Lecture 9

Measures and Metrics

Structural Metrics Degree distribution Average path length

Centrality Degree, Eigenvector, Katz, Pagerank, Closeness, Betweenness Hubs and Authorities

Transitivity Clustering coefficient

Reciprocity Signed Edges and Structural balance Similarity Homophily and Assortativity Mixing

2

Transitivity

3

Structural Metrics:Clustering coefficient

4

Local Clustering and Redundancy

5

Reciprocity

6

Signed Edges and Structural balance

7

Similarity

Structural Equivalence Cosine Similarity

Pearson Coefficient

Euclidian Distance

Regular Equivalence Katz Similarity

8

Homophily and Assortative Mixing Assortativity: Tendency to be linked with nodes that are

similar in some way Humans: age, race, nationality, language, income, education

level, etc. Citations: similar fields than others Web-pages: Language

Disassortativity: Tendency to be linked with nodes that are different in some way Network providers: End users vs other providers

Assortative mixing can be based on Enumerative characteristic Scalar characteristic

9

Modularity (enumerative) Extend to which a node is connected to a like in network

+ if there are more edges between nodes of the same type than expected value

- otherwise

is 1 if ci and cj are of same type, and 0 otherwise

err is fraction of edges that join same type of vertices ar is fraction of ends of edges attached to vertices type r

10

Assortative coefficient (enumerative) Modularity is almost always less than 1, hence we can

normalize it with the Qmax value

11

Assortative coefficient (scalar)

r=1, perfectly assortative r=-1, perfectly disassortative r=0, non-assortative

Usually node degree is used as scale

12

Assortativity Coefficient Various Networks

13M.E.J. Newman. Assortative mixing in networks