes in real time. Due to factors such as noise, fading, or interference, even when nodes are within each other’s transmission range, the probability of successful communication decreases as signal strength weakens, and communication failure may even occur. Given a certain amount of information processing capacity, channel capacity directly affects the receiving or transmitting duration of network nodes. When this data reception or transmission duration (determined by channel capacity) exceeds the link’s lifetime, communication failure can also result. Figure 1 illustrates the state classification of network links and routing paths under the influence of various factors.

Therefore, it is necessary to consider link quality and channel capacity, and adopt more accurate models to model and analyze the connectivity reliability of MANETs. Compared with existing studies, this study mainly focuses on the impacts of imperfect nodes and imperfect links on the connectivity reliability of MANETs. Specifically, the research objects are MANETs deployed and applied in clusters, such as VANETs, WSNs, and UAV swarms. These networks are widely used in scenarios such as disaster rescue, emergency communication networking, and surveillance patrols, with hierarchical leadership and various interaction rules within the network. The imperfect node assumption mainly considers four types of practical node failure modes: hardware/software failure, location failure, energy consumption failure, and isolation failure, each characterized by a corresponding failure model. The imperfect link assumption primarily accounts for information transmission failures in network links and routing paths: when signal attenuation or other factors prevent normal information transmission between nodes within the communication coverage of a link, the Free-Space-Two-Ray Ground (FS-TRG) model is introduced to quantify link quality and reliability; when a routing path (corresponding to multi-hop links) or a single link (corresponding to single-hop links) fails to complete information transmission within its lifetime due to limited channel capacity, the probability of successful information transmission under a given channel capacity is derived. The main contributions of this paper are as follows:

• Designing a simulation algorithm to calculate the probability of successful information transmission in MANETs under a specific channel capacity;
• Extending the existing assumptions of imperfect nodes and perfect links, comprehensively analyzing the impacts of internal and external network factors (e.g., fast node mobility, node failure rules, node information processing load, environment, and channel capacity), and designing a simulation-based algorithm for solving MANET connectivity reliability;
• Investigating, through simulation experiments and comparative result analysis, the influence mechanisms and laws of link quality and channel capacity on MANET connectivity reliability; exploring the correlation between node failure modes and the probability of successful information transmission in the network; and studying the correlation between node failure occurrence rules and the number of links/routing paths.

2. Link Communication Quality and Reliable Communication Distance

2.1. Analysis of Link Communication Quality

Based on the binary model of communication links [9], scholars are increasingly delving into link communication quality and reliability [10], conducting analyses from various perspectives. Link quality refers to the reliability and stability of wireless communication links between mobile nodes, specifically manifested as the effectiveness of signal transmission, including indicators such as signal strength, bit error rate, and transmission delay [9]. In MANETs, link quality directly determines whether data can be successfully and efficiently transmitted from the source node to the sink node. It is generally influenced by physical layer factors, network dynamics, and environmental factors. Physical layer factors include signal attenuation (obstruction by obstacles, multipath fading), interference (co-channel signal interference), channel noise, and other factors that degrade signal quality. Network dynamics involve frequent disconnections or reconstruction of links due to high-speed node movement, as well as changes in node density (sparse or dense scenarios) that affect connection stability. Environmental factors include weather conditions (such as rain, fog) that affect wireless signal propagation [7]. In MANETs, network connection disturbances caused by rapid node movement are considered one of the most challenging issues [6]. That is, frequent changes in network connections due to node movement are one of the main reasons for wireless link failures. The characteristic of node movement first results in the time-varying topology of MANETs, which is neither stable nor fixed but constantly changing. The intuitive manifestation of the topological structure between nodes is the link, which is frequently established or dissolved due to node movement, limiting the communication connectivity between nodes, especially between long-distance nodes [11]. When nodes move at high speed, the hop count between nodes also changes dramatically, and an increase in hop count often leads to an increased likelihood of data loss, thereby affecting network performance and reducing the probability of successful communication in a MANET [5].

Shinde et al. [9] used Signal-to-Noise Ratio (SNR) and Packet Delivery Ratio (PDR) as indicators to quantify link communication quality. Combining network topology, they optimized communication channel selection to avoid path failures and reduce transmission delays, thereby improving data transmission efficiency and speed. Regragui et al. [5] addressed the issue of network connectivity disturbances caused by vehicle movement, setting indicators such as Packet Delivery Ratio (PDR), End-to-End Delay (EED), Path Length, and Link Duration (LD) to quantify network performance. Mkongwa et al. [12] defined link reliability as the ratio of the standard deviation to the mean distance of node distribution. Lu et al. [13] analyzed the unreliability of links in wireless mobile ad hoc networks through the indicator of link transmission success rate, focusing on the retransmission phenomenon. Babu et al. [6] defined the duration of channel existence as link stability, quantifying it by calculating the Link Expiration Time (LET), identifying high-speed node movement as the main influencing factor, and optimizing the selection of relay nodes and links based on this metric. Shelly et al. [14] defined the effective transmission distance of a link by combining path loss, shadow fading, and multipath effects: the distance at which the probability of received power exceeding a set threshold is 99%.

2.2. Analysis of Reliable Communication Distance in MANETs

There are two ways for signal/communication propagation between nodes in MANETs [15,16]. The first is the possibility of a direct connection between the source node and the sink node, known as free-space propagation (FS) or line-of-sight (LOS) propagation. FS is suitable for short-range communication, in which the source node communicates with other nodes via wireless channels. The other is two-ray ground propagation (TRG), also known as non-line-of-sight (NLOS) propagation: when signals encounter obstacles, they reach the destination node via multiple routes, making TRG suitable for long-range communication. To precisely analyze the wireless signal propagation process within mobile ad hoc networks and evaluate their network performance, the paper further introduces the Free-Space-Two-Ray Ground (FS-TRG) propagation model, rather than the traditional binary communication model, to assess the reliability of links between nodes. Based on the reasonable assumptions of Liao et al. [17], this paper temporarily does not consider the mutual interference between nodes.

Based on the aforementioned analysis, the link reliability model primarily depends on the propagation quality of the signal, which is defined as a function of distance. At this point, the reliability of links between nodes $N_i$ and $N_j$ can be modeled as follows:

$$R_e\left(d_{ij}\left(t\right)\right)=\left\{\begin{array}{cc}1,\hfill & d_{ij}\left(t\right)\le \alpha d_{Th}\hfill \\ \left({\displaystyle \frac{{\alpha }^2}{1-{\alpha }^2}}\right)\left({\displaystyle \frac{d_{Th}^2}{d_{ij}^2\left(t\right)}}-1\right),\hfill & \alpha d_{Th}\le d_{ij}\left(t\right)\le \beta d_{Th}\hfill \\ \left({\displaystyle \frac{{\alpha }^2{\beta }^2}{\left(1-{\alpha }^2\right)\left(1+{\beta }^2\right)}}\right)\left({\displaystyle \frac{d_{Th}^4}{d_{ij}^4\left(t\right)}}-1\right),\hfill & \beta d_{Th}\le d_{ij}\left(t\right)\le d_{Th}\hfill \\ 0,\hfill & d_{ij}\left(t\right)\ge d_{Th}\hfill \end{array}\right.$$

(1)

In the formula, $d_{ij}\left(t\right)$ represents the Euclidean distance between nodes $N_i$ and $N_j$ at time $t$; $\alpha$ and $\beta$ are propagation model parameters, satisfying $0<\alpha <\beta <1$; $d_{Th}$ represents the node communication distance threshold, which is the radio coverage range of the node; $R_e\left(d_{ij}\left(t\right)\right)$ represents the reliability of the communication link between nodes $N_i$ and $N_j\left(i\ne j\right)$ at time $t$, with a value ranging from 0 to 1. According to the FS-TRG model, link reliability is divided into stage functions based on the 4 Euclidean distances between adjacent nodes, where:

(1)

Short-range ($d_{ij}\left(t\right)\le \alpha d_{Th}$): Link reliability is 1;

(2)

Medium range ($\alpha d_{Th}\le d_{ij}\left(t\right)\le \beta d_{Th}$): The signal adopts Free Space Propagation (FS) mode, and the reliability of the link is inversely proportional to $d_{ij}^2\left(t\right)$;

(3)

Long-range ($\beta d_{Th}\le d_{ij}\left(t\right)\le d_{Th}$): The signal employs the Two Ray Ground Propagation (TRG) method, and the reliability of the link is inversely proportional to $d_{ij}^4\left(t\right)$;

(4)

Exceeding communication distance threshold ($d_{ij}\left(t\right)\ge d_{Th}$): The nodes are beyond each other’s radio transmission range, and the link reliability is 0. The graphical representation of the link reliability model is shown in Figure 2.

The weighted adjacency matrix $A_R\left(t\right)$ between network nodes can be generated from the link reliability model. $A_R\left(t\right)$ is a two-dimensional $m\times m$ matrix, where the weight of each element in the matrix equals the reliability of the corresponding link

$$A_R\left(t\right)=\left[\begin{array}{cccc}{a}_{R11}\left(t\right)& a_{R12}\left(t\right)& \dots & a_{R1m}\left(t\right)\\ a_{R21}\left(t\right)& a_{R22}\left(t\right)& \dots & a_{R2m}\left(t\right)\\ ⋮& ⋮& \ddots & ⋮\\ a_{Rm1}\left(t\right)& a_{Rm2}\left(t\right)& \dots & a_{Rmm}\left(t\right)\end{array}\right]$$

(2)

The element $a_{Rij}\left(t\right)=R_e\left(d_{ij}\left(t\right)\right)$ in the matrix represents the reliability of the communication links established between nodes $N_i$ and $N_j$ at time $t$ with distance $d_{ij}\left(t\right)$, and based on the assumption of communication link symmetry, $a_{Rij}=a_{Rji}$ holds. $a_{Rji}\left(t\right)=0$ or $a_{Rij}=0 \left(i\ne j\right)$ indicates that nodes $N_i$ and $N_j$ are not connected at time $t$, and node self-loops are not considered, then:

$$a_{Rij}\left(t\right)=\left\{\begin{array}{l}{R}_e\left(d_{ij}\left(t\right)\right), i\ne j\\ 0, \mathrm{else} \end{array}\right.$$

(3)

The paper sets a link reliability threshold $R_e^{Th}$ [18] to further determine the status of established links between nodes within the MANETs at any given moment $t$. It stipulates that only when the link reliability is greater than or equal to $R_e^{Th}$ is the link deemed reliably connected—recorded as event $E^e\left(t\right)$. Otherwise, the link is considered to be disconnected (unreliable) due to failure to meet communication quality requirements. This process takes into account the signal randomness in transmission. Through the logical operations $A_R\left(t\right)\ge R_e^{Th}$, a binary adjacency matrix $A_{RB}\left(t\right)$ that considers the communication quality of links in the MANETs at any given moment $t$ is generated. Each element in the matrix $A_{RB}\left(t\right)$ is compared with $R_e^{Th}$: if $a_{Rij}\left(t\right)\ge R_e^{Th}$, $a_{RBij}\left(t\right)=1$; otherwise, $a_{RBij}\left(t\right)=0$. Thus, the binary adjacency matrix is defined as:

$$A_{RB}\left(t\right)=\left[\begin{array}{cccc}{a}_{RB11}\left(t\right)& a_{RB12}\left(t\right)& \dots & a_{RB1m}\left(t\right)\\ a_{RB21}\left(t\right)& a_{RB22}\left(t\right)& \dots & a_{RB2m}\left(t\right)\\ ⋮& ⋮& \ddots & ⋮\\ a_{RBm1}\left(t\right)& a_{RBm2}\left(t\right)& \dots & a_{RBmm}\left(t\right)\end{array}\right]$$

