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Power Law and Its Generative Models

Power Law and Its Generative Models. Bo Young Kim 2010-03-16. Contents. Recall The Definition of Power Law Recall Some Properties of Power Law Generative Models for Power Law - Power Laws via Preferential Attachment - Power Laws via Multiplicative Processes.

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Power Law and Its Generative Models

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  1. Power Law and Its Generative Models Bo Young Kim 2010-03-16

  2. Contents • Recall The Definition of Power Law • Recall Some Properties of Power Law • Generative Models for Power Law - Power Laws via Preferential Attachment - Power Laws via Multiplicative Processes Applied Algorithm Lab.

  3. Recall The Definition of Power Law • Recall Some Properties of Power Law • Generative Models for Power Law - Power Laws via Preferential Attachment - Power Laws via Multiplicative Processes Applied Algorithm Lab.

  4. 1. Recall The Definition of Power Law • X: a nonnegative random variable • Def Power Law X is said to have a power law distribution if Pr[X≥x]~cx-αfor constants c>0, α>0 • Def f(x)~g(x) ⇔ limx f(x)/g(x) = 1 • What does this mean? In a power law distribution, asymptotically the tails fall according to the power α. (heavier tail than exponential distribution) Applied Algorithm Lab.

  5. Recall The Definition of Power Law • Recall Some Properties of Power Law • Generative Models for Power Law - Power Laws via Preferential Attachment - Power Laws via Multiplicative Processes Applied Algorithm Lab.

  6. 2. Recall Some Properties of Power Law • E.g. The Pareto distribution Pr[X≥x]=(x/k)-α ln(Pr[X≥x])=-α(ln(x)-ln(k)) * Linear Log-log plot (complementary cumulative distribution function) - X has a power law distribution - Then a log-log plot behavior is a straight line. (asymptotic sense) Applied Algorithm Lab.

  7. 2. Recall Some Properties of Power Law “Scale Invariance” - Let f(x) := P[X≥x] - f(x) ~ cx-α - f(kx) ~ c(kx) -α = k-α(cx-α) = k’f(x) ∝ f(x) (k’=k-α) - Scaling by a constant simply multiplies the original power law relation by the constant k’. - If we change the measurement unit(=scale), it retains the same power law form w/ the same exponent.  We cannot decide what scale we’re observing. (like Fractals) Applied Algorithm Lab.

  8. 2. Recall Some Properties of Power Law • Web follows power law. [4] • Recall (Rank exponent) - dv: outdegree of a node v - rv: the rank of a node v dv=k*rvR (R,k: constant) • Designing random graph models that yield Web-like graphs? • i.e. that yields power law distributions for the indegree and outdegree? Applied Algorithm Lab.

  9. Recall The Definition of Power Law • Recall Some Properties of Power Law • Generative Models for Power Law - Power Laws via Preferential Attachment - Power Laws via Multiplicative Processes Applied Algorithm Lab.

  10. Generative Models for Power Law - Power Laws via Preferential Attachment • Def Preferential Attachment Process (=Yule Process) Any process s.t. some quantity (some form of wealth) is distributed among a number of individuals according to how much they already have,so that those who are already wealthy receive more than those who are not. • ”The rich get richer” Applied Algorithm Lab.

  11. Generative Models for Power Law - Power Laws via Preferential Attachment • The Chinese Restaurant Process - A Chinese restaurant has infinitely many tables - Each table can seat infinitely many customers - At each time step, customer Xtcomes into the restaurant. When Xt+1 comes into here… (CRP1) Sits at an already occupied table k w/ prob. Nk/(t+α) (Nk: # of customers at table k  ΣkNk=t) (CRP2)or, sits at the next unoccupied table w/ prob. α/(t+α) Applied Algorithm Lab.

  12. Generative Models for Power Law - Power Laws via Preferential Attachment When Xt+1 comes into here… (CRP1) Sits at an already occupied table k w/ prob. Nk/(t+α) (Nk: # of customers at table k  ΣkNk=t) (CRP2)or, sits at the next unoccupied table w/ prob. α/(t+α) Applied Algorithm Lab.

  13. Generative Models for Power Law - Power Laws via Preferential Attachment • CPR rule: Next customer sits at a table w/ prob. Proportional to # of customers already sitting at it(and sits at new table w/ prob. Proportional to α)  Customers tend to sit at most popular tables  Most popular tables attract the most new customers, and become even more popular • The concentration parameter α: how likely customer is to sit at a fresh table Applied Algorithm Lab.

  14. Generative Models for Power Law - Power Laws via Preferential Attachment • Generating Power law distribution via Preference Attachment (Most models are variations of this form) • Let’s say “Web Page Process” • Start w/ a single page • This single page has a link to itself • At each time step, a new page appears, w/ outdegree 1 (WPP1) The link of new page points to a page chosen u.a.r. w/ prob. α<1 (WPP2) The link of new page points to page chosen proportionally to the indegree of the page w/ prob. 1- α Applied Algorithm Lab.

