---- > [!definition] Definition. ([[eigenvector centrality]]) > For an undirected [[network]] $G$ with [[adjacency matrix]] $A$, the **eigenvector centrality** $x_{i}$ of node $i$ is the $i^{th}$ entry in the leading [[eigenvector]] [^1] of the [[block matrix|block]] of $A$ corresponding the [[component of a graph|component]] of $G$ to which $x_{i}$ belongs. [^1]: That is, the [[eigenvector]] corresponding to the largest (most positive) [[eigenvalue]]. > [!justification] Conceptual Motivation. > Useful though it is, [[degree centrality]] is quite a crude measure of centrality. In effect, it awards a node one “centrality point” for every neighbor it has. But not all neighbors are necessarily equivalent. In many circumstances a node’s importance in a network is increased by having connections to other nodes that are themselves important. For instance, you might have only one friend in the world, but if that friend is the president of the United States then you yourself may be an important person. Thus centrality is not only about how many people you know but also who you know. \ Eigenvector centrality is an extension of degree centrality that takes this factor into account. Instead of just awarding one point for every network neighbor a node has, eigenvector centrality awards a number of points proportional to the centrality scores of the neighbors. (Newman p.159) \ With eigenvector centrality, a node can achieve high centrality by *having a lot of neighbors with modest centrality* or by *having a few neighbors with high centrality*. > [!justification] Mathematical Motivation. > In light of the above discussion, the eigenvector [[centrality]] for node $i$ is defined to be proportional to the sums of eigenvector [[centrality|centralities]] of $is neighbors, so that $x_{i}=\kappa ^{-1} \sum_{\text{nodes $j$ that neighbor }i}^{} x_{j}$ where we have called the proportionality constant $\kappa ^{-1}$ for reasons soon clear. Using the [[adjacency matrix]] $A$ of $G$, this equals $x_{i}=\kappa ^{-1}\sum_{j=1}^{n}A_{ij}x_{j}=\kappa ^{-1}A \b x,$ or equivalently $A \b x = \kappa \b x,$ where $\b x$ is the [[vector]] of [[centrality]] scores $x_{i}$ of the nodes in $G$. We see that $\b x$ is an [[eigenvector]] of the [[adjacency matrix]] $A$. But $\b x$ could proportional to up to $n$ different [[eigenvector]]s — which should we choose? If $A$ is [[connected graph|connected]], there is actually *only one choice that ensures all centrality scores are nonnegative* by [[Perron-Frobenius Theorem for irreducible matrices]]: the [[eigenvector]] corresponding to the largest (most positive) [[eigenvalue]] of $A$. This also fixes $\kappa$ equal to the largest [[eigenvalue]] of $A$. In general it is not really relevant; we usually care only about which nodes have higher [[centrality]], not what that centrality is. \ If $A$ is not [[connected graph|connected]] the scores are just computed for each component; there is no issue. ^d1cc8e ---- #### ---- #### 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 > ```