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1.Peer-to-Peer and Social Networks Power law graphs

2.Random vs. Power-law Graphs The degree distribution in of the webpages in the World Wide Web follows a power-law Binomial distribution

3.Random vs. Power-Law networks

4.Example: Airline Routes Think of how new routes are added to an existing network

5.Examples of Power law distribution Also known as scale-free graph . Other examples are -- Airport network -- Income and number of people with that income -- Magnitude and number of earthquakes of that magnitude -- Population and number of cities with that population

6.Preferential attachment New node Existing network A new node connects with an existing node with a probability proportional to its degree . The sum of the node degrees = 8 This leads to a power-law distribution ( Barabási & Albert) Also known as “ Rich gets richer ” policy

7.Preferential attachment Barabási and Albert showed that when large networks are formed by the rules of preferential attachment , the resulting graph shows a power-law distribution of the node degrees. We will derive it in the class, so follow the lecture.

8.Preferential attachment The probability that the new node connects with an existing node = Since and so Degree of node = At t = 0, there are no nodes. At t = 1, one node appears. Thereafter, each time unit, a new node is added

9.Preferential attachment = number of nodes with degree k after time step t

10.Preferential attachment is then fraction of nodes with degree k at time t

11.Preferential attachment As Call it

12.Preferential attachment is of the order of * Before time step (t+1), the new node is the only node with degree 0, and its degree will change to 1 *

13.Other properties of power law graphs Graphs following a power-law distribution have a small diameter ( n = number of nodes). The clustering coefficient decreases as the node degree increases (power law again) Graphs following a power-law distribution tend to be highly resilient to random edge removal , but quite vulnerable to targeted attacks on the hubs.