  15. Generative Models for Power Law - Power Laws via Preferential Attachment • Xj(t): # of pages w/ indegree j when ∃ t pages in the system • Pr[Xj increase] = αXj-1/t+(1-α)(j-1)Xj-1/t • Pr[Xj decrease] = αXj/t+(1-α)jXj/t (WPP1) The link of new page points to a page chosen u.a.r. w/ prob. α<1 (WPP2) The link of new page points to page chosen proportionally to the indegree of the page w/ prob. 1- α Applied Algorithm Lab.

  16. Generative Models for Power Law - Power Laws via Preferential Attachment • Pr[Xj increase] = αXj-1/t+(1-α)(j-1)Xj-1/t • Pr[Xj decrease] = αXj/t+(1-α)jXj/t  dXj/dt = {α(Xj-1-Xj)+(1-α)((j-1)Xj-1-jXj-1)}/t • Intuitively appealing, BUT how continuous DE describes a discrete process?  This can be justified formally using martingales [Kumar et al 00] & theoretical frameworks of Kurtz, Wormald[Drinea et al. 00, Kurtz 81, Wormald 95]. Applied Algorithm Lab.

  17. Generative Models for Power Law - Power Laws via Preferential Attachment • dX0/dt=1-αX0/t • Suppose in the steady state limit: Xj(t)=cj*t (portion cj)  c0 =dX0/dt=1-αX0/t=1-αc0 ⇔ c0 = 1/(α+1) • Substitute this assumption for dXj/dt = {α(Xj-1-Xj)+(1-α)((j-1)Xj-1-jXj-1)}/t  cj(1+α+j(1-α))=cj-1(α+(j-1)(1-α))  We can determine cjexactly. • Focusing on the asymptotic, for large j cj/cj-1=1-(2-α)/(1+α+j(1-α))~1-{(2-α)/(1-α)}*(1/j) Applied Algorithm Lab.

  18. Generative Models for Power Law - Power Laws via Preferential Attachment • We have cj~cj^(- ) for some constant c, giving a power law. • Notecj~cj^(- ) implies WTS: Σj≥kcj behave the tail of power law distribution (Proof) For some constant c’. So, we’re done. Applied Algorithm Lab.

  19. Recall The Definition of Power Law • Recall Some Properties of Power Law • Generative Models for Power Law - Power Laws via Preferential Attachment - Power Laws via Multiplicative Processes Applied Algorithm Lab.

  20. Generative Models for Power Law - Power Laws via Multiplicative Processes • Pareto: income distribution obeys power law • [Champernowne 53] offered an explanation for this behavior. • Partition income in the following manner: • 1st range: between m and γm for some γ>1 • 2nd range: between γm and γ2m … • persons in class j: their income is between γj-1mand γjm • Pij: prob. of a person moving from class i to class j • At each time step, Pijdepends only on the value (j-i).  Under this assumption, Pareto distribution can be obtained. Applied Algorithm Lab.

  21. Generative Models for Power Law - Power Laws via Multiplicative Processes • E.g. γ=2, Pij=2/3 if j-i=-1 Pij=1/3 if j-i=1 • Special case: i=1  P11=2/3 • The equilibrium property of being in class k: 1/2k X: a person’s income  Pr[X≥2k-1m]=1/2k-1 Pr[X ≥ x]=m/xfor x= 2k-1m This is a power law distribution. Applied Algorithm Lab.

  22. References [1] M. Mitzenmacher, A Brief History of Generative Models for Power Law and Lognormal Distributions, Internet Mathematics, vol 1, No. 2, pp. 226-251, 2004. [2] Mark Johnson, Chinese Restaurant Processes(CG168 notes), cog.brown.edu/~mj/classes/cg168/.../ChineseRestaurants.pdf [3] The lecture notes of C. Faloutsos, Carnegie Mellon University, 15-826 Multimedia Databases and Data Mining, Spring 2008 http://www.cs.cmu.edu/~christos/courses/826.S08/FOILS-pdf/195_powerLaws.pdf [4] Bruno Bassetti, Mina Zarei, Marco Cosentino Lagomarsino, and Ginestra Bianconi., Statistical mechanics of the “Chinese restaurant process”: Lack of self-averaging, anomalous finite-size effects, and condensation, Phys. Rev. E 80, 066118 (2009) [4 pages] [5]http://en.wikipedia.org/wiki/Power_law, http://en.wikipedia.org/wiki/Chinese_restaurant_process, http://en.wikipedia.org/wiki/Preferential_attachment Applied Algorithm Lab.

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