(4)

$$a_{RBij}\left(t\right)=\left\{\begin{array}{l}1, a_{Rij}\left(t\right)\ge R_e^{Th}\\ 0, \mathrm{else} \end{array}\right.$$

(5)

3. Channel Capacity and Information Transmission Time

Channel capacity, a core concept in information theory proposed by Claude Shannon in 1948, is defined as the maximum theoretical rate of error-free information transmission under specific channel conditions (e.g., bandwidth and signal-to-noise ratio) [6,19]. In MANETs, channel capacity is determined by bandwidth, signal-to-noise ratio, and channel fading characteristics [20]. Its time-varying nature stems from channel fading (e.g., fast fading and Doppler frequency shift) caused by node movement, as well as from dynamic changes in network topology. Information transmission time encompasses the end-to-end delay of data from the source node to the destination node, including transmission delay (the ratio of data volume to transmission rate), propagation delay, queuing delay, and processing delay introduced by route switching [21]. In multi-hop transmission scenarios, the spatial distribution of channel capacity—specifically, capacity differences between consecutive hop links—creates transmission bottlenecks (i.e., the minimum capacity among hops limits the overall transmission rate). The paper further identifies information transmission delay as the primary factor influencing information transmission time in MANETs, and ignores propagation delay and queuing delay to focus on the impact of channel capacity on both the probability of successful information transmission and the connectivity reliability of MANETs. In this paper, based on the experimental data on unmanned aerial vehicle (UAV) communication by Liu et al. [22] as well as references [20,23], to simplify the analysis, the channel capacity at a specific distance is modeled as a truncated normal random variable.

3.1. Calculation of Information Transmission Time

At a given moment $t$, for a MANET with channel capacity $C\left(t\right)$ (unit: bps, $C\left(t\right)>0$ for error-free transmission), the transmission time for $l$-bit data over a single link $e_{ij}$ between nodes $N_i$ and $N_j$ is:

$$T_C^l\left(t\right)={\displaystyle \frac{l}{C\left(t\right)}}$$

(6)

where $T_C^l\left(t\right)$ denotes the single-link transmission time, and $l$ is the data volume sent by the source node (unit: bits).

For an $i$-hop routing path (composed of $i$ consecutive links), the formula for calculating the duration $T_{C,i}^l\left(t\right)$ of transmitting $l$ unit of data from the source node to the sink node is:

$$T_{C,i}^l\left(t\right)=i\times T_C^l\left(t\right)={\displaystyle \frac{i\times l}{C\left(t\right)}}$$

(7)

When $i=1$ (single-hop transmission), $T_{C,i}^l\left(t\right)$ reduces to $T_C^l\left(t\right)$.

3.2. Probability Density Function of Channel Capacity

Assuming that the channel capacity $C\left(t\right)\left(C\left(t\right)>0\right)$ within a MANET follows a truncated normal distribution $N\left(\mu ,{\sigma }^2\right)$, where the lower bound is 0 and there is no upper bound ($+\infty$), its probability density function (PDF) is:

$$f_C\left(c\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}\exp \left(-{\displaystyle \frac{{\left(c-\mu \right)}^2}{2{\sigma }^2}}\right)\cdot {\displaystyle \frac{1}{\mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}},\hfill & c>0\hfill \\ 0,\hfill & else\hfill \end{array}\right.$$

(8)

In the formula, $\mathsf{\Phi }\left(•\right)$ denotes the standard normal cumulative distribution function, defined as $\mathsf{\Phi }\left(x\right)={\displaystyle {\int }_{-\infty }^x\frac{1}{\sqrt{2\pi }}}\exp \left(-{\displaystyle \frac{t^2}{2}}\right)\mathrm{d}t$; $\mu >0$ and ${\sigma }^2$ respectively represent the mean and variance of channel capacity, with values determined according to network deployment and actual environment; $\mathsf{\Phi }\left({\displaystyle \frac{\mu }{\sigma }}\right)$ is a normalization constant.

3.3. Step-by-Step Derivation of Information Transmission Time PDF

To obtain the PDF of $T_{C,i}^l\left(t\right)$ (denoted as $A$ for simplicity), we use random variable transformation—a standard technique to derive the distribution of a function of a random variable. The derivation is split into 6 explicit steps with intermediate explanations:

Step 1: Define the Transformation Function

Let $k=i\times l$ (a constant determined by path hop count $i$ and data volume $l$). The transmission time $A=T_{C,i}^l$ is a function of channel capacity $C$:

$$A=g\left(C\right)={\displaystyle \frac{i\times l}{C}}={\displaystyle \frac{k}{C}}, k>0,C>0$$

(9)

Step 2: Derive the Inverse Transformation

The function $g\left(C\right)={\displaystyle \frac{k}{C}}$ is strictly monotonically decreasing (with a constant negative derivative) on $C>0$, and its inverse function is:

$$C=g^{-1}\left(A\right)={\displaystyle \frac{k}{A}}$$

(10)

At the time when $A=a$, $C={\displaystyle \frac{k}{a}},a>0$.

Step 3: Calculate the Absolute Derivative of the Inverse Function

Calculate the derivative of the inverse function $g^{-1}\left(A\right)$:

$${\displaystyle \frac{\mathrm{d}C}{\mathrm{d}A}}={\displaystyle \frac{\mathrm{d}}{\mathrm{d}A}}\left({\displaystyle \frac{k}{A}}\right)=-{\displaystyle \frac{k}{A^2}}$$

(11)

Take the absolute value:

$$\left|{\displaystyle \frac{\mathrm{d}C}{\mathrm{d}A}}\right|={\displaystyle \frac{k}{A^2}}\left(k>0,A>0\right)$$

(12)

Step 4: Apply the Random Variable Transformation Formula

The general formula for the PDF of a transformed random variable $A=g\left(C\right)$ is:

$$f_A\left(a\right)=f_C\left(g^{-1}\left(a\right)\right)\cdot \left|{\displaystyle \frac{\mathrm{d}C}{\mathrm{d}A}}\right|$$

(13)

Substitute $g^{-1}\left(a\right)={\displaystyle \frac{k}{a}}$ and $\left|{\displaystyle \frac{\mathrm{d}C}{\mathrm{d}A}}\right|={\displaystyle \frac{k}{a^2}}$:

$$f_A\left(a\right)=f_C\left({\displaystyle \frac{k}{a}}\right)\cdot {\displaystyle \frac{k}{a^2}}$$

(14)

Step 5: Substitute the Channel Capacity PDF

Due to $c={\displaystyle \frac{k}{a}}>0$, substitute the truncated normal probability density function:

$$f_C\left({\displaystyle \frac{k}{a}}\right)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma \cdot \mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\exp \left(-{\displaystyle \frac{{\left(\frac{k}{a}-\mu \right)}^2}{2{\sigma }^2}}\right)$$

(15)

Therefore,

$$f_A\left(a\right)=\left[{\displaystyle \frac{1}{\sqrt{2\pi }\sigma \cdot \mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\exp \left(-{\displaystyle \frac{{\left(\frac{k}{a}-\mu \right)}^2}{2{\sigma }^2}}\right)\right]\cdot {\displaystyle \frac{k}{a^2}}$$

(16)

Step 6: Simplify the Exponent Term and Substitute $k=i\times l$

Simplify the exponential expression:

$${\left({\displaystyle \frac{k}{a}}-\mu \right)}^2={\displaystyle \frac{{\left(k-\mu a\right)}^2}{a^2}}$$

(17)

Substitute $k=i\times l$ into the equation. The final PDF of the $i$-hop information transmission time $A=T_{C,i}^l\left(t\right)$ is:

$$f_A\left(a\right)={\displaystyle \frac{i\times l}{\sqrt{2\pi }\sigma a^2\cdot \mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\exp \left(-{\displaystyle \frac{{\left(i\times l-\mu a\right)}^2}{2{\sigma }^2{a}^2}}\right), a>0$$

(18)

The probability density function of information transmission time $T_{C,i}^l\left(t\right)$ for the $i$-hop path is:

$$f_{T_{C,i}^l}\left(t\right)={\displaystyle \frac{i\times l}{\sqrt{2\pi }\sigma t^2\cdot \mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\exp \left(-{\displaystyle \frac{{\left(i\times l-\mu t\right)}^2}{2{\sigma }^2{t}^2}}\right), t>0$$

(19)

4. Link Lifetime and Path Lifetime

Based on the aforementioned analysis, only when the “lifetime” of the link between node $N_i$ and $N_j$ is greater than or equal to the time $T_C^l\left(t\right)$ required for information transmission between these two nodes, does the link possess the conditions to perform its communication function and is it considered a reliable link. For mobile MANET nodes, when communication quality requirements are satisfied, the “lifetime” of link $e_{ij}$ refers to its normal operating time—i.e., the duration during which the link can effectively transmit data. Relevant studies have referred to this duration by various terms, including link lifetime [24], residual link lifetime [25], link duration [5], link residual lifetime [14], and link expiration time [6], among others. Despite the different naming conventions, there is a certain similarity in definition: terms such as “lifetime”, “survival time”, and “working time”—all referring to the same concept for inter-node links—denote the time interval from link establishment to disconnection—i.e., the duration for which two adjacent mobile nodes maintain communication connectivity. This paper uses “link lifetime” to describe the operating time of a link and denotes the survival time of link $e_{ij}$ as $T_{ij}^L$.

According to the analytical study on VANETs by Babu et al. [6], the formula for calculating the survival time $T_{ij}^L$ of a link $e_{ij}$ is:

$$\begin{array}{l}{T}_{ij}^L={\displaystyle \frac{-\left(a_2{b}_2+a_1{b}_1\right)+\sqrt{\left(a_2^2+a_1^2\right)d_{Th}^2-{\left(a_2{b}_1-b_2{a}_1\right)}^2}}{\left(a_2^2+a_1^2\right)}}\\ \left\{\begin{array}{l}{a}_1=v_i\sin {\theta }_i-v_j\sin {\theta }_j\\ a_2=v_i\cos {\theta }_i-v_j\cos {\theta }_j\\ b_1=y_i-y_j\\ b_2=x_i-x_j\end{array}\right.\end{array}$$

(20)

In the formula, $a_1$ denotes the relative speed difference between $N_i$ and $N_j$ along the $y$-axis; $a_2$ denotes the relative speed difference between $N_i$ and $N_j$ along the $x$-axis; $b_1$ denotes the distance difference between $N_i$ and $N_j$ along the $y$-axis; $b_2$ denotes the distance difference between $N_i$ and $N_j$ along the $x$-axis. From the formula, we can derive that if $N_i$ and $N_j$ travel in the same direction at the same speed (i.e., the relative speed is zero), then $a_1=a_2=0$, and the link lifetime $T_{ij}^L$ tends to infinity, and the link $e_{ij}$ remains connected at all times; if the relative speed between the $N_i$ and $N_j$ is non-zero or their directions differ, the lifetime $T_{ij}^L$ of link $e_{ij}$ will decrease as the relative speed increases, and the link will eventually be disconnected; at the same time, if the initial distance between the $N_i$ and $N_j$ is closer, and the motion trend remains within the communication distance threshold $d_{Th}$, then the survival time of link $e_{ij}$ will be higher.

