In a [[Network Types#^d09159|directed network]] you can have two kinds of special nodes: - *Sink node:* Only has incoming edges but no outgoing edges (i.e. out-degree is zero) - *Source node:* Only has outgoing edges but no incoming edges (i.e. in-degree is zero) When a [[Network Types#^7075f3|Walk]] passes such a kind of node, it will be stuck forever. This is problematic for [[Eigenvector Centrality]], as the sink / source node has centrality $0$, and this will be propagated through the whole network. We can circumvent this effect by giving every node at least some centrality $\beta$ for free. $ \begin{align} x_i^{(k)} &= \alpha \sum_{j=1}^n A_{ij}x_j^{(k-1)} + \beta \tag{1}\\[10pt] x^{(k)}&= \alpha Ax^{(k-1)} + \beta \tag{2}\\[8pt] x^{(k)} &=\alpha A^kx^{(0)}+ \beta \tag{3} \end{align} $ As $k$ goes to infinity the $x^{(k)}$ converges to a vector $v$, which is a generalization of the eigenvectors. $ x^{(k)}\xrightarrow[]{k \to \infty}v; \quad \quad v=(\mathbf I- \alpha A)^{-1}\beta $