This paper considers a 6G edge synergistic computing network with a total of N node devices and a degree distribution P ( k ) as shown in Figure 1. In this network, there are four types of node devices: Service state ( S ) , Auxiliary state ( A ) , Request state ( R ) , and Healing state ( H ) . The definitions of these four types of node devices are as follows:

S State: These refer to ES that have sufficient computing resources and capacity to provide offloading services for computational tasks, in addition to meeting their task requirements.

A State: These are UT that can meet their task requirements and also provide computational task offloading services.

R State: These UT experience task overload and need to initiate a synergistic computational task offloading request.

H State: These are UT that have recovered from the request state through task offloading and can meet their task requirements. However, due to prior task overload, they are no longer willing to offer computational task offloading services for safety reasons.

In 6G edge synergistic computing networks, communication between two neighboring node devices can be classified into two types: one is D2D communication between UT, and the other is wireless cellular communication between UT and ES. State UT can offload computational tasks to neighboring UT, neighboring ES, or cloud servers for processing. For the sake of analytical simplicity, this paper stipulates that during the task-offloading propagation process within the entire network, each pair of neighboring R state UT and A state UT (or S state devices) can only perform one synergistic computing task-offloading at a time, with each offloading corresponding to one computational task.

Regardless of the state of the node devices, potential computational tasks are categorized into two types: tasks generated by the device itself and tasks offloaded by neighboring UT. The task arrival and processing models for S state devices, A state terminals, and R state terminals are illustrated in Figure 2. It is assumed that the computational resources required to process a single data packet are identical across all node devices, denoted as W 0 ( c y c l e s ) . The set of computational resources available to each node device is represented as f ( c y c l e s / s ) = f 1 , f 2 , , f i , , f N , and the set of computational delays for processing a single data packet across the network is t = t 1 , t 2 , , t i , , t N , where t i = W 0 / f i .

For the locally generated computational tasks, an M / M / 1 / N queuing model is assumed with an arrival rate of λ i and a service rate of μ i . Here, the arrival rate λ i represents the number of data packets arriving at the node per second, while the service rate μ i denotes the number of data packets the node can process per second. Accordingly, the service rate is given by μ i = 1 / t i . Let ρ = λ / μ , then, according to the queuing model, the steady-state queue length for the data packet sequence generated by the node device itself is L i = ρ / ( 1 − ρ ) , where L i < N p , N q , with N p and N q representing the maximum queue lengths that can be accommodated by each S state device and A state UT, respectively. For tasks offloaded from neighboring R state UT, since the task-offloading transmission process is not stable or continuous, the task processing model differs from that for the node’s tasks. The specific handling mechanism is detailed in the S A R H task-offloading propagation model.

In the 6G edge synergistic computing network scenario, as shown in Figure 3, this paper establishes a S A R H model to describe the task-offloading propagation process. It is assumed that the network is in a stable state initially, with two types of node devices in the network: S state and A state. The proportions of S state ES and A state UT are denoted as p and q , respectively, with p + q = 1 . At a certain time t 0 , the base station in the network dispatched an excessive number of computational tasks, resulting in a proportion ρ 0 of UT experiencing task overload and subsequently transitioning to a R state. In each time step, R state UT may transmit offloading tasks to neighboring A state UT or neighboring ES with a probability α . For A state UT, when the accumulated packet sequence exceeds the threshold, they will transition to the R state with a probability h q ( m , T q ) . Similarly, S state devices will transition to the R state with a probability h p ( m , T p ) . It is generally assumed that ES can provide sufficient computing resources and system capacity, so S state devices remain in the S state. Meanwhile, each R state UT can transition to the H state with a probability γ , and no longer participate in the task-offloading propagation. The task-offloading propagation process terminates when there are no more R state UT in the network, and the network reaches stability.

For analysis, this paper proposes two linear threshold functions to describe the state transition threshold models for UT and ES. For A state UT, the set of stable queue lengths of all UT in the network can be represented as L q = { L q , 1 , … , L q , n , … , L q , q N } , where the maximum value is denoted as L q , max . The threshold for the number of data packets that each A state UT can accommodate is defined as T h q = N q − L q , max . The number of offloaded data packets received by the A state UT must not exceed this threshold, otherwise, the terminal will undergo a state transition. Assuming that the number of data packets contained in each offloaded task P satisfies T h q / ( T q + 1 ) < P < T h q / T q , this indicates that the A state UT will not undergo a state transition as long as it receives no more than T q offloading tasks. Therefore, for A state terminals, the following threshold function is used to represent the relationship between the state transition probability and the number of offloading tasks received: h q m , T q = 0 , 0 < m ≤ T q 1 , m > T q (1)

Let m represent the number of computational tasks received by an A state terminal within a time step, and let h q ( m , T q ) denote the state transition probability of the A state UT.