Let the path $P_i$ be represented as a set of links $P_i=\left\{e_{ij}|i\ne j,e_{ij}\in E\right\}$, $P_i\subseteq E$, that is, the path $P_i$ is a subset of the edge set of the MANET. We further introduce the concept of path lifetime [25,26], which is defined as the time interval from the formation of a path (composed of several links) to its interruption caused by the disconnection of any constituent link. This lifetime is denoted as $T_{P_i}^L$ for path $P_i$. By definition, the lifetime of path $P_i$ is determined by the minimum lifetime of its constituent links—specifically, $\min \left\{T_e^L|e\in P_i\right\}$. This is because the path is interrupted as soon as any constituent link disconnects, making the shortest link lifetime the bottleneck for path survival:

$$T_{P_i}^L=\min \left\{T_e^L|e\in P_i\right\}$$

(21)

5. Available Duration and Transmission Success Probability of Routing Transmission

When an $i$-hop path $P_k$ is selected as the routing path for transmitting information from the source node $N_s$ to the sink node $N_t$, the available duration for routing-based information transmission—denoted as $T_{ava}^l$—is equal to the lifetime of the path $P_k$. For an $i$-hop routing path $P_k$, the necessary and sufficient condition for successful transmission of $l$-bit of information is $T_{ava}^l\ge T_{C,i}^l\left(t\right)$, where $T_{C,i}^l\left(t\right)$ denotes the information transmission time of the $i$-hop path. This is because the information be successfully transmitted only if every link in the routing path $P_k$ remains connected when data is sent from the source node to the sink node. At this point, the success probability of an $i$-hop routing path $P_k$ is equivalent to the probability that the available duration is longer than the transmission time, expressed as:

$$\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)=\mathrm{Pr}\left(T_{ava}^l\ge T_{C,i}^l\left(t\right)\right)=\mathrm{Pr}\left(T_{P_k}^L\ge T_{C,i}^l\left(t\right)\right)$$

(22)

In the formula, $E_{P_k}^{trans}=1$ denotes the event that path $P_k$ is selected as the routing path and successfully transmits information from the source node to the sink node; $E_{P_k}^{trans}=0$ denotes the event that routing path $P_k$ fails to do so, and it satisfies:

$$\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)+\mathrm{Pr}\left(E_{P_k}^{trans}=0\right)=1$$

(23)

When transmitting information, the shortest routing path $P_k$ selected for transmission has only two possible outcomes: success ($E_{P_k}^{trans}=1$) or failure ($E_{P_k}^{trans}=0$). Furthermore, based on the aforementioned analysis, the probability density function (PDF) of the $i$-hop information transmission time $T_{C,i}^l\left(t\right)$ is:

$$f_{T_{C,i}^l}\left(t\right)={\displaystyle \frac{i\times l}{\sqrt{2\pi }\sigma t^2\cdot \mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\exp \left(-{\displaystyle \frac{{\left(i\times l-\mu t\right)}^2}{2{\sigma }^2{t}^2}}\right), t>0$$

(24)

The probability of successful transmission over an $i$-hop routing path $P_k$ can be expressed as:

$$\begin{array}{l}\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)=\mathrm{Pr}\left(T_{C,i}^l\left(t\right)\le T_{P_k}^L\right)\\ ={\displaystyle {\int }_0^{T_{P_k}^L}{f}_{T_{C,i}^l}\left(t\right)\mathrm{d}t}={\displaystyle {\int }_0^{T_{P_k}^L}\frac{i\times l}{\sqrt{2\pi }\sigma t^2\mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}\exp \left(-\frac{\left(i\times l-\mu t\right)}{2{\sigma }^2{t}^2}\right)\mathrm{d}t}\\ ={\displaystyle \frac{\mathsf{\Phi }\left(\frac{\mu -\frac{i\times l}{T_{P_k}^L}}{\sigma }\right)}{\mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\end{array}$$

(25)

The simplification process of the above formula is as follows:

(1)

Variable substitution

Let $h={\displaystyle \frac{i\times l}{t}}$ (where $t>0$), so $i\times l>0$. As $t\to 0^{+}$, $h\to +\infty$; When $t=T_{P_k}^L$, $h={\displaystyle \frac{i\times l}{T_{P_k}^L}}$. The differential relationship is:

$$t={\displaystyle \frac{i\times l}{h}}$$

(26)

$$\mathrm{d}t=-{\displaystyle \frac{i\times l}{h^2}}\mathrm{d}h$$

(27)

When $t$ varies from 0 (minimum transmission time) to $T_{P_k}^L$ (maximum available time), $h$ varies from $+\infty$ to ${\displaystyle \frac{i\times l}{T_{P_k}^L}}$—this transformation converts the integral over $t$ into an integral over $h$, which reduces computational complexity. Swap the limits of integration to eliminate the negative sign:

$${\displaystyle {\int }_0^{T_{P_k}^L}\dots \mathrm{d}t}={\displaystyle {\int }_{\infty }^{\frac{i\times l}{T_{P_k}^L}}\left(-\frac{i\times l}{h^2}\right)\mathrm{d}h}={\displaystyle {\int }_{\frac{i\times l}{T_{P_k}^L}}^{\infty }\frac{i\times l}{h^2}\mathrm{d}h}$$

(28)

(2)

Substitute the integrand

Substitute $t={\displaystyle \frac{i\times l}{h}}$ and $\mathrm{d}t={\displaystyle \frac{i\times l}{h^2}}\mathrm{d}h$ to obtain:

$$f_{T_{C,i}^l}\left(t\right)\mathrm{d}t={\displaystyle \frac{i\times l}{\sqrt{2\pi }\sigma \mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\cdot {\displaystyle \frac{1}{t^2}}\cdot \exp \left(-{\displaystyle \frac{{\left(i\times l-\mu t\right)}^2}{2{\sigma }^2{t}^2}}\right)\cdot {\displaystyle \frac{i\times l}{h^2}}\mathrm{d}h$$

(29)

among which:

$${\displaystyle \frac{1}{t^2}}={\displaystyle \frac{h^2}{{\left(i\times l\right)}^2}}$$

(30)

The integrand simplifies to:

$$\begin{array}{l}{\displaystyle \frac{i\times l}{\sqrt{2\pi }\sigma \mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\cdot {\displaystyle \frac{h^2}{{\left(i\times l\right)}^2}}\cdot {\displaystyle \frac{i\times l}{h^2}}\cdot \exp \left(-{\displaystyle \frac{{\left(i\times l-\mu \cdot \frac{i\times l}{h}\right)}^2}{2{\sigma }^2{\left(\frac{i\times l}{h}\right)}^2}}\right)\mathrm{d}h\\ ={\displaystyle \frac{1}{\sqrt{2\pi }\sigma \mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\cdot \exp \left(-{\displaystyle \frac{{\left(h-\mu \right)}^2}{2{\sigma }^2}}\right)\mathrm{d}h\end{array}$$

(31)

And there are:

$$f_{T_{C,i}^l}\left(t\right)\mathrm{d}t={\displaystyle \frac{1}{\sqrt{2\pi }\sigma \mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\cdot \exp \left(-{\displaystyle \frac{{\left(h-\mu \right)}^2}{2{\sigma }^2}}\right)\mathrm{d}h$$

(32)

(3)

Simplification of integrals

The integral transform becomes:

$$\begin{array}{l}\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)=\mathrm{Pr}\left(T_{C,i}^l\left(t\right)\le T_{P_k}^L\right)\\ ={\displaystyle {\int }_{\frac{i\times l}{T_P^L}}^{\infty }\frac{1}{\sqrt{2\pi }\sigma \mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}\exp \left(-\frac{{\left(h-\mu \right)}^2}{2{\sigma }^2}\right)\mathrm{d}h}\\ ={\displaystyle \frac{1}{\mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}{\displaystyle {\int }_{\frac{i\times l}{T_{P_k}^L}}^{\infty }\varphi \left(h\right)\mathrm{d}h}\\ ={\displaystyle \frac{1}{\mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\left[1-\mathsf{\Phi }\left({\displaystyle \frac{\frac{i\times l}{T_{P_k}^L}-\mu }{\sigma }}\right)\right]\\ ={\displaystyle \frac{1}{\mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\cdot \mathsf{\Phi }\left({\displaystyle \frac{\mu -\frac{i\times l}{T_{P_k}^L}}{\sigma }}\right)\\ ={\displaystyle \frac{\mathsf{\Phi }\left(\frac{\mu -\frac{i\times l}{T_{P_k}^L}}{\sigma }\right)}{\mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}\end{array}$$

(33)

That is:

$$\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)=\mathrm{Pr}\left(T_{C,i}^l\left(t\right)\le T_{P_k}^L\right)={\displaystyle \frac{\mathsf{\Phi }\left(\frac{\mu -\frac{i\times l}{T_{P_k}^L}}{\sigma }\right)}{\mathsf{\Phi }\left(\frac{\mu }{\sigma }\right)}}$$

(34)

At this stage, the actual value of $\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)$ for the $i$-hop routing path $P_k$ can be obtained by looking up the standard normal distribution table. According to the analysis in the paper, the link state between nodes in MANETs does not follow a simple binary model determined by Euclidean distance, but rather follows different attenuation laws at different transmission distances. Given that the link state in MANETs follows distance-dependent attenuation laws, theoretical calculations of link reliability and path transmission success probability require numerical implementation. Therefore, we propose a simulation algorithm (Algorithm 1) to compute link reliability and the probability of successful routing transmission. The parameters adopted in the subsequent simulation algorithms are detailed in

Table 1. Parameters.

Parameter Category	Parameter Name	Parameter Meaning
Basic Parameters	$m$	Number of Nodes
	$T_{mi}$	Simulation Duration
	$\Delta t$	Time Step
	$N_{simu}$	Number of Simulation Runs
	$M_d\left(t\right)$	Euclidean Distance Matrix
	$A_R\left(t\right)$	Weighted Adjacency Matrix
	$A_{RB}\left(t\right)$	Binary Adjacency Matrix
Link/Channel Parameters	$\alpha$	Short-to-Medium Range Coefficient
	$\beta$	Medium-to-Long Range Coefficient
	$d_{Th}$	Communication Distance Threshold
	$R_e^{Th}$	Link Reliability Threshold
	$R_e\left(d_{ij}\left(t\right)\right)$	Reliability of link eij
	$C\left(t\right)$	Channel Capacity
	$\mu$	Channel Capacity Mean
	$\sigma$	Channel Capacity Standard Deviation
	$T_{ij}^L$	Lifetime of Link eij
	$T_{P_k}^L$	Lifetime of Path Pk
Mobility Model Parameters	$Lo{c}_i\left(t\right)$	Coordinate of Node Ni
	$M{d}_i\left(t\right)$	Motion Direction of Node Ni
	$d_{ij}$	Euclidean Distance between Node Ni and Nj
	$d_{CO}$	Repulsion Distance
	$d_{AT}$	Attraction Distance
	$Ve{l}_{\min }~Ve{l}_{\max }$	Node Speed Range
Node Failure Parameters	$MTTF$	Hardware/Software MTTF
	$Lo{c}_f$	Hazard Source Coordinates
	$E\left(0\right)$	Initial Energy
	$E_{elec}$	Circuit Energy Consumption
	${\epsilon }_{amp}$	Power Amplification Coefficient
	$E_{Th}$	Minimum Operating Energy
Information Transmission Parameters	$l_I$	Periodic Information I Data Volume
	$l_R^{N_i}$	Received Information Processing Capacity of Node Ni
	$l_T^{N_i}$	Sent Information Processing Capacity of Node Ni
	$N_s$	Source Node
	$N_t$	Sink Node
	$T_{C,i}^l\left(t\right)$	Information Transmission Time
	$\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)$	Transmission Success Probability of Path Pk

.

