## 2 Materials and Methods
### Graph Construction  
Like the traditional PageRank method, the network of sports teams is considered as a weighted directed graph. We construct the graph from the viewpoint of Bayesian correction, which is similar to the Elo method. It will be shown that the building process can provide a more statistical explanation for the model. The construction process is illustrated as follows. Let \(N(a,b)\) represent the normal distribution with mean \(a\) and variance \(b\). For a game \(k\) between team \(i\) and team \(j\), we assume:  
\[h(S_{j};\gamma_{j})\rho(X_{i}^{k},X_{j}^{k};\theta)\sim N(S_{i},\sigma^{2}), \tag{1}\]  
where \(X_{i}^{k}(X_{j}^{k})\) and \(S_{i}(S_{j})\) denote the performance vector and strength of team \(i(j)\), respectively. \(\rho(X_{i}^{k},X_{j}^{k};\theta)\) is set as the link function with parameter \(\theta\) to integrate game information including the performance of two teams. \(h(S_{j};\gamma_{j})\) is a discount factor dependent on \(S_{j}\) with a team-wise parameter \(\gamma_{j}\) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Team** & **Assists** & **Rebounds** & **Score** & **Fouls** & **Turnovers** \\ \hline PHI & 18 & 47 & 87 & 16 & 20 \\ BOS & 21 & 55 & 105 & 15 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The box score of the game between Philadelphia 76 ers (PHI) and Boston Celtics (BOS) in the 2018–2019 season. which allows taking the opponent's strength into account in the measurement of the effect of one game. Furthermore, the strength \(S_{i}\) is considered as a stochastic variable with prior distribution \(N(S_{i}^{0},\tau^{2})\). Let \(R(i)\) denote the set of games that team \(i\) has played and \(g(k,i)\) denote the opponent team of team \(i\) in game \(k\). Then, the posterior mean of \(S_{i}\) can be given by:  
\[\hat{S}_{i}=\frac{\tau^{2}\sum_{k\in R(i)}h(S_{g(k,i)};\gamma_{g(k,i)})\rho(X_{ i}^{k},X_{g(k,i)}^{k};\theta)+\sigma^{2}S_{i}^{0}}{\|R(i)\|\tau^{2}+\sigma^{2}}, \tag{2}\]  
where \(\|R(i)\|\) is the cardinal number of the game set \(R(i)\). The posterior mean of each team's strength can be calculated in the same fashion. Then, we will have a network with strengths that flow from team to team. Suppose there are \(n\) teams to be ranked. Let \(S=(S_{1},S_{2},\cdots,S_{n})^{\top},S^{0}=(S_{1}^{0},S_{2}^{0},\cdots,S_{n}^{0 })^{\top}\), \(\gamma=(\gamma_{1},\gamma_{2},\cdots,\gamma_{n})^{\top}\), and \(H(S,\gamma)=(h(S_{1},\gamma_{1}),h(S_{2},\gamma_{2}),\cdots,h(S_{n},\gamma_{n }))^{\top}\). Then, the rating vector is recursively defined as follows. \[S=\text{Diag}(\alpha)WH(S;\gamma)+(1-\alpha)S^{0}, \tag{3}\]  
where \(\text{Diag}(\alpha)\) is the diagonal matrix composed of the vector \(\alpha\) whose \(i\)-th element is equal to \(\frac{\|R(i)\|\tau^{2}}{\|R(i)\|\tau^{2}+\sigma^{2}}\), and matrix \(W\) is defined by:  
\[W_{ij}=\left\{ \begin{array}{ll}\frac{1}{\|R(i)\|}\sum_{k\in R(i),g(k,i)=j} \rho(X_{i}^{k},X_{j}^{k};\theta),&\text{team $i$ has played with team $j$,}\\ 0,&\text{otherwise.}\end{array} \right. \tag{4}\]  
Assume \(H(S,\gamma)=\text{Diag}(\gamma)S\), and let \(A=W\text{Diag}(\gamma)\). Then, we obtain the generalized PageRank rating system,  
\[S=\alpha AS+(1-\alpha)S^{0}. \tag{5}\]  
To preserve the column stochastic property of matrix A, we design \(\gamma_{j}\) as a normalization constant in practice, i.e., \(\gamma_{j}=\sum_{i}A_{ij}\). The rating vector for teams is retrieved from the solution of the linear system. Then, the team's ranking can be obtained directly from the rating vector. We hope the link function \(\rho(\cdot,\cdot;\theta)\) is set to ensure the non-negativity of matrix \(A\). Hence, an exponential function is considered, i.e.,  
\[\rho(X_{i}^{k},X_{j}^{k};\theta)=\exp\left[\theta^{T}(X_{i}^{k}-X_{j}^{k}) \right]. \tag{6}\]  
With the link function, the problem of dangling nodes in PageRank can be avoided, and such a setting ensures that even the strength of the losing team will be corrected.