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: Erds Number (Paul Erds)
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-
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