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Eigenvector Centrality

The idea of the eigenvector centrality is to not only look at the number of neighbours a node has, but to also look at the importance of the node's neighbours. Therefore, a node has a high eigenvector centrality if it either has a high number of neighbours, or a few, but important neighbours.

Because the eigenvector centrality is calculated for a single node, it is of ego-centric scope. The metric was proposed by Smith et al. (2009), Berger et al. (2014) and discussed in Wasserman et al. (1994).

Calculation

The calculation is based on Bonacich (1987) and explained by Newman (2010). It makes use of the adjacency matrix $A$.

Let $x(0)$ be the initial guess for the vector of centralities. Because it is unknown which node may be important, we assume an arbitrary value for the centralities. This value has to be equal for all elements in the vector. A common approach is to set all elements in $x(0)$ to 1.

Now the vector is multiplied with the adjacency matrix $A$ to calculate the update value $x(1)$: $$ x(1) = A * x(0). $$

Multiplying with the adjacency matrix assigns each element in the centrality vector the sum of the values of its neighbours. This can be repeated any number of times, thus we take the matrix $A$ to the power of $t$. $$ x(t) = A^t * x(0). $$

To get a final value, we let $t$ approach infinity. For this we write $x(0)$ as a linear combination of the eigenvectors according to Newman (2010): $$ x(t) = A^t \sum_i c_i v_i = \sum_i c_i K^t_i v_i = K^t_1 \sum_i c_i \left[\frac{K_i}{K_1}\right]^t v_i. $$

He describes it as follows: "$K_i$ are the eigenvalues of $A$, and $K_j$ is the largest of them. Since $K_i/K_j$ < 1 for all $i \neq 1$, all terms in the sum decay exponentially as $t$ becomes large, and hence in the limit $t \rightarrow \infty$ we get $x(t) \rightarrow c_j K^t_i v_j$. In other words, the limiting vector of centralities is proportional to the leading eigenvector of the adjacency matrix. Equivalently we could say that the centrality $x$ satisfies the formula" (Newman, 2010): $$ Ax = K_jx, $$

which is the eigenvector centrality first proposed by Bonacich (1987). Newman (2010) mentions that the centrality $x_i$ of the node $v_i$ is proportional to the sum of the centralities of $x_i$'s neighbours.

This means that the eigenvector centrality can be of a high value, because of the node's neighbours importance or the node's own importance. Therefore final formula looks like: $$ x_i = K^{-1}_j \sum{A_{ij}x_j}. $$

A visual explanation of the calculation was created by Dan Ryan^dr.

Interpretation

Smith et al. (2009) claim the that eigenvector centrality is interpreted similar to the betweenness centrality. Users that form relationships with important neighbours, become relevant themselves (Newman, 2010). Due to their social relationships they fill a structural hole in the network leading to Bridging Social Capital. The neighbours pass information along a user with high eigenvector centrality. Therefore, such a user sees a lot of information and can disseminate the information to more users. Berger et al. 2014 identify this as a characteristic of key users, who contribute the most to the network. A low eigenvector centrality implies that a user is not well connected with other users. Thus, he is not in a position, where he receives a lot of information and does not establish Bridging Social Capital.