The detailed steps and pseudocode of Algorithm 1 are as follows.

Input: Euclidean distance matrix $M_d\left(t\right)$ at time $t$, information volume $l$, source node $N_s$, sink node $N_t$, channel capacity distribution function $N\left(\mu ,{\sigma }^2\right)$.

Output: The probability $\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)$ of the routing path $P_k$.

(1)

Generate weighted adjacency matrix $A_R\left(t\right)$: Substitute the elements in the Euclidean distance matrix $M_d\left(t\right)$ into the link reliability model to calculate the reliability values $R_e\left(d_{ij}\left(t\right)\right)$, and generate a weighted adjacency matrix $A_R\left(t\right)$.

(2)

Determine the connectivity status of links: Compare the link reliability $R_e\left(d_{ij}\left(t\right)\right)$ with the threshold $R_e^{Th}$ in turn, traverse all links within the MANET, and generate a binary adjacency matrix $A_{RB}\left(t\right)$.

(3)

Calculate the lifetime $T_{ij}^L$ of reliable links: For the reliable links, substitute them into the link lifetime calculation formula to obtain the lifetime $T_{ij}^L$.

(4)

Identify the minimum path set and routing path: In response to the information transmission and reception requirements, the minimum path set and shortest routing path between source node and sink node is obtained by using Dijkstra algorithm.

(5)

Calculate the information processing capacity of nodes: Determine the received information processing capacity $l_R^{N_i}$ and the sent information processing capacity $l_T^{N_i}$ of each node in different routing paths.

(6)

Calculate the information transmission time $T_{C,i}^l\left(t\right)$: For links and routing paths, substitute them into the information transmission time calculation formula to obtain the transmission time $T_C^l\left(t\right)$ and $T_{C,i}^l\left(t\right)$ under the premise of channel capacity $C\left(t\right)$.

(7)

Calculate the lifetime $T_{P_k}^L$: Based on the lifetime time $T_{ij}^L$, substitute the path lifetime formula to calculate the lifetime $T_{P_k}^L$ of each routing path $P_k$ separately.

(8)

Calculate the transmission success probability $\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)$: Based on the information transmission time $T_{C,i}^l\left(t\right)$ and lifetime $T_{P_k}^L$, the transmission success probability $\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)=\mathrm{Pr}\left(T_{C,i}^l\left(t\right)\le T_{P_k}^L\right)$ of the routing path is obtained by looking up the standard normal distribution table.

(9)

Algorithm ends: Output $\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)$.

Algorithm 1. Simulation-based solution algorithm for probability of successful routing transmission.
Input:	$M_d\left(t\right)$, $l$$, N_s$$, N_t$$, N\left(\mu ,{\sigma }^2\right)$
Output:	$\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)$
1.	$\mathrm{Generate} A_R\left(t\right)$
2.	// Based on FS-TRG model
3.	$\mathrm{Generate} A_{RB}\left(t\right)=A_R\left(t\right)\ge R_e^{Th}$
4.	// Determine the connectivity status of links
5.	$\mathrm{Calculate} T_{ij}^L$
6.	$\mathrm{Identify} P_k$
7.	$// \mathrm{Use} \mathrm{Dijkstra} \mathrm{algorithm} \mathrm{to} \mathrm{find} \mathrm{MP} \mathrm{from} N_s$ $\mathrm{to} N_t$ $\mathrm{based} \mathrm{on} A_{RB}\left(t\right)$
8.	Calculate l
9.	// $\mathrm{Determine} \mathrm{the} \mathrm{received} \mathrm{information} \mathrm{processing} \mathrm{capacity} l_R^{N_i}$ $\mathrm{and} \mathrm{the} \mathrm{sent} \mathrm{information} \mathrm{processing} \mathrm{capacity} l_T^{N_i}$
10.	$\mathrm{Calculate} T_{C,i}^l\left(t\right)$
11.	$\mathrm{Calculate} T_{P_k}^L$
12.	$\mathrm{Calculate} \mathrm{Pr}\left(E_{P_k}^{trans}=1\right)=\mathrm{Pr}\left(T_{C,i}^l\left(t\right)\le T_{P_k}^L\right)$
13.	// Look up the standard normal distribution table
14.	$\mathrm{Output} \mathrm{Pr}\left(E_{P_k}^{trans}=1\right)$

The analysis of the computational complexity of Algorithm 1 is presented in

Table 2. Computational complexity of Algorithm 1.

Main Steps	Core Operations	Time Complexity (TC)
Matrix Generation	$\mathrm{Generating} \mathrm{the} \mathrm{matrix} A_R\left(t\right)$$\mathrm{and} A_{RB}\left(t\right)$$\mathrm{requires} \mathrm{traversing} \mathrm{all} \mathrm{node} \mathrm{pairs}, \mathrm{resulting} \mathrm{in} O\left(m^2\right)$$(\sin \mathrm{ce} \mathrm{there} \mathrm{are} O\left(m^2\right)$ potential node pairs).	$O\left(m^2\right)$
Link Lifetime Calculation	$\mathrm{For} \mathrm{reliable} \mathrm{links} (\mathrm{up} \mathrm{to} O\left(m^2\right)$$\mathrm{in} \mathrm{the} \mathrm{worst} \mathrm{case}, \mathrm{as} \mathrm{each} \mathrm{node} \mathrm{pair} \mathrm{may} \mathrm{form} \mathrm{a} \mathrm{link}), \mathrm{calculating} T_{ij}^L$$\mathrm{takes} O\left(n\right)\approx O\left(m^2\right)$, where $n$ is the number of links.	$O\left(n\right)\approx O\left(m^2\right)$
Shortest Path Finding	$\mathrm{Dijkstra}’\mathrm{s} \mathrm{algorithm} (\mathrm{used} \mathrm{to} \mathrm{find} \mathrm{the} \mathrm{source}\text{-}\mathrm{sink} \mathrm{shortest} \mathrm{path}) \mathrm{runs} \mathrm{in} O\left(m^2\right)$ with an adjacency matrix implementation.	$O\left(m^2\right)$
Probability calculation	$\mathrm{Probability} \mathrm{calculation} (\mathrm{e}.\mathrm{g}., \mathrm{normal} \mathrm{distribution} \mathrm{lookup}) \mathrm{is} O\left(1\right)$ (constant time).	$O\left(1\right)$

.

Overall, the dominant term is $O\left(m^2\right)$, so TC of Algorithm 1 = $O\left(m^2\right)$.

6. Network Node Failure Modes Analysis

Based on the assumption of imperfect nodes, this paper identifies four types of node failure modes in MANETs, each with distinct triggering mechanisms: (1) hardware/software failure (sporadic hardware or software malfunctions) [27,28,29], (2) location failure (distance-dependent node failure) [30,31,32], (3) energy consumption failure (battery depletion-induced offline) [20,26,33], and (4) isolation failure (no neighboring nodes within radio coverage) [20,34].

Hardware/software failure refers to sporadic hardware or software malfunctions of MANET nodes (e.g., malfunctions in vehicular communication modules). Its failure model is defined as:

$$F_{N_i}^1\left(t\right)=1-e^{-{\lambda }_{i}t}$$

(35)

In the formula, $F_{N_i}^1\left(t\right)$ denotes the hardware/software failure probability of node $N_i$ at time $t$, and ${\lambda }_i$ is the failure rate of $N_i$.

Location failure occurs when MANET nodes performing tasks (e.g., disaster rescue, emergency communication networking) are more likely to fail as they approach the hazard source (such as the point of ignition or the epicenter of an earthquake). The failure model is as follows:

$$F_{N_i}^2\left(t\right)=1-e^{-{\displaystyle \frac{t}{d_{iD}}}}$$

(36)

In the formula, $F_{N_i}^2\left(t\right)$ denotes the probability of location failure occurring at the node $N_i$ at time $t$, and $d_{iD}$ represents the Euclidean distance between the node $N_i$ and the hazard source.

Energy consumption failure occurs when MANET nodes (e.g., drone swarm nodes) continuously consume energy (battery) without replenishment during task execution, ultimately leading to offline failure due to insufficient remaining battery power to support normal operation. Its failure model is defined as:

$$\begin{array}{l}{F}_{N_i}^3\left(t\right)=\mathrm{Pr}\left(E_i\left(t\right)<E_{Th}\right)\\ \left\{\begin{array}{l}{E}_i\left(t\right)=E_i\left(0\right)-{\displaystyle {\int }_0^t\left(E_T^{N_i}\left(t\right)+E_R^{N_i}\left(t\right)\right)\mathrm{d}t}\\ E_T^{N_i}\left(t\right)=l_T^{N_i}\left(t\right)\cdot \left(E_{elec}+{\epsilon }_{amp}\cdot d^{{\gamma }_i\left(t\right)}\right)\\ E_R^{N_i}\left(t\right)=l_R^{N_i}\left(t\right)\cdot E_{elec}\end{array}\right.\end{array}$$

(37)

In the formula, $F_{N_i}^3\left(t\right)$ denotes the probability of energy consumption failure for node $N_i$ at time $t$, $E_i\left(t\right)$ denotes the remaining energy of $N_i$ at time $t$ (unit: J); $E_{Th}$ denotes the minimum energy threshold for $N_i$ to operate normally; $E_T^{N_i}\left(t\right)$ and $E_R^{N_i}\left(t\right)$ denote the energy consumed by $N_i$ to transmit and receive information at time $t$, respectively; $l_T^{N_i}\left(t\right)$ and $l_R^{N_i}\left(t\right)$ denote the volume of information transmitted and received by $N_i$ at time $t$, respectively; $E_{elec}$ denotes the circuit energy consumption for transmitting 1 bit of information, ${\epsilon }_{amp}$ denotes the power consumption coefficient of the power amplifier circuit; ${\gamma }_i\left(t\right)$ denotes the environment interference factor for node $N_i$ at time $t$.

Isolation failure occurs when a MANET node goes offline due to the absence of neighboring nodes within its radio coverage—without experiencing hardware/software failures, location failures, or energy consumption failures. Specifically, there are no nodes within the radio coverage range of the node $N_i$, and the node $N_i$ is not within the radio coverage range of any other nodes. Its failure model is as follows:

$$F_{N_i}^4\left(t\right)=\mathrm{Pr}\left({\displaystyle \sum _{j\in V}{a}_{RBij}\left(t\right)}={\displaystyle \sum _{j\in V}{a}_{RBji}\left(t\right)}=0\right)$$

(38)

In the formula, $F_{N_i}^4\left(t\right)$ denotes the probability of isolation failure for node $N_i$ at time $t$. $a_{RBij}$ represents the element in the $i$-th row and the $j$-th column of the binary adjacency matrix $A_{RB}\left(t\right)$ in MANETs, $a_{RBij}=1$ indicating the existence of a communication link between nodes $N_i$ and $N_j$, and $a_{RBij}=0$ indicating the absence of a communication link between nodes $N_i$ and $N_j$.

