## 3 Bounds via Partial Spectral Factorization  
This section recalls how to derive bounds for expressions of the form (8), with \(\|\mathbf{u}\|=\|\mathbf{v}\|=1\), using a partial spectral factorization of \(A\). Introduce the spectral factorization  
\[A=Q\Lambda Q^{T},\]  
where the eigenvector matrix \(Q=[\mathbf{q}_{1},\mathbf{q}_{2},\ldots,\mathbf{q}_{n}]\in\mathbb{R}^{n\times n}\) is orthogonal and the eigenvalues in the diagonal matrix \(\Lambda=\text{diag}[\lambda_{1},\lambda_{2},\ldots,\lambda_{n}]\in\mathbb{R}^ {n\times n}\) are ordered according to \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\). Then  
\[f(A)=Qf(\Lambda)Q^{T}=\sum_{k=1}^{n}f(\lambda_{k})\mathbf{q}_{k}\mathbf{q}_{k }^{T},\]  
so that  
\[F_{\mathbf{u},\mathbf{v}}(A)=\mathbf{u}^{T}f(A)\mathbf{v}=\sum_{k=1}^{n}f( \lambda_{k})\bar{u}_{k}\bar{v}_{k},\]  
where \(\bar{u}_{k}=\mathbf{u}^{T}\mathbf{q}_{k}\) and \(\bar{v}_{k}=\mathbf{v}^{T}\mathbf{q}_{k}\). Let the first \(N\) eigenpairs \(\{\lambda_{k},\mathbf{v}_{k}\}_{k=1}^{N}\) of \(A\) be known. Then \(F_{\mathbf{u},\mathbf{v}}(A)\) can be approximated by  
\[F_{\mathbf{u},\mathbf{v}}(A)\approx F_{\mathbf{u},\mathbf{v}}^{(N)}:=\sum_{k= 1}^{N}f(\lambda_{k})\bar{u}_{k}\bar{v}_{k}. \tag{12}\]  
The following results from [20] shows how upper and lower bounds for \(F_{\mathbf{u},\mathbf{v}}(A)\) can be determined with the aid of the first \(N\) eigenpairs of \(A\). Theorem 3.1: _Let the function \(f\) be nondecreasing and nonnegative on the convex hull of the spectrum of \(A\) and let \(F_{\mathbf{u},\mathbf{v}}^{(N)}\) be defined by (12). Let \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}\) be the \(N\) largest eigenvalues of \(A\) and let \(\mathbf{q}_{1},\mathbf{q}_{2},\ldots,\mathbf{q}_{N}\) be associated orthonormal eigenvectors. Then we have_  
\[L_{\mathbf{u},\mathbf{v}}^{(N)}\leq F_{\mathbf{u},\mathbf{v}}(A)\leq U_{ \mathbf{u},\mathbf{v}}^{(N)}, \tag{13}\]  
_where_  
\[L_{\mathbf{u},\mathbf{v}}^{(N)} :=F_{\mathbf{u},\mathbf{v}}^{(N)}-f(\lambda_{N})\Bigg{(}1-\sum_ {k=1}^{N}\bar{u}_{k}^{2}\Bigg{)}^{1/2}\Bigg{(}1-\sum_{k=1}^{N}\bar{v}_{k}^{2} \Bigg{)}^{1/2},\] \[U_{\mathbf{u},\mathbf{v}}^{(N)} :=F_{\mathbf{u},\mathbf{v}}^{(N)}+f(\lambda_{N})\Bigg{(}1-\sum_{k =1}^{N}\bar{u}_{k}^{2}\Bigg{)}^{1/2}\Bigg{(}1-\sum_{k=1}^{N}\bar{v}_{k}^{2} \Bigg{)}^{1/2}.\]  
_When \(\mathbf{u}=\mathbf{v}\), we have_  
\[F_{\mathbf{u},\mathbf{u}}^{(N)}\leq F_{\mathbf{u},\mathbf{u}}(A)\leq U_{ \mathbf{u},\mathbf{u}}^{(N)} \tag{14}\]\[F_{\mathbf{u},\mathbf{u}}^{(N)}\leq F_{\mathbf{u},\mathbf{u}}^{(N+1)},\qquad U_{ \mathbf{u},\mathbf{u}}^{(N)}\geq U_{\mathbf{u},\mathbf{u}}^{(N+1)},\qquad 1\leq N<n. \tag{15}\]  
To determine which nodes have the largest \(f\)-subgraph centrality (4), we use the inequalities (14) and (15). The \(N\) leading eigenpairs \(\{\lambda_{k},\mathbf{q}_{k}\}_{k=1}^{N}\) of \(A\) and the bounds (13) and (14) can be used to determine a subset of nodes that contains the vertices with the largest value of the node metric we are considering. Let \(L_{\mathbf{u},\mathbf{v}}^{(N)}\) and \(U_{\mathbf{u},\mathbf{v}}^{(N)}\) be the lower and upper bounds defined in Theorem 1. Since we seek an approximation of the centrality value for all the nodes of the network, we will either set \(\mathbf{u}=\mathbf{v}=\mathbf{e}_{i}\), as in (4), or \(\mathbf{u}=\mathbf{e}\) and \(\mathbf{v}=\mathbf{e}_{i}\), as in (6). So, \(F_{\mathbf{u},\mathbf{v}}^{(N)}\) will be a quantity depending on an index \(i=1,\ldots,n\). We will write \(F_{i}^{(N)}=F_{\mathbf{u},\mathbf{v}}^{(N)}\) to simplify the notation when we are computing either \(F_{\mathbf{e}_{i},\mathbf{e}_{i}}^{(N)}\) or \(F_{\mathbf{e},\mathbf{e}_{i}}^{(N)}\). When approximating (5), we will fix a value of \(j\) and consider \(F_{i}^{(N)}=F_{\mathbf{e}_{j},\mathbf{e}_{i}}^{(N)}\) for \(i=1,\ldots,n\). Let \(\mathcal{F}_{m}^{(N)}\) denote the \(m\)th largest value of the vector \((F_{i}^{(N)})_{i=1}^{n}\).