---- > [!definition] Definition. ([[Katz centrality]]) > For a [[network]] $G$ with $|G|=n$ and [[adjacency matrix]] $A$, the **Katz centrality** $x_{i}$ of node $i$ is equal to the $i^{th}$ entry of the [[vector]] $\b x := (I_{n}-\alpha A)^{-1} \v 1= \b 1 + \alpha A \b 1 + \alpha^{2}A^{2}\b 1+ \dots$ > where $0 < \alpha < \frac{1}{\kappa_{1}}$ is a [[hyperparameter]]. ($\kappa_{1}$ is the leading [[eigenvalue]] of $A$.) > [!note] Remark. > The [[series]] expansion is obtained as follows. The [[Perron-Frobenius Theorem for irreducible matrices]] implies that $\b 1$ is the [[eigenvector]] of $A$ with largest [[eigenvalue]]; call that [[eigenvalue]] $\kappa_{1}$. > First write the LHS as $\b x=\alpha A \b x + \b 1.$ > Now since $A \b 1=\kappa_{1} \b 1$, we equivalently have $\begin{align} \b x= \alpha A \b x + \frac{A \b 1}{\kappa_{1}}. \end{align}$ Common denominators yields $\b x = \frac{\alpha \kappa_{1} A \b x + A \b 1}{\kappa_{1}},$ > [!justification] Motivation. > Recall the **warning** regarding [[eigenvector centrality]]: ![[eigenvector centrality#^d1cc8e]] > One solution to such issues is the following: simply give each node a small amount of [[centrality]] 'for free', regardless of other factors. That is, define $x_{i}=\alpha \sum_{j=1}^{n} A_{ij}x_{j} + \beta$ > for constants $\alpha, \beta > 0$. In [[matrix]] terms, this is $\b x = \alpha A \b x + \beta \b 1.$ > Rearranging for $\b x$ yields the expression $\b x = \beta (I_{n} - \alpha A)^{-1} \b 1,$ > but we don't care about the scale factor $\beta$ so we just write $\b x = (I - \alpha A)^{-1} \b 1.$ > > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```