Similarly, for S state devices, the set of stable queue lengths of all ES in the network can be represented as L p = { L p , 1 , L p , 2 , … , L p , n , … , L p , p N } , where the maximum value is denoted as L p , max . The threshold for the number of data packets that each S state device can accommodate is defined as T h p = N p − L p , max , with T h p ≫ T h q . The threshold T p for the number of offloading tasks that a S state device can receive satisfies T h p / ( T p + 1 ) < P < T h p / T p , where T p is sufficiently large. Therefore, for service S devices, the following threshold function is used to represent the relationship between the state transition probability and the number of offloading tasks received: h p m , T p = 0 , 0 < m ≤ T p 1 , m > T p (2) m is the number of computational tasks received by the S state device in a time step, and h p ( m , T p ) is the S state device state transition probability.

Considering a large number of nodes and the complex topology in 6G edge synergistic computing networks, extensive simulation and numerical analysis were conducted on both Erdos-Renyi ( E R ) networks [36] and Scale-Free ( S F ) networks [37] to verify the universality and applicability of the theoretical analysis. In E R random networks, node degrees follow a Poisson distribution, i.e., P ( k ) = e − ⟨ k ⟩ ⟨ k ⟩ / k ! , where random connections are generated based on a probability mechanism, reflecting the characteristic of random node distribution. In S F networks, the degree distribution follows a power-law, i.e., P ( k ) ̃ k − v , where the scale-free property highlights the phenomenon where a few nodes take on a large number of connections, which corresponds to the importance of certain key nodes in 6G edge synergistic computing networks. The parameter v represents the degree distribution exponent of the S F network, where the degree distribution is negatively correlated with v ; the smaller the value of v , the more heterogeneous the network, leading to stronger network heterogeneity. The fixed parameters, including network size, offloading propagation probability, and the healing rate of R state UT, are set as N = 10 4 , α ∈ [ 0,1 ] and γ = 1.0 , respectively. The network size of 10 4 refers to 10,000 independent dynamic entities within the mobile edge network, including UT and ES. Based on the system model and the characteristics of practical mobile edge communication systems, it can be concluded that each R state UT will ultimately be able to process its accumulated computational tasks through edge synergistic computing offloading, cloud computing offloading, or other offloading methods, thereby transitioning back to the idle state. However, due to prior experiences of task overload, for safety considerations and the inherent selfishness of UT, these terminals are no longer willing to provide computational task-offloading services and instead transition into the H state. Therefore, this study assumes the healing rate of state UT as γ = 1.0 .

H ( ∞ ) represents the proportion of H state UT in the network when the entire network reaches a stable state after a sufficiently long period. It indirectly reflects the total number of R state nodes that experienced task overload and computation offloading requests throughout the entire synergistic task-offloading propagation process. A sudden surge in H ( ∞ ) at a certain moment indicates a large number of devices experiencing task overload within a short period, signifying a network paralysis scenario.

The parameters influencing the offloading propagation process can be categorized into dynamic parameters and network structural parameters. The dynamic parameters include the initial proportion of R state UT ρ 0 , the maximum number of R state UT that a single A state UT can serve T q , the initial proportion of A state UT q , and the maximum number of R state UT that a single S state ES can serve T p . The network structural parameters include the average degree of E R networks ⟨ k ⟩ and the degree distribution exponent v for S F networks. The following sections will discuss the impact of these dynamic and network structural parameters on the offloading propagation process.

