Introduction to the Modeling and Analysis of Complex Systems

17.3. CENTRALITIES AND CORENESS 381Degree centrality is simply a normalized node degree, i.e., the actual degreedivided by the maximal degree possible (n − 1). For directed networks, you candefine in-degree centrality and out-degree centrality separately.Betweenness centralityc B (i) =1(n − 1)(n − 2)∑j≠i,k≠i,j≠kN sp (j i −→ k)N sp (j → k)(17.20)where N sp (j → k) is the number of shortest paths from node j to node k, andiN sp (j −→ k) is the number of the shortest paths from node j to node k thatgo through node i. Betweenness centrality of a node is the probability for theshortest path between two randomly chosen nodes to go through that node.This metric can also be defined for edges in a similar way, which is called edgebetweenness.Closeness centrality(∑ )jd(i → j) −1c C (i) =(17.21)n − 1This is an inverse of the average distance from node i to all other nodes. Ifc C (i) = 1, that means you can reach any other node from node i in just onestep. For directed networks, you can also define another closeness centralityby swapping i and j in the formula above to measure how accessible node i isfrom other nodes.Eigenvector centralityc E (i) = v i(i-th element of the dominant eigenvector v of thenetwork’s adjacency matrix) (17.22)Eigenvector centrality measures the “importance” of each node by consideringeach incoming edge to the node an “endorsement” from its neighbor. Thisdiffers from degree centrality because, in the calculation of eigenvector centrality,endorsements coming from more important nodes count as more. Anothercompletely different, but mathematically equivalent, interpretation of eigenvectorcentrality is that it counts the number of walks from any node in the network