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# Complex Network Analysis

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### Complex Network Analysis

1. 1. Complex Network Analysis<br />
2. 2. What will you get to know ?<br />To stop the fire you have to create fire<br />Why do your friends seem to be more popular than you are<br />Are we living in a “Small World”<br />How do we detect epidemics early<br />Friendship network in BITS<br />Behavior in Online Social Networking Sites<br />How popular is something on DC++<br />
3. 3. Complex Networks<br />Non-trivial real-life networks<br />Observed in most Social, Biological and Computer networks.<br />
4. 4. The Friendship Paradox<br />“On an average, your friends have more friends than you do”<br />True for all networks (or graphs).<br />Prominent in real life networks.<br />
5. 5. The Small World Phenomenon<br />Any two persons in the world are connected by at most six links of acquaintances.<br />Among Mathematicians: Erdӧs Number (Paul Erdӧs)<br />Among Actors: Bacon Number (Kevin Bacon)<br />
6. 6. http://findthebacon.com/Play.aspx<br />
7. 7. Complex Network Analysis<br />Diameter: Then number of links in the shortest path between furthest nodes. (Small World)<br />Average path-length<br />Degree: Number of links on a particular node(Number of neighbors)<br />
8. 8. Network Density: The ratio of edges in the network to the max possible number of edges.<br /> Density of a social network with large number of nodes is highly unlikely to exceed 0.5<br />
9. 9. Clustering Coefficient: Likelihood that two associates of a node are associates themselves<br />Lies between 0 and 1<br />Y<br />X<br />A<br />
10. 10. Centrality Measures (Betweenness): The number of shortest path that passes through a node.<br />Synonymous with importance.<br />Important in study of spreading of forest fires, rumors, information, epidemics etc.<br />Revisit “Friendship Paradox”<br />
11. 11. BITSian Friendship Network<br />
12. 12. BITSian Friendship Network<br />Network Density: 0.37<br />Diameter: 4<br />Average Path-length: 1.99<br />Average Clustering Coefficient: 0.51<br />
13. 13. Twitter Growth Model<br />With probability p, a new node(user) enters the network and links with one existing node.<br />With probability q = 1-p, an existing user gets linked to an existing node.<br />Preferential Selection:<br />P(deg i -> deg i+1) proportional to (i+constant)<br />
14. 14. The Twitter growth model<br />The rate equations are:<br />
15. 15. Formula vs Model Simulation<br />
16. 16. Model vs Twitter Data<br />
17. 17. Power Law!!!<br />Degree distribution: n(j) = c.j-γ<br />Straight line in log-log plot.<br />Scale free networks.<br />Many networks conjectured(and many found) to follow power law.<br />Eg.-Online Social Networks, Friendship Network, Collaboration Network (Movie-Actor, Research-Scientists), World Wide Web, Protien-Protien Interaction, Airline Networks<br />Pareto Principle: 80-20 rule.<br />
18. 18. DC++ Search Spy<br />A similar approach can be applied to find out number of searches vs “rank” of search query.<br />query<br />keyword<br />
19. 19. Power Law !!!<br />
20. 20. Rank of a keyword (node) = number of nodes with degree greater than its degree.<br />The inverse function gives the frequency of a keyword ranked r:<br />POWER LAW !!!<br />
21. 21. Formula matches with the Real DC++ data<br />