Figure 4 explores the variation of the proportion of UT in the final H state, H ( ∞ ) , with the offloading propagation probability, α , as the initial proportion of R state UT, ρ 0 , changes in E R networks. The other parameters are set as N = 10 4 , α ∈ [ 0,1 ] , γ = 1.0 , ⟨ k ⟩ = 10 , T q = 2 , T p = 50 and q = 0.95 . As observed in the figure, when the initial proportion of R state UT is relatively small ρ 0 = 0.02 , the proportion of UT in the final H state, H ( ∞ ) , shows almost no increase as the offloading propagation probability, α , gradually increases. This indicates that the task-offloading process only affects a very small portion of A state terminals that transition to R state terminals, implying that the computational resources and capacity provided by the 6G edge synergistic computing network can handle the task-offloading of a certain proportion of R state UT. However, when the initial proportion of R state UT increases to a certain level ρ 0 = 0.05 , 0.10 , the proportion of UT in the final H state, H ( ∞ ) , initially increases slowly within a certain range as the offloading propagation probability, α , gradually increases. When α reaches a certain value, H ( ∞ ) experiences a sudden increase, reaching a significantly higher proportion. This phenomenon is referred to as a first-order discontinuous phase transition in the field of propagation dynamics [38], and the offloading propagation probability at which this sudden surge in H ( ∞ ) occurs is known as the critical offloading probability, α c . When such a first-order discontinuous phase transition occurs during the offloading process in the network, it indicates that a large number of A state terminals in the network have quickly experienced task overload and resource shortages, transitioning to R state terminals, which ultimately leads to network paralysis. It is worth noting that the larger the proportion of R state UT selecting the D2D-assisted MEC synergistic task-offloading method α , the more likely network paralysis will occur. A higher initial proportion of R state UT ρ 0 indicates a greater volume of computational tasks that need to be offloaded in a short period at the initial stage, thus requiring a smaller α to cause network paralysis. Comparing the variation of H ( ∞ ) with α for ρ 0 = 0.05 , ρ 0 = 0.10 , and ρ 0 = 0.15 in Figure 5, it can be observed that the larger the initial proportion of R state UT ρ 0 , the smaller the critical offloading probability corresponding to network paralysis caused by task overload on a large number of A UT. This demonstrates that the theoretical analysis results are consistent with the simulation outcomes.

Figure 5 shows the variation pattern of the proportion of UT in the final H state, H ( ∞ ) , with the offloading propagation probability, α , as the maximum number of R state UT that a single A state UT can serve, T q , changes. The other parameters are set as N = 10 4 , α ∈ [ 0,1 ] , γ = 1.0 , ⟨ k ⟩ = 10 , ρ 0 = 0.10 , T p = 50 and q = 0.98 . As observed in Figure 5, when a single A state terminal can only serve one R state terminal T q = 1 , H ( ∞ ) gradually increases in a small range before reaching the critical offloading propagation probability, α c , at which point a discontinuous phase transition occurs, leading to network paralysis. As the maximum number of R state terminals that a single A state terminal can serve increases T q = 2 and T q = 3 , the critical offloading propagation probability at which task overload causes a large number of A state terminals to fail, resulting in network paralysis, also increases. Additionally, by comparing the variation trends of H ( ∞ ) with the offloading propagation probability α for T q = 1 , T q = 2 and T q = 3 , it can be observed that the stronger the task-offloading service capacity of a single A state terminal, the greater the increase in the proportion of H state terminals when the network stabilizes at the critical offloading propagation probability. However, when T q = 4 , H ( ∞ ) increases slowly within a small range as α increases, and no matter how large α becomes, H ( ∞ ) will not undergo a discontinuous phase transition, meaning that network paralysis does not occur. From this, it can be inferred that the stronger the service capacity of a single A state terminal, the larger the proportion of R state terminals allowed to select mobile edge synergistic task-offloading before network paralysis occurs. If the service capacity of a single A state terminal is sufficiently strong, the task-offloading capacity of the mobile edge synergistic computing network will be able to handle the task load, keeping the network in a stable state without the risk of paralysis. Simulation results indicate that if the proportion of user terminals opting for D2D-assisted MEC synergistic task-offloading remains below the critical offloading propagation probability, network paralysis can be avoided. Therefore, controlling the proportion of user terminals selecting D2D-assisted MEC synergistic task-offloading is essential to ensuring the stability and efficiency of the synergistic task-offloading network.