7. Simulation Algorithm for Solving Connectivity Reliability of MANETs

7.1. Evaluation Metrics

7.1.1. Mean Link Lifetime

The mean link lifetime is the statistical mean of the survival time of all reliable links within a MANET, reflecting the average lifespan of wireless links in achieving communication functions. A higher value indicates a more stable wireless link and network, as well as a higher probability of successful information transmission between nodes. This metric is primarily influenced by node movement, speed, direction, and spacing. It distinguishes between instantaneous (e.g., at a given moment $t$) mean link lifetime and task-duration (e.g., during a simulation experiment $\left[0,\hspace{0.33em}{T}_{mi}\right]$) mean link lifetime. Here, $T_{mi}$ represents the running time of a single simulation experiment.

Instantaneous mean link lifetime: Assume that in the $i$-th simulation experiment, there are a total of $N_{Links}^i\left(t\right)$ connected links existing in the MANET at time $t$, and the lifetime of each link is $T_{e_k}^L\left(t\right)$ ($k=1,\hspace{0.33em}2,\hspace{0.33em}\dots ,\hspace{0.33em}{N}_{Links}^i\left(t\right)$). $N_{Links}^i\left(t\right)$ can be calculated using a binary adjacency matrix $A_{RB}\left(t\right)$ that considers the communication quality of the links:

$$N_{Links}^i\left(t\right)={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=i+1}^m{a}_{RBij}\left(t\right)}}$$

(39)

In the $i$-th simulation experiment, the formula for calculating the instantaneous mean link lifetime of all connected links within the MANET at time $t$ is as follows:

$$M{T}_{Links}^i\left(t\right)={\displaystyle \frac{1}{N_{Links}^i\left(t\right)}}{\displaystyle \sum _{k=1}^{N_{Links}^i\left(t\right)}{T}_{e_k}^L\left(t\right)}$$

(40)

After running $N_{simu}$ simulation experiments, the formula for calculating the instantaneous mean link lifetime of all connected links in the MANET at any given moment $t$ is as follows:

$$M{T}_{Links}\left(t\right)={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}M{T}_{Links}^i\left(t\right)}={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}{\displaystyle \sum _{k=1}^{N_{Links}^i\left(t\right)}\frac{T_{e_k}^L\left(t\right)}{N_{Links}^i\left(t\right)}}}$$

(41)

Task-duration mean link lifetime: After the completion of the $i$-th simulation experiment, the formula for calculating the task-duration mean link lifetime $M{T}_{Links}^i$ during the period $\left[0,\hspace{0.33em}{T}_{mi}\right]$ of all connected links in the MANET is:

$$M{T}_{Links}^i={\displaystyle \frac{{\displaystyle \sum _{t=0}^{T_{simu}}{\displaystyle \sum _{k=1}^{N_{Links}^i\left(t\right)}{T}_{e_k}^L\left(t\right)}}}{{\displaystyle \sum _{t=0}^{T_{simu}}{N}_{Links}^i\left(t\right)}}}$$

(42)

After running $N_{simu}$ simulation experiments, the formula for calculating the task-duration mean link lifetime $M{T}_{Links}$ during the period $\left[0,\hspace{0.33em}{T}_{mi}\right]$ of all connected links in the MANET can be expressed as:

$$M{T}_{Links}={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}M{T}_{Links}^i}={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}\frac{{\displaystyle \sum _{t=0}^{T_{simu}}{\displaystyle \sum _{k=1}^{N_{Links}^i\left(t\right)}{T}_{e_k}^L\left(t\right)}}}{{\displaystyle \sum _{t=0}^{T_{simu}}{N}_{Links}^i\left(t\right)}}}$$

(43)

7.1.2. Mean Routing Path Lifetime

The mean routing path lifetime is the statistical mean of the survival time of all routing paths within a MANET, reflecting the average lifespan of routing paths when transmitting information from the source node to the sink node. Similarly, a higher value indicates that the constituent links and the network are more stable, and the probability of successful information transmission between nodes is higher. This metric is primarily influenced by node movement, speed, direction, and spacing. It distinguishes between instantaneous (e.g., at a specific moment $t$) mean routing path lifetime and task-duration (e.g., during a simulation experiment $\left[0,\hspace{0.33em}{T}_{mi}\right]$) mean routing path lifetime.

Instantaneous mean routing path lifetime: Assume that in the $i$-th simulation experiment, under various information transmission and reception requirements, there are a total of $N_{Paths}^i\left(t\right)$ routing paths existing within the MANET at time $t$. The lifetime of each routing path is $T_{P_k}^L\left(t\right)$($k=1,\hspace{0.33em}2,\hspace{0.33em}\dots ,\hspace{0.33em}{N}_{Paths}^i\left(t\right)$). Here, the routing path $P_k$ is represented as a set of links $P_k=\left\{e_{ij}|i\ne j,e_{ij}\in E\right\}$, $P_k\subseteq E$.

In the $i$-th simulation experiment, the formula for calculating the instantaneous mean routing path lifetime of all shortest paths within the MANET at time $t$ is as follows:

$$M{T}_{Paths}^i\left(t\right)={\displaystyle \frac{1}{N_{Paths}^i\left(t\right)}}{\displaystyle \sum _{k=1}^{N_{Paths}^i\left(t\right)}{T}_{P_k}^L\left(t\right)}$$

(44)

After running $N_{simu}$ simulation experiments, the formula for calculating the instantaneous mean routing path lifetime $M{T}_{Paths}\left(t\right)$ of all shortest paths within the MANET at any given moment is:

$$M{T}_{Paths}\left(t\right)={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}M{T}_{Paths}^i\left(t\right)}={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}{\displaystyle \sum _{k=1}^{N_{Paths}^i\left(t\right)}\frac{T_{P_k}^L\left(t\right)}{N_{Paths}^i\left(t\right)}}}$$

(45)

Task-duration mean routing path lifetime: After the completion of the $i$-th simulation experiment, the formula for calculating the task-duration mean routing path lifetime during the period $\left[0,\hspace{0.33em}{T}_{mi}\right]$ of all shortest paths in the MANET is as follows:

$$M{T}_{Paths}^i={\displaystyle \frac{{\displaystyle \sum _{t=0}^{T_{simu}}{\displaystyle \sum _{k=1}^{N_{Paths}^i\left(t\right)}{T}_{P_k}^L\left(t\right)}}}{{\displaystyle \sum _{t=0}^{T_{simu}}{N}_{Paths}^i\left(t\right)}}}$$

(46)

After running $N_{Simu}$ simulation experiments, the formula for calculating the task-duration mean routing path lifetime during the period $\left[0,\hspace{0.33em}{T}_{mi}\right]$ of all shortest paths in a MANET can be expressed as:

$$M{T}_{Paths}={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}M{T}_{Paths}^i}={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}\frac{{\displaystyle \sum _{t=0}^{T_{simu}}{\displaystyle \sum _{k=1}^{N_{Paths}^i\left(t\right)}{T}_{P_k}^L\left(t\right)}}}{{\displaystyle \sum _{t=0}^{T_{simu}}{N}_{Paths}^i\left(t\right)}}}$$

(47)

7.1.3. Mean Success Rate of Information Transmission

The mean information successful transmission rate is represented by the ratio of the number of successful information transmissions to the total number of information transmissions within a MANET. It reflects the network’s ability to achieve communication connectivity within the MANET. A higher ratio indicates higher network connectivity reliability, enabling real-time, accurate, and reliable transmission of information under existing channel capacity and environmental conditions. This results in fewer instances of information transmission failure, effectively supporting the successful execution of various tasks by the cluster network. This metric distinguishes between instantaneous (e.g., at a given moment $t$) mean success rate of information transmission and task-duration (e.g., during a simulation experiment $\left[0,\hspace{0.33em}{T}_{simu}\right]$) success rate of information transmission.

Instantaneous mean success rate of information transmission: Assuming that in the $i$-th simulation experiment, under the consideration of various information transmission and reception requirements at each moment $t$, there are a total of $N_{Paths}^i\left(t\right)$ routing paths within the MANET; thus, there are $N_{Paths}^i\left(t\right)$ information transmission requests and transmission times. An information transmission indicator function $I_{P_k}^i\left(t\right)$ is introduced:

$$I_{P_k}^i\left(t\right)=\left\{\begin{array}{l}1, \mathrm{Pr}\left(E_{P_k}^{trans}=1\right)\ge r\\ 0, else\end{array}\right.$$

(48)

In the formula, if the probability value $\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)$ at time $t$ is greater than or equal to a random number $r$ with a value between the interval $\left[0,\hspace{0.33em}1\right]$, it indicates that the routing path $P_k$ successfully sends data information from the source node to the sink node, and the value of $I_{P_k}^i\left(t\right)$ is set to 1; otherwise, the value is set to 0. In the $i$-th simulation experiment, the calculation formula for the instantaneous mean success rate of information transmission within the MANET at time $t$ is:

$$M{R}_{IT}^i\left(t\right)={\displaystyle \frac{1}{N_{Paths}^i\left(t\right)}}{\displaystyle \sum _{k=1}^{N_{Paths}^i\left(t\right)}{I}_{P_k}^i\left(t\right)}$$

(49)

After running $N_{simu}$ simulation experiments, the formula for calculating the instantaneous success rate of information transmission within the MANET at any given moment $t$ is:

$$M{R}_{IT}\left(t\right)={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}M{R}_{IT}^i\left(t\right)}={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}{\displaystyle \sum _{k=1}^{N_{Paths}^i\left(t\right)}\frac{I_{P_k}^i\left(t\right)}{N_{Paths}^i\left(t\right)}}}$$

(50)

Task-duration mean success rate of information transmission: After the completion of the $i$-th simulation experiment, the formula for calculating the task-duration mean success rate of information transmission of the MANET during the period $\left[0,\hspace{0.33em}{T}_{mi}\right]$ is:

$$M{R}_{IT}^i={\displaystyle \frac{{\displaystyle \sum _{t=0}^{T_{simu}}{\displaystyle \sum _{k=1}^{N_{Paths}^i\left(t\right)}{I}_{P_k}^i\left(t\right)}}}{{\displaystyle \sum _{t=0}^{T_{simu}}{N}_{Paths}^i\left(t\right)}}}$$

(51)

After running $N_{Simu}$ simulation experiments, the formula for calculating the task-duration mean success rate of information transmission of MANET during the period $\left[0,\hspace{0.33em}{T}_{mi}\right]$ is:

$$M{R}_{IT}={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}M{R}_{IT}^i}={\displaystyle \frac{1}{N_{simu}}}{\displaystyle \sum _{i=1}^{N_{simu}}\frac{{\displaystyle \sum _{t=0}^{T_{simu}}{\displaystyle \sum _{k=1}^{N_{Paths}^i\left(t\right)}{I}_{P_k}^i\left(t\right)}}}{{\displaystyle \sum _{t=0}^{T_{simu}}{N}_{Paths}^i\left(t\right)}}}$$

(52)

7.2. Simulation Algorithm for Solving Connectivity Reliability of MANETs Considering Link Status and Capacity

By analyzing the reliability of links and the probability of successful transmission of information, the paper proposes a simulation algorithm (Algorithm 2) for solving the connectivity reliability of MANETs, considering link status and capacity. For each simulation experiment, the specific process and pseudocode are as follows:

Input: Graph theory model of MANETs $G=\left(V,E\right)$, simulation task duration $t_{mi}$, propagation model parameters $\alpha$ and $\beta$, link reliability threshold $R_e^{Th}$, channel capacity (following a truncated normal distribution $N\left(\mu ,{\sigma }^2\right)$).

