Lecture 8 Measures and Metrics

1. Lecture 8 Measures and Metrics

2. Cocitation and Bibliographic coupling • Cocitation of two vertices i and j is the number of vertices that have outgoing edges to both • Bibliographic coupling is the number of vertices to which both point

3. Independent paths • Edge independent paths: if they share no common edge • Vertex independent paths: if they share no common vertex except start and end vertices • Vertex-independent => Edge-independent • Also called disjoint paths • These set of paths are not necessarily unique • Connectivity of vertices: the maximal number of independent paths between a pair of vertices • Used to identify bottlenecks and resiliency to failures

4. Cut Sets and Maximum Flow • A minimum cut set is the smallest cut set that will disconnect a specified pair of vertices • Need not to be unique • Menger’s theorem: If there is no cut set of size less than n between a pair of vertices, then there are at least n independent paths between the same vertices. • Implies that the size of min cut set is equal to maximum number of independent paths • for both edge and vertex independence • Maximum Flow between a pair of vertices is the number of edge independent paths times the edge capacity.

5. Transitivity •  is said to be transitive if a  b and b  c together imply a  c • Perfect transitivity in network → cliques • Partial transitivity • u knows v and v knows w → =

6. Structural Metrics:Clustering coefficient

7. Local Clustering and Redundancy • Redundancy

8. Reciprocity • How likely is it that the node you point to will point to you as well.

9. Signed Edges and Structural balance • Friends / Enemies • Friend of friend → • Enemy of my enemy → • Structural balance: only loops of even number of “negative links” • Structurally balanced → partitioned into groups where internal links are positive and between group links are negative

10. Similarity • Structural Equivalence: share many of the same neighbors • Cosine Similarity: • Pearson Coefficient: Given degree of two nodes, how many common neighbors they have () • Euclidian Distance: • Regular Equivalence: neighbors are the same • Katz Similarity:

11. Homophily and AssortativeMixing • 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

12. 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 ciand 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

13. Assortativecoefficient (enumerative) • Modularity is almost always less than 1, hence we can normalize it with the Qmax value

14. Assortativecoefficient (scalar) • r=1, perfectly assortative • r=-1, perfectly disassortative • r=0, non-assortative • Usually node degree is used as scale

15. Assortativity Coefficientof Various Networks M.E.J. Newman. Assortative mixing in networks