1. The Very Small World of the Well-connected. (19 june 2008 ) Lada Adamic School of Information University of Michigan Ann Arbor, MI 48109-1107 ladamic@umich.edu Anna C. Gilbert Department of Mathematics University of Michigan Ann Arbor, MI 48109-1043 annacg@umich.edu Xiaolin Shi Department of EECS University of Michigan Ann Arbor, MI 48109-2121 shixl@umich.edu Matthew Bonner Department of EECS University of Michigan Ann Arbor, MI 48109-2121 mabonner@umich.edu School of Information. University of Michigan Ann Arbor, MI 48109-1107

2. PRELIMINARIES Importance measures Network datasets proprieties description IMPORTANT VERTICES Network properties and important vertices Original vs. subgraph properties Summary Introduction The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

3. Importance measures Let the graph G (V,E ) have |V | = n vertices 1 Degree D (vi ): Is the number of edges incident to vi. Degree reflects a local property of the vertices in the graph. PRELIMINARIES The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

4. Importance measures Let the graph G (V,E ) have |V | = n vertices 1 Degree D (vi ). 2 Betweenness B (vi ) : a measure of how many pairs of vertices go through vi in order to connect through shortest paths in G: PRELIMINARIES The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

5. Importance measures Let the graph G (V,E ) have |V | = n vertices 1 Degree D (vi ). 2 Betweenness B (vi ). 3 Closeness C (vi ): a measure of the distances from all other vertices in G to vertex vi closeness means that vertices that are in the “middle” of the network are important. PRELIMINARIES The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

6. Importance measures Let the graph G (V,E ) have |V | = n vertices 1 Degree D (vi ). 2 Betweenness B (vi ). 3 Closeness C (vi ). 4 PageRank : a variant of the Eigenvector centrality measure and assigns greater importance to vertices that are themselves neighbors of important vertices PRELIMINARIES The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

7. Network datasets proprieties description Data sets is a representative of web. Data sets as an online social network data. Data sets will be interested in examining the properties of important vertices and their graph synopsis. PRELIMINARIES The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

8. Network datasets proprieties description prototypical random graph 1 Erdos-Renyi random graph : each pair of vertices having an equal probability p of being joined by an edge. |V | = 10000 ; p = 0.001 ; d = p × |V | = 10. PRELIMINARIES The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

9. Network datasets proprieties description prototypical random graph 1 Erdos-Renyi random graph. 2 Budyzoo dataset : Considered as the first real-world network producing an undirected graph from AOL Instant Messenger (AIM) Users >> Nodes Contact list >> edges PRELIMINARIES The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

10. Network datasets proprieties description prototypical random graph 1 Erdos-Renyi random graph. 2 Budyzoo dataset. 3 TREC (Text REtrieval Conference). Considered as the second real-world graph is a network of blog connections It is a crawl of 100,649 RSS and Atom feeds collected The TREC dataset contains Hyperlinks, comments, trackbacks, etc. removed feeds and feeds without a homepage or permalinks are. over 300 Technorati tags. which are in fact automatically generated are not true indicators of social linking. PRELIMINARIES The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

11. Network datasets proprieties description prototypical random graph 1 Erdos-Renyi random graph. 2 Budyzoo dataset. 3 TREC (Text REtrieval Conference). 4 Web graph dataset 259,794 websites 50 million pages Collected in 1998 PRELIMINARIES The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

12. Network datasets proprieties description prototypical random graph 1 Erdos-Renyi random graph. 2 Budyzoo dataset. 3 TREC (Text REtrieval Conference). 4 Web graph dataset PRELIMINARIES The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107 recent blog datasets the decade old website-level data set == Similarity == applicable to larger, ore current webcrawls

13. Network properties and important vertices 1 Degree distributions. The degree distributions of online networks IMPORTANT VERTICES The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107 Type : social networks due to the limitation of The data sampling

14. Network properties and important vertices 1 Degree distributions. 2 Correlation of importance values of different measures. relationships of importance measures in different networks. Analysis of correlation Higher : degree, betweenness and PageRank Lower : closeness. The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107 IMPORTANT VERTICES

15. Network properties and important vertices 1 Degree distributions. 2 Correlation of importance values of different measures. 3 Assortativity. The concept of assortativity or assortative mixing is defined as the preference of the vertices in a network to have edges with others that are similar. The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107 IMPORTANT VERTICES

16. The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107 Assortativity :

17. Important vertices in their subgraphs. The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

18. Connectivity The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

19. Density The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

20. Network properties and important vertices 1 Degree distributions. 2 Correlation of importance values of different measures. Assortativity. 3 Important vertices in their subgraphs. Connectivity Density The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107 IMPORTANT VERTICES

21. The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107 IMPORTANT VERTICES Original vs. subgraph properties 1 Density 2 distance. 3 Relative importance.

22. Density The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

23. distance The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

24. Relative importance The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107

25. Original vs. subgraph properties 1 Density. 2 distance. 3 Relative importance. The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107 IMPORTANT VERTICES

26. two overall observations about the four networks: Different importance measures yield subgraphs of varying density and topology However, in spite of these differences, “important vertices” in the online networks have some properties that agree with each other Thus, we know that in the real online networks, in contrast to random graph model the important vertices tend to preserve information about the relationships among important vertices we can use the subgraphs to study the properties of important vertices in the original graphs. The Very Small World of the Well-connected. School of Information. University of Michigan Ann Arbor, MI 48109-1107 Summary and conclusion