Output: Instantaneous mean link lifetime $M{T}_{Links}^i\left(t\right)$ and task-duration mean link lifetime $M{T}_{Links}^i$(over $\left[0,\hspace{0.33em}{t}_{mi}\right]$); instantaneous mean routing path lifetime $M{T}_{Paths}^i\left(t\right)$ and task-duration mean routing path lifetime $M{T}_{Paths}^i$; information transmission indicator function $I_{P_k}^i\left(t\right)$, Instantaneous mean success rate of information transmission $M{R}_{IT}^i\left(t\right)$ and task-duration mean success rate of information transmission $M{R}_{IT}^i$.

(1)

Initialize the graph theory model of the MANET at $t=0$: Based on task and deployment information, determine the number of nodes $m=\left|V\right|$ in the MANET and the index of each node $N_i\left(i=1,\hspace{0.33em}2,\hspace{0.33em}\dots ,\hspace{0.33em}m\right)$; assign initial coordinates, directions, and velocities to all nodes; and set the node sets as follows: $V_1=V$ (nodes with normal communication), $V_2=\varnothing$ (hardware/software failure), $V_3=\varnothing$ (energy consumption failure), $V_4=\varnothing$ (location failure), and $V_5=\varnothing$ (isolation failure).

(2)

Determine whether a node has experienced a hardware/software failure: For all nodes in the current $V_1$, calculate the probability of hardware/software failure at each node $N_i\in V_1$ according to the formula $F_{N_i}^1\left(t\right)=1-e^{-{\lambda }_{i}t}$, and compare it with a random number $r\in \left[0,\hspace{0.33em}1\right]$. If $F_{N_i}^1\left(t\right)>r$, the node is identified as failed. We update the node sets as $V_2=V_2{\displaystyle \cup \left\{N_i\right\}}$ (add to hardware/software failure set) and $V_1=V_1\backslash \left\{N_i\right\}$ (remove from normal node set). After checking all nodes in $V_1$, if $V_1\ne \varnothing$ (there are still normal nodes), proceed to step (3); otherwise (no normal nodes remain, and the network is non-functional), proceed to step (12) to terminate the algorithm.

(3)

Determine whether a node has experienced a location failure: For all nodes in the current $V_1$, calculate the location failure probability for each node $N_i\in V_1$ using the formula $F_{N_i}^2\left(t\right)=1-e^{-{\displaystyle \frac{t}{d_{iD}}}}$, and compare it with a random number $r\in \left[0,\hspace{0.33em}1\right]$. If $F_{N_i}^2\left(t\right)>r$, then the node is determined to be failed, and updates the sets as $V_4=V_4{\displaystyle \cup \left\{N_i\right\}}$ and $V_1=V_1\backslash \left\{N_i\right\}$. After traversing all normal nodes in the current MANET, if $V_1\ne \varnothing$, then proceed to step (4); otherwise, proceed to step (12).

(4)

Generate inter-node distance matrix $M_d\left(t\right)$ and binary adjacency matrix $A_{RB}\left(t\right)$: For all nodes in the current $V_1$, generate the inter-node Euclidean distance matrix $M_d\left(t\right)$ using the formula $d_{ij}\left(t\right)=\sqrt{{\left|x_i\left(t\right)-x_j\left(t\right)\right|}^2+{\left|y_i\left(t\right)-y_j\left(t\right)\right|}^2}$, and further obtain the binary adjacency matrix $A_{RB}\left(t\right)$ that accounts for link communication quality in the MANET at this moment $t$.

(5)

Identify the current information transmission and reception requirements of MANETs: Determine the volume of information $l$, the source nodes $N_s$ for information processing, and the corresponding sink nodes $N_t$.

(6)

Calculate the successful transmission probability $\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)$ of each routing path: Utilize Algorithm 1 proposed in the paper to compute the successful transmission probability $\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)$ of each routing path under the current full information transmission and reception requirements of the MANET. Compare this probability with a random number $r\in \left[0,\hspace{0.33em}1\right]$. If $\mathrm{Pr}\left(E_{P_k}^{trans}=1\right)\ge r$, it is determined that the routing path $P_k$ successfully transmits information, achieving communication connectivity from the source node to the sink node, and updating $I_{P_k}^i\left(t\right)=1$. Otherwise, it is determined that the routing path $P_k$ fails to transmit information, communication connectivity between the source node and the sink node is not established, and updating $I_{P_k}^i\left(t\right)=0$. This process is repeated for all routing paths within the MANET at the moment $t$.

(7)

Determine whether a node has experienced an energy consumption failure: Calculate the remaining energy of each node $N_i$ in the MANET at time $t$ using the formula $E_i\left(t\right)=E_i\left(0\right)-{\displaystyle {\int }_0^t\left(E_T^{N_i}\left(t\right)+E_R^{N_i}\left(t\right)\right)\mathrm{d}t}$ and compare each node’s remaining energy with the energy threshold $E_{Th}$. If $E_i\left(t\right)<E_{Th}$, node $N_i$ is identified as failed, and updates $V_3=V_3{\displaystyle \cup \left\{N_i\right\}}$ and $V_1=V_1\backslash \left\{N_i\right\}$ are made. After traversing all normal nodes in the current MANET, if $V_1\ne \varnothing$, then proceed to step (8); otherwise, proceed to step (12).

(8)

Update distance matrix and adjacency matrix: For all nodes in the current $V_1$ network, generate an Euclidean distance matrix between nodes $M_d\left(t\right)$, and further obtain a binary adjacency matrix considering the link communication quality in the MANET at this moment $A_{RB}\left(t\right)$.

(9)

Determine whether a node has experienced isolation failure: If a node $N_i\in V_1$ exists and meets the specified criteria ${\displaystyle \sum _{j\in V_1}{a}_{RBij}}={\displaystyle \sum _{j\in V_1}{a}_{RBji}}=0$, it is determined that isolation failure has occurred. Update $V_5=V_5{\displaystyle \cup \left\{N_i\right\}}$ and $V_1=V_1\backslash \left\{N_i\right\}$, if $V_1\ne \varnothing$, proceed to step (10); otherwise, proceed to step (12).

(10)

Update node position information: For all nodes in the current $V_1$, calculate and update the direction vectors, coordinates, and other information of each node at the next moment $t+\Delta t$ based on the Couzin-leader model.

(11)

Update the simulation clock: Update $t=t+\Delta t$ If $t<t_{mi}$, proceed to step (2) and continue the iterative operation; otherwise, proceed to step (12).

(12)

Algorithm ends: Output the result data.

Algorithm 2. Simulation algorithm for solving connectivity reliability of MANETs considering link status and capacity.
Input:	$G=\left(V,E\right)$, $t_{mi}$, $\alpha$, $\beta$, $R_e^{Th}$, $N\left(\mu ,{\sigma }^2\right)$
Output:	$M{T}_{Links}^i\left(t\right)$, $M{T}_{Links}^i$, $M{T}_{Paths}^i\left(t\right)$$, M{T}_{Paths}^i$$, I_{P_k}^i\left(t\right)$$, M{R}_{IT}^i\left(t\right)$$, M{R}_{IT}^i$
1.	$\mathrm{Initialize} G=\left(V,E\right)$
2.	$// \mathrm{Determine} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{nodes} m=\left|V\right|$ $\mathrm{and} \mathrm{the} \mathrm{index} \mathrm{of} \mathrm{each} \mathrm{node} N_i\left(i=1,2,\dots ,m\right)$
3.	$\mathrm{Set} V_1=V$$, V_2=\varnothing$$, V_3=\varnothing$$, V_4=\varnothing$$, \mathrm{and} V_5=\varnothing$
4.	// Corresponding to the 5 node sets
5.	For $N_i\in V_1$$, \mathrm{calculate} F_{N_i}^1\left(t\right)$
6.	$\mathbf{If} F_{N_i}^1\left(t\right)>r$$, r\in \left[0,1\right]$
7.	$\mathrm{Update} V_2=V_2{\displaystyle \cup \left\{N_i\right\}}$$, V_1=V_1\backslash \left\{N_i\right\}$
8.	End if
9.	End for
10.	// Determine whether a node has experienced a hardware/software failure
11.	For $N_i\in V_1$$, \mathrm{calculate} F_{N_i}^2\left(t\right)$
12.	$\mathbf{If} F_{N_i}^2\left(t\right)>r$$, r\in \left[0,1\right]$
13.	$\mathrm{Update} V_4=V_4{\displaystyle \cup \left\{N_i\right\}}$$, V_1=V_1\backslash \left\{N_i\right\}$
14.	End if
15.	End for
16.	// Determine whether a node has experienced a location failure
17.	$\mathrm{Generate} M_d\left(t\right)$ $\mathrm{and} A_{RB}\left(t\right)$
18.	Determine $l$$, N_s$$, \mathrm{and} N_t$
19.	// Identify the information transmission and reception requirements
20.	$\mathrm{Calculate} \mathrm{Pr}\left(E_{P_k}^{trans}=1\right)$
21.	// Utilize Algorithm 1
22.	For $N_i\in V_1$$, \mathrm{calculate} E_i\left(t\right)$
23.	$\mathbf{If} E_i\left(t\right)<E_{Th}$
24.	$\mathrm{Update} V_3=V_3{\displaystyle \cup \left\{N_i\right\}}$$, V_1=V_1\backslash \left\{N_i\right\}$
25.	End if
26.	End for
27.	// Determine whether a node has experienced an energy consumption failure
28.	$\mathrm{Update} M_d\left(t\right)$ $\mathrm{and} A_{RB}\left(t\right)$
29.	For $N_i\in V_1$
30.	$\mathbf{If} {\displaystyle \sum _{j\in V_1}{a}_{RBij}}={\displaystyle \sum _{j\in V_1}{a}_{RBji}}=0$$, j\ne i$
31.	$\mathrm{Update} V_5=V_5{\displaystyle \cup \left\{N_i\right\}}$$, V_1=V_1\backslash \left\{N_i\right\}$
32.	End if
33.	End for
34.	// Determine whether a node has experienced isolation failure
35.	If $V_1\ne \varnothing$
36.	$t=t+\Delta t$
37.	$\mathbf{If} t<t_{mi}$
38.	Proceed to step (5)
39.	Else
40.	Proceed to step (45)
41.	End if
42.	Else
43.	Proceed to step (45)
44.	End if
45.	Output the result data

Algorithm 2 is an iterative simulation that evaluates MANET connectivity reliability over time. Its complexity depends on $m$ (node count) and $N_{Steps}$ (total simulation steps, $N_{Steps}=T_{mi}/\Delta t$, where $\Delta t$ is the time step). Iterative Framework: The algorithm runs for $N_{Steps}$ steps. Each step integrates core operations of Algorithm 1 ($O\left(m^2\right)$) plus additional tasks: node failure checks (hardware/location/energy/isolation, $O\left(m\right)$ per step) and node position updates (Couzin-leader model, $O\left(m\right)$ per step). Dominant Term: The $O\left(m^2\right)$ cost per step (from Algorithm 1) dominates the $O\left(m\right)$ auxiliary operations. Thus, TC of Algorithm 2 = $O\left(N_{Steps}\times m^2\right)$.

