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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++
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Complex Networks Non-trivial real-life networks Observed in most Social, Biological and Computer networks.
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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.
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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)
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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)
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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
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Clustering Coefficient: Likelihood that two associates of a node are associates themselves Lies between 0 and 1 Y X A
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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”
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BITSian Friendship Network Network Density: 0.37 Diameter: 4 Average Path-length: 1.99 Average Clustering Coefficient: 0.51
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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)
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The Twitter growth model The rate equations are:
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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.
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DC++ Search Spy A similar approach can be applied to find out number of searches vs “rank” of search query. query keyword
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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 !!!
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