## 2 Constructing the Probability Model of Typical Faults  
During the manufacturing process of large-scale integrated circuit (LSI) systems, virtual soldering, de-soldering, and network short-circuit fault caused by soldering can easily occur. For an integrated circuit system with \(n\) networks, the probability of short-circuit fault between any two networks can be described by a symmetric matrix, as shown in Equation (1). \[ \begin{array}{cccc}0&p_{12}&\cdots&p_{1n}\\ p_{21}&0&\cdots&p_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ p_{n1}&p_{n2}&\cdots&0\end{array}  \tag{1}\]  
The values of matrix elements \(p_{ij}\) and \(p_{ji}\) are equal, indicating the probability of a short-circuit fault occurring between the network numbered \(i\) and the network numbered \(j\), and their values are calculated according to Equation (2). \[p_{ij}= \begin{cases}a_{0}\cdot A^{(1-\frac{t_{ij}}{t_{0}})}&L_{0}\leq L_{ij} \leq L_{M}\\ 0&\text{otherwise}\end{cases}  \tag{2}\]  
\(a_{0}\) is the probability of a short-circuit between the nearest two networks, \(L_{0}\) is the minimum distance between any two networks, \(L_{ij}\) is the distance between net \(N_{i}\) and net \(N_{j}\), \(L_{M}\) is the maximum physical distance between two nets capable of short-circuit failure, and \(A\) is an exponential decay function. The above parameters are determined according to the actual conditions of the integrated circuit system. In this paper, it is called m-order short-circuit fault when the number of networks participating in the short-circuit fault is m. For a circuit with n networks (\(m\leq n\)), if each network is treated as a node of an undirected graph, and if a short-circuit fault occurs between networks, there is an edge between the corresponding nodes, the occurrence of an m-order short-circuit fault in this circuit is equivalent to the formation of a connected graph in which the number of nodes is according to the properties of connected graphs, and the minimum number of edges of m-nodes connected graphs is \(m-1\)[30; 31]. Therefore, the number of possible m-order short-circuit faults is the number of connected graphs with m nodes and \(m-1\) edges, which equals the number of graphs with m nodes and \(m-1\) edges minus the number of all non-connected graphs with \(m\) nodes and \(m-1\) edges. Let the number of unconnected nodes in these graphs be t (\(1\leq t<m\)). According to the graph theory, the remaining \(m-t\) nodes can form \(C_{m-t}^{2}\) edges at most. If the remaining \(m-t\) nodes form a graph with at least \(m-1\) edges, it must satisfy that \(C_{m-t}^{2}\geq m-1\), so that \(t\) must satisfy the following conditions. \[1\leq t\leq\frac{2m-1-\sqrt{8m-7}}{2} \tag{3}\]  
If the largest integer value of \(t\) which satisfies Equation (3) is \(t_{0}\), and this paper defines the number of graphs which have \(m-1\) edges and \(m-t_{0}\) nodes as \(K_{0}\), then \(K_{0}\) can be calculated as follows:  
\[K_{0}=C_{C_{m-t_{0}}^{2}}^{m-1} \tag{4}\]  
Likewise, as this paper defines the number of graphs which have \(m-1\) edges and \(m-t_{0}+1\) nodes as \(K_{1}\), \(K_{1}\) can be calculated as follows:  
\[K_{1}=C_{C_{m-t_{0}+1}}^{m-1}-C_{m-t_{0}+1}^{1}\cdot K_{0} \tag{5}\]And similarly, as this paper defines the number of graphs which have \(m-1\) edges and \(m\) nodes as \(K_{t_{0}}\), \(K_{t_{0}}\) can be calculated as follows:  
\[K_{t_{0}}=C_{C_{m-i_{0}}^{m-1}}^{m-1}-\sum_{j=1}^{t_{0}}C_{m-t_{0}+j}^{j}\cdot K _{n-j} \tag{6}\]  
The probabilities of these \(K_{t_{0}}\) groups of \(m-1\) short-circuit faults in each group are taken out from Equation (1) and put into matrix \(Q_{m}\), which is shown in Equation (7). Each line of \(Q_{m}\) represents a combination of network short-circuit faults that can cause m-order short-circuit faults, and element \(q_{ij}\) is the probability of these network short-circuit faults occurring. \[ \begin{array}{cccc}q_{11}&q_{12}&\cdots&q_{1(m-1)}\\ q_{21}&q_{22}&\cdots&q_{2(m-1)}\\ \vdots&\vdots&\ddots&\vdots\\ q_{K_{t_{0}}1}&q_{K_{t_{0}}2}&\cdots&q_{K_{t_{0}}(m-1)}\end{array}