8. Experiments and Analysis

8.1. Simulation Experiment Parameter Setting

The simulation case designed in the paper is a MANET consisting of 150 nodes, with each network node numbered sequentially from $N_1$ to $N_m=N_{150}$. The simulation task duration $t_{mi}=1\hspace{0.33em}\mathrm{h}$ and simulation time step length $\Delta t=1 \mathrm{s}$ are specified. We repeated the simulation 50 times using MATLAB R2023, and estimated the connectivity reliability and related parameters of the MANET by averaging the results. For the arbitrarily end-to-K ends (s-Kt) connectivity reliability $Arb{R}_{sKt}$, $K=120$ is set; for the appointed end-to-K ends (s-Kt) connectivity reliability $App{R}_{sKt}$, the target node set is defined as $\left\{N_i|i=2,\hspace{0.33em}3,\hspace{0.33em}\dots ,\hspace{0.33em}K+1\right\}$; for the arbitrarily end-to-end (s-t) connectivity reliability $Arb{R}_{st}$ and the appointed end-to-end (s-t) connectivity reliability $App{R}_{st}$, it is a special case when the value of K is 2. For link quality and channel capacity parameters, set $\alpha =0.5$, $\beta =0.7$, link reliability threshold $R_e^{Th}=0.8$, channel capacity (following a truncated normal distribution) with mean $C_{\mu }=2\times {10}^6\hspace{0.33em}\mathrm{bps}$ and standard deviation $C_{\sigma }=2\times {10}^5\hspace{0.33em}\mathrm{bps}$.

For the Couzin-leader model and the isolation failure model, it is assumed that there are 7 leader nodes in the MANET $\left\{N_1,\hspace{0.33em}{N}_2,\hspace{0.33em}\dots ,\hspace{0.33em}{N}_7\right\}$, and $N_1$ is the highest-level control node of the network. The remaining 143 nodes are follower nodes $\left\{N_8,\hspace{0.33em}{N}_9,\hspace{0.33em}\dots ,\hspace{0.33em}{N}_{150}\right\}$. The repulsion distance $d_{CO}=20\hspace{0.33em}\mathrm{m}$, attraction distance $d_{AT}=100\hspace{0.33em}\mathrm{m}$, minimum speed $Ve{l}_{\min }=5.56\hspace{0.33em}\mathrm{m}/\mathrm{s}$ and maximum speed $Ve{l}_{\max }=16.67\hspace{0.33em}\mathrm{m}/\mathrm{s}$ of node movement are set (the stability proof of the Couzin-leader model parameters based on Jacobian eigenvalue analysis is provided in the Appendix A). At the initial moment $t=0$, the location $Lo{c}_i\left(t\right)$, velocity ${Vel}_i\left(t\right)$, and motion direction $M{d}_i\left(t\right)$ of each node are randomly generated with $N_1$ as the center and twice the repulsion distance as the radius. The communication distance threshold of network nodes is set $d_{Th}=70\hspace{0.33em}\mathrm{m}$. The energy parameters are set as follows: $E_{elec}={10}^{-9}\hspace{0.33em}\mathrm{J}/\mathrm{bit}$ (circuit energy consumption per bit), ${\epsilon }_{amp}={100}^{-12}\hspace{0.33em}\mathrm{J}/\left(\mathrm{bit}\cdot {\mathrm{m}}^2\right)$ (power amplification coefficient), ${\gamma }_i=2$ (environmental interference factor), and $E\left(0\right)=500\hspace{0.33em}\mathrm{J}$ (initial node energy). There are three types of periodic information that need to be processed in the MANET: the first type $I_1$ includes the transmission and reception cycle $Cy{c}_1=5\hspace{0.33em}\mathrm{s}$, data packet size $l_{I1}=10\hspace{0.33em}\mathrm{KB}$, source node $N_1$, and sink node (all remaining nodes in the MANET); the second type $I_2$ includes the transmission and reception cycle $Cy{c}_2=5\hspace{0.33em}\mathrm{s}$, data packet size $l_{I2}=10\hspace{0.33em}\mathrm{KB}$, source node (all outermost nodes in the MANET, also known as convex hull nodes), and sink node $N_1$; the third type $I_3$ includes the transmission and reception cycle $Cy{c}_3=10\hspace{0.33em}\mathrm{s}$, data packet size $l_{I3}=5\hspace{0.33em}\mathrm{KB}$, sink node $N_1$, and source node (all remaining nodes in the MANET). For the hardware/software failure and location failure models, $MTTF=\mathrm{10,000}\hspace{0.33em}\mathrm{h}$ and the coordinates $Lo{c}_f=\left(\mathrm{19,000},\mathrm{19,000}\right)$ of the hazard source (also known as the task point coordinates) are set.

8.2. Analysis of Simulation Experiment Results

Figure 3 presents the analysis of MANET node movement consistency. At the initial moment $t=0$, the movement directions of nodes are randomly generated, so the node movement consistency index $O_{Net}\left(0\right)=0.5610$. As the simulation time progresses, interactions occur between MANET nodes, especially the increasingly evident influence of leader nodes on follower nodes. At time $t=1000$, the network node consistency reaches $O_{Net}\left(1000\right)=1.3758$, and thereafter converges to a certain interval $\left[1.3687,\hspace{0.33em}1.3833\right]$, indicating that the MANET modeled based on the Couzin-leader model moves in a highly consistent cluster form. The above process depicts the evolution of the MANET from a random and disorderly state at the initial moment of the simulation experiment to a regular and orderly state, which is consistent with the research hypothesis of the paper and will further support the analysis of connectivity reliability and information transmission quality of the MANET.

Observe the communication connectivity function of MANETs from metrics such as connectivity reliability, network lifetime, and transmission quality. Five connectivity reliability metrics are shown in Figure 4a. Among them, (1) the all-terminal (s-At) connectivity reliability $Arb{R}_{sAt}$ drops to 0 at $t=940$ (see Figure 4f)—this is the strictest metric, as it requires all nodes to be connected; (2) the appointed end-to-K ends (s-Kt) connectivity reliability $App{R}_{sKt}$ drops to 0 at time $t=1015$ (see Figure 4d); (3) the arbitrarily end-to-K ends (s-Kt) connectivity reliability $Arb{R}_{sKt}$ drops to 0 at time $t=3480$ (see Figure 4e); (4) the appointed end-to-end (s-t) connectivity reliability $App{R}_{st}$ drops to 0.7801 at the end of the simulation experiment $t=3600$ (see Figure 4b); and (5) the arbitrarily end-to-end (s-t) connectivity reliability $Arb{R}_{st}$ is 0.9740 at the end of the simulation experiment (see Figure 4c). As the most stringent evaluation metric in the network, $Arb{R}_{sAt}$ shows the most obvious downward trend. Conversely, $Arb{R}_{st}$, as the most lenient evaluation metric, remains nearly constant at 0.97 throughout the simulation experiment, which is consistent with existing research [34].

Considering that managers often focus on network connectivity reliability at specific task moments, this paper presents the corresponding statistical means and 95% confidence intervals at time $t=1200$, $t=2400$, and the end of the task ($t=3600$) respectively. At $t=1200$, the $App{R}_{st}$, $Arb{R}_{st}$, and $Arb{R}_{sKt}$ have a mean value of 0.9740, with a 95% confidence interval of [0.9710, 0.9770]. At $t=2400$, the $App{R}_{st}$ has a mean value of 0.9156 (95% confidence interval: [0.8508, 0.9803]); the $Arb{R}_{st}$ has a mean value of 0.9740 (95% confidence interval: [0.9710, 0.9770]); and the $Arb{R}_{sKt}$ has a mean value of 0.9545 (95% confidence interval: [0.9163, 0.9927]). At $t=3600$ (end of the task), the $App{R}_{st}$ has a mean value of 0.7792 (95% confidence interval: [0.6701, 0.8883]); the $Arb{R}_{st}$ has a mean value of 0.9740 (95% confidence interval: [0.9710, 0.9770]); and the $Arb{R}_{sKt}$ has a mean value of 0.

A comparative analysis was conducted on five connectivity reliability metrics in sequence. The blue curve represents the reliability obtained through simulation under the assumption of imperfect nodes and perfect links (denoted as Assumption 1), serving as the control data. The yellow curve represents the reliability obtained through simulation under the assumption of imperfect nodes and imperfect links (denoted as Assumption 2), which are the data obtained from Figure 4a. Since the appointed end-to-end connectivity reliability $App{R}_{st}$ and arbitrarily end-to-end connectivity reliability $Arb{R}_{st}$ are always greater than 0 in simulation experiments, transmission quality (the integral area of the reliability curve) is introduced for quantitative analysis. Since appointed end-to-K ends connectivity reliability $App{R}_{sKt}$, arbitrarily end-to-K ends connectivity reliability $Arb{R}_{sKt}$, and all-terminal connectivity reliability $Arb{R}_{sAt}$ all eventually drop to 0 in simulation experiments, network lifetime (the time required for the reliability curve to drop to 0) and transmission quality are introduced for analysis.

Transmission Quality Analysis. We calculate the communication quality of the MANET using the transmission quality formula: $TQ={\displaystyle {\int }_0^{+\infty }R\left(t\right)\mathrm{d}t}$, where $TQ$ denotes the integral area of the reliability curve, and $R\left(t\right)$ is a specific connectivity reliability metric of MANET at time $t$. The relevant results are shown in

Table 3. Transmission quality of MANET characterized by 5 reliability metrics under 2 assumptions.

Reliability Metrics	Transmission Quality (Assumption 1)	Transmission Quality (Assumption 2)	Difference Value (Difference Ratio)
$App{R}_{sKt}$	279.1	272.3355	−6.7645 (−2.4237%)
$App{R}_{st}$	3416.1	3331.0636	−85.0364 (−2.4893%)
$Arb{R}_{sAt}$	225.3	219.8399	−5.4601 (−2.4235%)
$Arb{R}_{sKt}$	2972.1	2897.913	−74.187 (−2.4961%)
$Arb{R}_{st}$	3595	3505.5358	−89.4642 (−2.4886%)

. In this example, if link reliability and channel capacity are not considered, the connectivity reliability of MANET will be overestimated by 2.42–2.50%.

Network Lifetime Analysis. Based on simulation experiment data and Figure 4d–f, the lifetime of MANET characterized by three connectivity reliability metrics under Assumption 1 and Assumption 2 is queried, as shown in

Table 4. Lifetime of MANET characterized by 3 reliability metrics under 2 assumptions.

Reliability Metrics	Network Lifetime (Assumption 1)	Network Lifetime (Assumption 2)	Difference Value (Difference Ratio)
$App{R}_{sKt}$	1015	1012	−3 (−0.2956%)
$Arb{R}_{sAt}$	940	940	0 (0%)
$Arb{R}_{sKt}$	3480	3479	−1 (−0.0003%)

. It can be seen that regardless of whether under the assumption of imperfect nodes and perfect links, or under the assumption of imperfect nodes and imperfect links, there is no difference (0–0.3%) in the lifetime of MANET characterized by the three connectivity reliability metrics.