Figure 6 illustrates the growth trend and pattern of the proportion of terminal users in the final H state, H ( ∞ ) , with the offloading propagation probability, α , as the network topology of the E R network changes. The other parameters are set as N = 10 4 , α ∈ [ 0,1 ] , γ = 1.0 , T q = 2 , ρ 0 = 0.08 , T p = 50 and q = 0.95 . As shown in Figure 6, when the average degree is relatively low ( ⟨ k ⟩ = 5 ) , the connections between the network nodes are sparse, and H ( ∞ ) shows almost no increase with increasing offloading propagation probability, α , and no phase transition is observed. As the average degree increases ( ⟨ k ⟩ = 8 , ⟨ k ⟩ = 10 , and ⟨ k ⟩ = 15 ), H ( ∞ ) increases continuously within a small range as α increases, until a discontinuous phase transition occurs at the critical offloading propagation probability. Moreover, the larger the ⟨ k ⟩ , the smaller the critical offloading propagation probability at which the first-order discontinuous phase transition of H ( ∞ ) occurs. In practical mobile communication networks, the average degree ⟨ k ⟩ represents the average number of communication links established by each dynamic entity (UT or ES) with ⟨ k ⟩ other UT (or ES), enabling the transmission of offloaded computational tasks. A larger average degree ⟨ k ⟩ means that, on average, each UT (or ES) has more communication links with neighboring devices, resulting in a more tightly connected network, which is more conducive to task offloading propagation. Under constant conditions of other parameters, a larger network average degree ⟨ k ⟩ leads to faster and earlier global diffusion of task offloading, while reducing the maximum proportion of R state UT allowed to choose mobile edge synergistic task-offloading. Therefore, controlling the average number of communication links (i.e., the network connectivity density) among devices (UT and ES) can help prevent network paralysis caused by task overloads from a large number of UT.

Figure 7 illustrates the impact of the degree distribution heterogeneity of the S F network on the variation trend of the proportion of UT in the final H state, H ( ∞ ) , for the offloading propagation probability, α . The other parameters are set as N = 10 4 , α ∈ [ 0,1 ] , γ = 1.0 , T q = 2 , ρ 0 = 0.08 , T p = 100 and q = 0.95 . It can be observed from the figure that when the degree distribution exponents, v , are 2.0, 3.0, and 3.9, there are two trends in the growth of H ( ∞ ) as α increases: when the probability of R state UT opting for D2D-assisted MEC synergistic task-offloading is relatively low, H ( ∞ ) decreases as v increases; conversely, when the probability is relatively high, H ( ∞ ) increases with the growth of v . This is because, for scale-free networks with a power-law degree distribution (such as S F networks), the smaller the degree distribution exponent v , the stronger the heterogeneity of the network’s degree distribution. In such networks, there is a large number of nodes with very small degrees and a few nodes with very large degrees. When the propagation probability is relatively low, the nodes with smaller degrees, which are more numerous, play a major role in information dissemination and are more likely to receive information fragments exceeding the threshold, thereby undergoing state transitions. However, as the propagation probability increases, the influence of high-degree nodes on the dissemination process gradually becomes more significant, reducing the probability of smaller-degree nodes adopting the behavior, ultimately resulting in a smaller propagation scale [39]. Additionally, it can be observed from the figure that as the degree distribution exponent v increases, the variation trend of H ( ∞ ) with the offloading propagation probability α transitions from a second-order continuous phase transition to a first-order discontinuous phase transition. Through extensive simulations, a critical degree distribution exponent, v c = 3.9 , was identified. When v < v c , H ( ∞ ) exhibits continuous growth, while when v > v c , H ( ∞ ) shows discontinuous growth. These two phenomena indicate that as the network becomes more uneven, deploying high-degree nodes as S state devices can prevent the network from crashing prematurely due to task overload in a large number of UT. The higher the maximum proportion of R state UT allowed to request D2D-assisted MEC synergistic task-offloading, the greater the network’s stability and efficiency.

As shown in Figure 8, this paper further investigates the impact of the maximum number of R state UT, T q , that can be served by a single A state UT on the proportion of final H state UT, H ( ∞ ) , as it changes with the task offloading propagation probability, α , in a S F network with a power-law degree distribution. When T q = 1 , T q = 2 , and T q = 3 , H ( ∞ ) increases continuously with α within a small range, and starts to increase sharply at the critical offloading propagation probability. Moreover, as T q increases, the growth trend of H ( ∞ ) with α shifts from continuous to discontinuous, and the critical offloading propagation probability, at which H ( ∞ ) increases sharply, becomes larger. When T q = 4 , H ( ∞ ) only increases continuously within a small range without a sudden increase as α rises. By comparing Figures 5, 8, it can be seen that the effect of T q on the variation of H ( ∞ ) with α in an S F network is generally consistent with its effect in an E R network. This demonstrates that, regardless of whether the network structure is E R or S F , the stronger the service capability of a single A state UT, the more stable the network becomes. Therefore, improving the task-offloading service capability of UT is an effective method to enhance the efficiency and stability of 6G edge synergistic computing networks.