# Eigenvector centrality¶

Eigenvector centrality measure give us information about how given node is important in network. It is based on degree centrality. In here we have more sophisticated version, where connections are not equal.

\(E(x)=\frac{1}{\lambda}\sum_{j=1}^{n}{A_{ij}x_j}\)

Eigenvector centrality is more general approach than PageRank. For further informations please refer to [Newman].

Library uses pregel operator in order to do computations.

```
import ml.sparkling.graph.operators.OperatorsDSL._
import org.apache.spark.SparkContext
import org.apache.spark.graphx.Graph
implicit ctx:SparkContext=???
// initialize your SparkContext as implicit value
val graph =???
// load your graph (for example using Graph loading API)
val centralityGraph: Graph[Double, _] = graph.eigenvectorCentrality()
// Graph where each vertex is associated with its eigenvector centrality
```

You can also compute eigenvector centrality for graph treated as undirected one:

```
import ml.sparkling.graph.operators.OperatorsDSL._
import org.apache.spark.SparkContext
import ml.sparkling.graph.api.operators.measures.VertexMeasureConfiguration
import org.apache.spark.graphx.Graph
implicit ctx:SparkContext=???
// initialize your SparkContext as implicit value
val graph =???
// load your graph (for example using Graph loading API)
val centralityGraph: Graph[Double, _] = graph.eigenvectorCentrality(VertexMeasureConfiguration(treatAsUndirected=true))
// Graph where each vertex is associated with its eigenvector centrality computed for undirected graph
```

Eigenvector centrality is implemented using iterative approach and Pregel operator. Because of that you can provide your own computation stop predicate:

```
import org.apache.spark.graphx.GraphLoader
import org.apache.spark.sql.SparkSession
import org.apache.spark.SparkContext
import org.apache.spark.graphx.Graph
import ml.sparkling.graph.api.operators.measures.VertexMeasureConfiguration
import ml.sparkling.graph.operators.measures.vertex.eigenvector.EigenvectorCentrality
import ml.sparkling.graph.operators.OperatorsDSL._
// initialize your SparkContext as implicit value
val graph =???
val eic = EigenvectorCentrality.computeEigenvector(graph,VertexMeasureConfiguration(),(iteration,oldValue,newValue)=>iteration<999).vertices
```

As you can see, you can also use average values of Eigenvector centrality in consecutive iterations.

References:

[Newman] | Newman, M. E. (2008). The mathematics of networks. The new palgrave encyclopedia of economics, 2(2008):1–12., PDF |