Complex Network Analysis

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  • 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.
  • 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
  • 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.
  • 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