Graph Theory: Part III (Facebook)

  g = new Neo4jGraph('/tmp/neo4j')

  // calculate basic collaborative filtering for vertex 1
  m = [:]
  g.v(1).out('likes').in('likes').out('likes').groupCount(m)
  m.sort{a,b -> a.value <=> b.value}

  // calculate the primary eigenvector (eigenvector  centrality) of a graph
  m = [:]; c = 0;
  g.V.out.groupCount(m).loop(2){c++ < 1000}
  m.sort{a,b -> a.value <=> b.value}

In case others are/were likewise confused: assume you've been given the influence of all but one of the vertices in the graph. You want to assign the last vertex an influence. What lambda do you pick? You might think you could pick any lambda and arbitrarily set the influence decay across edges.

On the other hand, we want a consistent measure of influence. After you assign that last vertex some influence, you can now recalculate the influence of any other node using your equation. If the recalculated value is different, you picked the wrong lambda.

The eigenvalue problem formalizes the problem of finding a consistent lambda before assigning an influence to any of the vertices.