# Reliability of Social Networks on Activity-on-Node Binary-State with Uncertainty Environments
## 1 Introduction  
In social networks and many other industrial types of networks based on binary-state networks (BNs), each unreliable component has only two states. The BN architecture is one of the most common structures used to model issues in not only SNs [1, 2] but also numerous industrial types of networks that have become an integral part of our daily life, such as networks involving multi-terminal network [3], transportation [4], grid/cloud computing [5, 6], general network [7], Internet of things [8, 9], network resilience problems [10, 11], directed acyclic network [12], transition [13], and wireless sensor network [14]. Hence, an increasing amount of BN applications have recently emerged [15, 16]. BNs thus play an important role in the planning, design, execution, management, and control of all the aforementioned systems [17, 18, 19]. Assume that \(G(V,\,E,\,\mathbf{D})\) is a BN, where \(\mathbf{D}\) is the state distribution of the components, and \(V=\{1,\,2,\,\ldots,\,n\}\) and \(E=\{a_{1},\,a_{2},\,\ldots,\,a_{m}\}\) are the sets of nodes and arcs, respectively [20, 21]. In the activity-on-arc (AOA) BN, each node is considered perfect, whereas each arc may fail based on a predefined probability in \(\mathbf{D}\)[22, 23]. On the contrary, in the activity-on-node (AON) BN, each arc is considered perfect, whereas each node (except the source node 1 and the sink node \(n\)) may fail based on a predefined probability in \(\mathbf{D}\)[24, 25, 26, 27]. Note that a BN can be a mixture of AOA and AON; and a component can be a node, an arc, cable, the Internet, machine, process station, workstation, tool, unit, or worker [29]. In this study, we focus on the AON--and the component is a node in the AON, and an arc in the AOA. For example, Figure 1 shows an AOA BN \(G(V,\,E,\,\mathbf{D})\), where \(V=\{1,\,2,\,3,\,4,\,5\}\), \(E=\{a_{1}=e_{1,2},\,a_{2}=e_{1,3},\,a_{3}=e_{1,A},\,a_{4}=e_{2,A},\,a_{5}=e_{2,5 },\,a_{6}=e_{3,5}\}\). Node 1 is the source node and node 5 is the sink node; the state distribution \(\mathbf{D}\) is listed in Table 1.  
The traditional reliability, \(R\), of BNs is defined as the success probability that there is at least one simple path from node 1 to node \(n\). It is one of the most popular indices for evaluating the performance of real-world BNs. For example, the reliability of propagation is studied in SNs based on BN [1]. An algorithm is proposed for the transportation BN [4]. A binary-code genetic algorithm and the simplified swarm optimization (SSO) is hybrid to optimize the grid BN reliability [6]. The SSO is employed to optimize the BN reliability in the Internet of Things [8]. The telecom BN reliability is evaluated by considering network resiliency [10]. The BN reliability is solved using the unscented transformation [13]. Additionally, a squeezed artificial neural network is proposed to evaluate the BN reliability [25]. In the traditional BN, the reliability of each component must be known in advance [19; 22], for example, the reliability \(\mathbf{D}(a)\) is already known in Table 1 for all \(a\in E\). It is NP-Hard and #P-Hard to calculate the BN reliability [17; 18; 19]. Various algorithms have been proposed to calculate BN reliability and can be categorized into direct and indirect algorithms [30; 31; 32; 33]. The former, for example, the quick binary-addition-tree algorithm (BAT) [34] and binary-decision diagram (BDD) [35], can calculate the BN reliability directly; the latter is separated further into two main steps:  
1. Find all simple paths or cuts based on path-based algorithms [5; 8; 9; 21] or cut-based algorithms [20; 22; 23], respectively; and
2. Calculate the BN reliability using inclusion-exclusion technology (IET) [36; 37; 38] or the sum-of-disjoint product (SDP) [39; 40; 41; 42]. The direct algorithm is more efficient than the indirect algorithm in calculating the BN reliability because the two main steps in all indirect algorithms are both NP-hard and #P-hard [34].