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Power Laws By Cameron Megaw 3/11/2013

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What is a Power Law?

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Measuring Power Laws Sampling Errors

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Measuring Power Laws Sampling errors Solution 1: Throw out the data in the tail of the curve Statistically significant information lost Some distributions only follow a power law distribution in their tail Not recommended

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Measuring Power Laws Sampling errors Solution 2: Very the width of the bins Normalize the data Results in a count per unit interval of x Very bin size by a fixed multiplier (for example 2) Bins become: 1 to 1.1, 1.1 to 1.3, 1.3 to 1.7 and so on Called logarithmic binning

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Measuring Power Laws Sampling errors

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Measuring Power Laws Unknown exponent

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Mathematics of Power Laws Calculating C

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Mathematics of Power Laws Moments

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Mathematics of Power Laws Largest Value

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Mathematics of Power Laws Scale Free Distribution

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Mechanisms for Generating Power Laws Some examples : Combinations of exponents Inverses of quantities Random Walks The Yule process Critical phenomena

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The Topology of the Internet Some Key Questions What does the internet look like? Are there any topological properties that stay constant in time? How can I generate Internet-like graphs for simulation?

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Internet Instances Three Inter-domain topologies November 1997, April and December 1998 One Router topology from 1995

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Metrics

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Outdegree of a Node and it’s Rank

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Inter domain topologies Correlation coefficient above.974 Exponents -.81, -.82, -.74 Router Correlation coefficient.948 Exponent -.48

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Outdegree and it’s Rank Power Law Analysis

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Frequency of the Outdegree

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Inter domain topologies Correlation coefficient above.968 Exponents -2.15, -2.16, and -2.2 Router Correlation coefficient.966 Exponent -2.48

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The exponent is relatively fixed for the three inter-domain topologies Topological property is fixed in time Could be used to generate models or test authenticity Similar exponent value for the router topology Could suggest a fundamental property of the network Frequency of the Outdegree Power Law Analysis

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Eigenvalues and their Ordering

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Inter domain topologies Correlation coefficient.99 Exponents -.47, -.50, and -.48 Router Correlation coefficient.99 Exponent -.1777

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Eigenvalues are closely related to many topological properties Graph diameter Number of edges Number of spanning trees… The exponent is relatively fixed for the three inter-domain topologies Topological property seems fixed in time Can be used to generate models Significant difference in the exponent value for the router topology Can characterize different families of graphs Eigenvalues and their Ordering Power Law analysis

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Hop Plot Exponent

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Inter domain topologies First 4 hops Correlation coefficient above.96 Exponents -4.6, -4.7, -4.86 Router First 12 hops Correlation coefficient.98 Exponent -2.8

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The exponent is relatively fixed for the three inter-domain topologies Topological property seems fixed in time Can be used to generate models Significant difference in the exponent value for the router topology Can characterize different families of graphs Hop Plot Exponent Power Law analysis

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The Effective Diameter

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Average Neighborhood Size

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Conclusions Power Law and Internet topology Can assess realism of synthetic graphs Provide important parameters for graph generators Help with network protocols Help answer “what if” questions What would the diameter be if the number of nodes doubles? What would be the average neighborhood size be?

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Questions?

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