1. Importance Measures on Nodes Lecture 2 Srinivasan Parthasarathy 1

2. Importance of Nodes Not all nodes are equally important Centrality Analysis: – Find out the most important nodes in one network Commonly-used Measures – Degree Centrality – Closeness Centrality – Betweenness Centrality – Eigenvector Centrality – PageRank/Prestige Centrality – Clustering Coefficient 2

3. Degree Centrality The importance of a node is determined by the number of nodes adjacent to it – The larger the degree, the more import the node is – Only a small number of nodes have high degrees in many real-life networks Degree Centrality Normalized Degree Centrality: For node 1, degree centrality is 3; Normalized degree centrality is 3/(9-1)=3/8. 3

4. Closeness Centrality “Central” nodes are important, as they can reach the whole network more quickly than non-central nodes Importance measured by how close a node is to other nodes Average Distance: Normalized Closeness Centrality Closeness Centrality (un-normalized) 4

5. Closeness Centrality Example Node 4 is more central than node 3 5

6. Harmonic Centrality What happens when we have disconnected components? Key Idea – Sum of reciprocal distances instead of reciprocal of sum of distances! – SUM (1/dist(u,v)) In the limit – 1/inf  0. 6

7. Betweenness Centrality Node betweenness counts the number of shortest paths that pass one node Nodes with high betweenness are important in communication and information diffusion Betweenness Centrality The number of shortest paths between s and t The number of shortest paths between s and t that pass v i 7

8. Betweenness Centrality Example The number of shortest paths between s and t The number of shortest paths between s and t that pass v i What’s the betweenness centrality for node 5? 8

9. Eigenvector Centrality One’s importance is determined by his friends’ If one has many important friends, he should be important as well. The centrality corresponds to the top eigenvector of the adjacency matrix A. A variant of this eigenvector centrality is the PageRank score. 9

10. Primitives and Notations G = (V, E) – E can also be represented as an adjacency matrix Undirected vs. directed graph Degree (Shortest) distance between two vertices 10

11. Clustering coefficient (local) Clustering coefficient – "all-my-friends-know-each-other" property – Measures the interconnectivity of a node’s neighbors. 5 1 2 3 4 6 vivi vjvj Network Average of (local) clustering coefficient is simply: – AVG (CC(v)) for all vertices in graph

12. Properties of Nodes Clustering coefficient: how much does a node cluster with neighbors – Global clustering coefficient Connected triple – any group of three nodes with two connected edges. Each triangle will have 3 connected triples – hence the 3 in numerator ! 12

13. PAGERANK Centrality Besides the keywords, what other evidence can one use to rate the importance of a webpage ? Solution: Use the hyperlink structure E.g. a webpage linked by many webpages is probably important. – but this method is not global (comprehensive). PageRank is developed by Larry Page in 1998. 13

14. Idea A graph representing WWW – Node: webpage – Directed edge: hyperlink A user randomly clicks the hyperlink to surf WWW. – The probability a user stop in a particular webpage is the PageRank value. A node that is linked by many nodes with high PageRank value receives a high rank itself; If there are no links to a node, then there is no support for that page. 14

15. A simple version u: a webpage B u : the set of u’s backlinks N v : the number of forward links of page v Initially, R(u) is 1/N for every webpage Iteratively update each webpage’s PR value until convergence. 15

16. Example 1 16 PageRank Calculation: first iteration

17. Example 1 17 PageRank Calculation: second iteration

18. Example 1 18 Convergence after some iterations

19. A little more advanced version Adding a damping factor d Imagine that a surfer would stop clicking a hyperlink with probability 1-d R(u) is at least (1-d)/(N-1) – N is total num. of nodes. 19