The above results indicate that while link reliability and channel capacity do have a certain impact on the connectivity reliability evaluation of MANETs, they are not decisive factors, and their effects may even be limited. However, reliability, as a science aimed at combating failures, cannot be ignored when analyzing and evaluating MANETs, a special network system. Otherwise, the connectivity reliability of MANETs may be overestimated, leading to decisions that deviate from reality and incurring additional losses. Furthermore, as can be intuitively observed from Figure 4, during the slow decline phase of the connectivity reliability, the impact of link reliability and channel capacity on network communication functions is more pronounced; whereas during the sharp decline phase, the impact of link reliability and channel capacity on network communication functions is relatively weakened. For example, in Figure 4b, when the $App{R}_{st}$ decreases from 1 to 0.9, it takes a total of 3055 s. Within this time interval, there is a significant difference between the two curves; when the $App{R}_{st}$ decreases from 0.9 to 0.8, it takes a total of 322 s. Within this time interval, the difference between the two curves narrows significantly. Based on observations, the paper fits two-stage piecewise linear models to formalize the $App{R}_{st}$ and $Arb{R}_{sKt}$, respectively, and conducts Chow tests for structural breaks, as shown in Figure 5 and Figure 6. At the significance level of p < 0.05, the data exhibit significant structural breaks, with the breakpoints of the $App{R}_{st}$ and $Arb{R}_{sKt}$ being 2785 and 2495, respectively.

This result echoes the theoretical analysis mentioned earlier in the paper: when the connectivity reliability of MANETs sharply declines, it indicates that network nodes are offline due to various failure reasons, and the network topology structure changes constantly, leading to frequent disconnection or reconstruction of network links and restricting communication connectivity between network nodes. At this time, the importance of node reliability to network connectivity reliability exceeds that of link reliability. The more stable the network topology is, the more significant the impact of link reliability and channel capacity on network connectivity reliability; the more dramatic the fluctuations in network topology, the more ordinary the impact of link reliability and channel capacity on network connectivity reliability.

Based on scenarios involving link reliability, node density, and node movement patterns, this paper conducts sensitivity analysis on the following parameters: $\alpha$, $\beta$, and link reliability threshold $R_e^{Th}$ in the FS-TRG model; the number of nodes $m$ in the basic network model; and the maximum movement speed $Ve{l}_{\max }$ in the node movement characteristic model. The dependent variables are the network lifetime and transmission quality calculated based on $Arb{R}_{sKt}$. The visualization results of the sensitivity analysis are shown in Figure 7.

The abscissa values 1–5 in Figure 7a correspond to five $\left(\alpha ,\beta \right)$ combinations: (0.4, 0.6), (0.5, 0.7), (0.5, 0.8), (0.6, 0.8), and (0.9, 0.1). As shown in Figure 7a, network lifetime and transmission quality both increase with the increase of $\left(\alpha ,\beta \right)$ values—but this trend is only significant when $\left(\alpha ,\beta \right)$ changes from (0.4, 0.6) to (0.5, 0.7). From Figure 7b, it can be observed that as the link reliability threshold gradually increases from 0.5 to 0.9, network lifetime and transmission quality decrease by 5.17% and 7.66%, respectively. Although there is a value (0.6, 3490) with statistical randomness in the curve of network lifetime changes, it does not alter the overall downward trend. Meanwhile, the magnitude of the decrease in network lifetime and transmission quality indicates that, compared with the link reliability threshold, the FS-TRG parameters $\left(\alpha ,\beta \right)$ set in the simulation case of this paper and the strict judgment of link status have a more significant impact on network lifetime and transmission quality. Based on Figure 7a,b, future studies need to determine the values of $\left(\alpha ,\beta \right)$ (of the FS-TRG model) and the link reliability threshold $R_e^{Th}$ for specific deployment scenarios by combining experimental data.

As shown in Figure 7c, the increase in the number of nodes (density) effectively improves network lifetime and transmission quality. Under the condition that other parameters of the Couzin-leader model and the geographical area of network deployment remain unchanged, a larger number of network nodes directly increases the number of links in the network, thereby improving the success probability of single-hop or multi-hop communication between different nodes and ultimately affecting network connectivity reliability. However, it should be noted that the impact of the number of nodes (density) on network lifetime and transmission quality exhibits a diminishing marginal effect. Figure 7d shows an opposite trend to Figure 7c: as the maximum movement speed of network nodes increases (with the minimum node movement speed remaining unchanged), network lifetime and transmission quality continuously decrease, and the marginal impact becomes increasingly significant. This trend also confirms the previous theoretical analysis: high-speed node movement significantly disrupts network topology, reduces link reliability and stability, and ultimately degrades network connectivity reliability—consistent with the link lifetime model in Section 4. Based on Figure 7c,d, future studies on the deployment optimization of MANETs can focus on factors such as node density and node speed.

The success rate of information transmission at different moments in the simulation experiment of MANETs are shown in Figure 8. Figure 8a–c correspond to three types of periodic information. Figure 8d corresponds to the mean success rate of the three types of information combined within the MANET. It can be observed that the mean success rates of the three information types and the overall network all fall within a narrow range, with no significant difference and small standard deviations (0.0108–0.0537). The success rate of information transmission maintains a compact convergence pattern throughout. In the simulation experiment, the mean success transmission rate of the first type of information is 97.06% (95% confidence interval: [96.59%, 97.54%]); the mean success transmission rate of the second type of information is 97.75% (95% confidence interval: [96.26%, 99.24%]); the mean success transmission rate of the third type of information is 97.71% (95% confidence interval: [97.28%, 98.13%]); and the overall network success transmission rate has a mean value of 97.40% (95% confidence interval: [97.10%, 97.69%]). It is worth noting that the success rate of information transmission does not exhibit a similar decreasing correlation with the decrease in connectivity reliability of the MANET, indicating that the occurrence of node failures and changes in the topological structure within the MANET do not have a significant effect on the information transmission of existing network links and routing paths. This finding is also consistent with the previous analysis in the paper.

Analyze the statistical data of links and paths in MANETs from simulation experiments. Figure 9 illustrates the real-time count of network links and paths. In the simulation experiments, the number of network links first increases and then decreases. At the initial moment, nodes in the MANET are randomly and sparsely distributed, with an average of 2425.24 links (95% confidence interval: [2364.51, 2485.97]). As the attraction and alignment rules between network nodes take effect, the spacing between network nodes gradually approaches the repulsion distance. At this point, more and more reliable links are established between network nodes. In this example, the number of network links peaks at 8393.92 at a certain moment $t=234$. Subsequently, with the implementation of repulsion rules and the occurrence of multiple node failure modes, the number of network links gradually decreases. By the end of the simulation experiment, only 3769.94 links remain (95% confidence interval: [3647.34, 3892.54]), but this is still greater than the initial value, as shown in Figure 9a. Meanwhile, the number of paths in the MANET consistently shows a steady downward trend, which aligns with the characteristics of active networks. As network nodes go offline due to failure events, the number of working nodes that the source node can connect to decreases, leading to a reduction in the number of paths. By the end of the simulation experiment, the mean number of paths is 212.32 (95% confidence interval: [209.19, 215.45]).

Figure 10 shows the statistical results of link and path lifetime, as well as path hop count in MANETs. The mean link lifetime is 26.50 s (95% confidence interval: [25.96, 27.05]). The simulation experiment starts with a minimum value of 12.9361 s, and then converges to the mean value. The mean network path lifetime is 24.41 s (95% confidence interval: [21.16, 27.65]), indicating a larger fluctuation range compared to link lifetime. Considering that path lifetime is mainly influenced by link lifetime and the number of links, Figure 10a and the analysis suggest that during most of the simulation experiment, link lifetime remains relatively constant, with only a few random values. Therefore, further analysis is conducted on the time-varying pattern of path hop count. Statistical data indicate that the mean network path hop count is 1.33 (95% confidence interval: [1.26, 1.41]), with a maximum value of 1.74 at the beginning of the simulation experiment. At the beginning of the simulation experiment, there are fewer reliable links established between network nodes (see Figure 10a), so the source node needs to pass through more relay nodes to achieve communication connectivity with a certain sink node, resulting in a higher routing path hop count and a lower path lifetime. Later in the simulation experiment, a negative correlation and variation pattern between path lifetime and path hop count can be clearly observed by comparing Figure 10a,b.

The occurrence patterns of four types of network node failure modes are shown in Figure 11. At the termination of the simulation experiment $t=3600$, the number of failed nodes, the four failure modes are ranked as follows (ascending order): hardware/software failure (22.18), location failure (11.6), isolation failure (9.04), and energy consumption failure (1.44), as shown in Figure 11a. Upon further observation of the proportion of nodes in each failure mode to the total number of failed nodes, it is evident that energy consumption failure consistently ranks last; prior to time $t=914$, the failure mode with the highest proportion within the network is isolation failure, demonstrating the characteristics of MANETs that differ from general engineering systems; hardware/software failure rapidly increases in proportion prior to time $t=914$ and subsequently occupies the top position. the proportion of location failure ranks between second and third, with a noticeable upward trend at the end of the simulation—this aligns with the location failure model (Section 7), which states that failure probability increases as nodes approach the hazard source, indicating the MANET is moving closer to the hazard source, as shown in Figure 11b.

9. Conclusions

Based on the assumption of imperfect nodes, this paper further incorporates link quality and channel capacity to investigate the impact of both imperfect nodes and links on the connectivity reliability of MANETs deployed in a clustered form. By analyzing the simulation results, we find that link quality and channel capacity primarily affect the quality of information transmission (rather than network lifetime) with respect to network connectivity reliability, and this impact exhibits a clear stage-dependent pattern: when the MANET topology is relatively stable, the impact of links is more pronounced; when the network topology changes drastically, the impact of links weakens. Neglecting link quality and channel capacity when evaluating the connectivity reliability of MANETs will lead to overestimated reliability metrics, which in turn misleads subsequent management decisions. Meanwhile, this paper finds no significant correlation between the probability of successful information transmission within MANETs and the failure pattern of network nodes. In MANETs, the number of links increases in the early stage of the simulation due to the attraction and alignment rules of the Couzin-leader model. However, as the Euclidean distance between nodes approaches the repulsion distance, the implementation of repulsion rules and the effects of various node failure modes cause the number of links to decrease. The variation in the number of network paths is associated with the number of active nodes and exhibits a monotonically decreasing trend.

Besides yielding more practical connectivity reliability evaluation results, the proposed connectivity reliability model for MANETs also offers good scalability. The node energy consumption failure considered in this paper is actually determined by the amount of information processed by network nodes during the mission cycle. Based on known routing paths and information volume, the cascading failure phenomenon based on node capacity–load can be further considered. Herein, node capacity refers to the instantaneous maximum amount of information that a node can process without failure, and node load refers to the amount of information that a node needs to process at time $t$.

The limitations of this study lie in the assumptions that link failures and channel capacity are mutually independent, and that omnidirectional antennas have perfect circular coverage. Future research will further explore the introduction of a joint copula-based model to analyze link reliability and channel capacity, as well as the incorporation of a realistic 3-dB beamwidth pattern to optimize the antenna coverage model—thereby improving the analysis and evaluation of connectivity reliability for MANETs.