This document discusses complex network analysis and several concepts related to social networks such as the small world phenomenon, friendship paradox, centrality measures, clustering coefficient, and degree distribution. It provides examples of applying complex network analysis to a friendship network of BITS students and a Twitter growth model. Power law distributions are found to describe properties of many real-world networks like degree distributions.

1. Complex Network Analysis

2. What will you get to know ? To stop the fire you have to create fire Why do your friends seem to be more popular than you are Are we living in a “Small World” How do we detect epidemics early Friendship network in BITS Behavior in Online Social Networking Sites How popular is something on DC++

3. Complex Networks Non-trivial real-life networks Observed in most Social, Biological and Computer networks.

4. The Friendship Paradox “On an average, your friends have more friends than you do” True for all networks (or graphs). Prominent in real life networks.

5. The Small World Phenomenon Any two persons in the world are connected by at most six links of acquaintances. Among Mathematicians: Erdӧs Number (Paul Erdӧs) Among Actors: Bacon Number (Kevin Bacon)

6. http://findthebacon.com/Play.aspx

7. Complex Network Analysis Diameter: Then number of links in the shortest path between furthest nodes. (Small World) Average path-length Degree: Number of links on a particular node(Number of neighbors)

8. Network Density: The ratio of edges in the network to the max possible number of edges. Density of a social network with large number of nodes is highly unlikely to exceed 0.5

9. Clustering Coefficient: Likelihood that two associates of a node are associates themselves Lies between 0 and 1 Y X A

10. Centrality Measures (Betweenness): The number of shortest path that passes through a node. Synonymous with importance. Important in study of spreading of forest fires, rumors, information, epidemics etc. Revisit “Friendship Paradox”

11. BITSian Friendship Network

12. BITSian Friendship Network Network Density: 0.37 Diameter: 4 Average Path-length: 1.99 Average Clustering Coefficient: 0.51

13. Twitter Growth Model With probability p, a new node(user) enters the network and links with one existing node. With probability q = 1-p, an existing user gets linked to an existing node. Preferential Selection: P(deg i -> deg i+1) proportional to (i+constant)

14. The Twitter growth model The rate equations are:

15. Formula vs Model Simulation

16. Model vs Twitter Data

17. Power Law!!! Degree distribution: n(j) = c.j-γ Straight line in log-log plot. Scale free networks. Many networks conjectured(and many found) to follow power law. Eg.-Online Social Networks, Friendship Network, Collaboration Network (Movie-Actor, Research-Scientists), World Wide Web, Protien-Protien Interaction, Airline Networks Pareto Principle: 80-20 rule.

18. DC++ Search Spy A similar approach can be applied to find out number of searches vs “rank” of search query. query keyword

19. Power Law !!!

20. Rank of a keyword (node) = number of nodes with degree greater than its degree. The inverse function gives the frequency of a keyword ranked r: POWER LAW !!!

21. Formula matches with the Real DC++ data