Mechanisms for Generating Power-Law Distributions Random Walks The - PowerPoint PPT Presentation

Power-Law Mechanisms Mechanisms for Generating Power-Law Distributions Random Walks The First Return Problem Examples Principles of Complex Systems Variable Course CSYS/MATH 300, Fall, 2009 transformation Basics Holtsmark’s Distribution PLIPLO Growth Prof. Peter Dodds Mechanisms Random Copying Words, Cities, and the Web Dept. of Mathematics & Statistics References Center for Complex Systems :: Vermont Advanced Computing Center University of Vermont Frame 1/88 .

Power-Law Outline Mechanisms Random Walks Random Walks The First Return Problem Examples The First Return Problem Variable transformation Examples Basics Holtsmark’s Distribution PLIPLO Variable transformation Growth Mechanisms Basics Random Copying Holtsmark’s Distribution Words, Cities, and the Web References PLIPLO Growth Mechanisms Random Copying Words, Cities, and the Web References Frame 2/88

Power-Law Mechanisms Mechanisms Random Walks The First Return Problem Examples Variable transformation Basics Holtsmark’s Distribution PLIPLO A powerful theme in complex systems: Growth Mechanisms Random Copying ◮ structure arises out of randomness. Words, Cities, and the Web References ◮ Exhibit A: Random walks... ( ⊞ ) Frame 3/88

Power-Law Random walks Mechanisms Random Walks The First Return Problem Examples Variable The essential random walk: transformation Basics ◮ One spatial dimension. Holtsmark’s Distribution PLIPLO ◮ Time and space are discrete Growth Mechanisms ◮ Random walker (e.g., a drunk) starts at origin x = 0. Random Copying Words, Cities, and the Web ◮ Step at time t is ǫ t : References � + 1 with probability 1/2 ǫ t = − 1 with probability 1/2 Frame 4/88

Power-Law Random walks Mechanisms Random Walks The First Return Problem Examples Displacement after t steps: Variable transformation Basics t Holtsmark’s Distribution � PLIPLO x t = ǫ i Growth Mechanisms i = 1 Random Copying Words, Cities, and the Web References Expected displacement: � � t t � � � x t � = ǫ i = � ǫ i � = 0 i = 1 i = 1 Frame 5/88

Power-Law Random walks Mechanisms Random Walks The First Return Problem Examples Variable Variances sum: ( ⊞ ) ∗ transformation Basics Holtsmark’s Distribution � � t PLIPLO � Var ( x t ) = Var ǫ i Growth Mechanisms i = 1 Random Copying Words, Cities, and the Web t t References � � = Var ( ǫ i ) = 1 = t i = 1 i = 1 ∗ Sum rule = a good reason for using the variance to measure spread Frame 6/88

Power-Law Random walks Mechanisms Random Walks The First Return Problem Examples Variable transformation Basics Holtsmark’s Distribution So typical displacement from the origin scales as PLIPLO Growth Mechanisms σ = t 1 / 2 Random Copying Words, Cities, and the Web References ⇒ A non-trivial power-law arises out of additive aggregation or accumulation. Frame 7/88

Power-Law Random walks Mechanisms Random Walks The First Return Problem Examples Random walks are weirder than you might think... Variable transformation For example: Basics Holtsmark’s Distribution PLIPLO ◮ ξ r , t = the probability that by time step t , a random Growth Mechanisms walk has crossed the origin r times. Random Copying Words, Cities, and the Web ◮ Think of a coin flip game with ten thousand tosses. References ◮ If you are behind early on, what are the chances you will make a comeback? ◮ The most likely number of lead changes is... 0. See Feller, [3] Intro to Probability Theory, Volume I Frame 8/88

Power-Law Random walks Mechanisms Random Walks The First Return Problem Examples Variable transformation Basics In fact: Holtsmark’s Distribution PLIPLO Growth Mechanisms ξ 0 , t > ξ 1 , t > ξ 2 , t > · · · Random Copying Words, Cities, and the Web References Even crazier: The expected time between tied scores = ∞ ! Frame 9/88

Power-Law Random walks—some examples Mechanisms Random Walks The First Return Problem Examples 50 Variable 0 transformation x −50 Basics Holtsmark’s Distribution −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 PLIPLO t Growth 50 Mechanisms Random Copying 0 x Words, Cities, and the Web −50 References −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 t 200 100 x 0 −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 t Frame 10/88

Power-Law Random walks—some examples Mechanisms Random Walks The First Return Problem Examples 50 Variable 0 transformation x −50 Basics Holtsmark’s Distribution −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 PLIPLO t Growth 200 Mechanisms Random Copying 100 x Words, Cities, and the Web 0 References −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 t 200 100 x 0 −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 t Frame 11/88

Power-Law Random walks Mechanisms Random Walks The First Return Problem Examples Variable transformation Basics The problem of first return: Holtsmark’s Distribution PLIPLO ◮ What is the probability that a random walker in one Growth Mechanisms dimension returns to the origin for the first time after t Random Copying Words, Cities, and the Web steps? References ◮ Will our drunkard always return to the origin? ◮ What about higher dimensions? Frame 12/88

Power-Law First returns Mechanisms Random Walks The First Return Problem Examples Variable transformation Reasons for caring: Basics Holtsmark’s Distribution PLIPLO 1. We will find a power-law size distribution with an Growth Mechanisms interesting exponent Random Copying Words, Cities, and the Web 2. Some physical structures may result from random References walks 3. We’ll start to see how different scalings relate to each other Frame 13/88

Power-Law Random Walks Mechanisms Random Walks The First Return Problem Examples Variable 50 transformation 0 Basics x Holtsmark’s Distribution −50 PLIPLO −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Growth t Mechanisms 200 Random Copying Words, Cities, and the Web 100 x References 0 −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 t Again: expected time between ties = ∞ ... Let’s find out why... [3] Frame 15/88

Power-Law First Returns Mechanisms Random Walks The First Return Problem Examples Variable transformation 4 Basics Holtsmark’s Distribution PLIPLO 2 Growth Mechanisms 0 x Random Copying Words, Cities, and the Web References −2 −4 0 5 10 15 20 t Frame 16/88

Power-Law First Returns Mechanisms Random Walks The First Return Problem Examples For random walks in 1-d: Variable transformation ◮ Return can only happen when t = 2 n . Basics Holtsmark’s Distribution PLIPLO ◮ Call P first return ( 2 n ) = P fr ( 2 n ) probability of first return Growth Mechanisms at t = 2 n . Random Copying ◮ Assume drunkard first lurches to x = 1. Words, Cities, and the Web References ◮ The problem P fr ( 2 n ) = 2 Pr ( x t ≥ 1 , t = 1 , . . . , 2 n − 1 , and x 2 n = 0 ) Frame 17/88

Power-Law First Returns Mechanisms Random Walks 4 4 The First Return Problem Examples 3 3 Variable transformation Basics 2 2 x x Holtsmark’s Distribution PLIPLO 1 1 Growth Mechanisms Random Copying 0 0 Words, Cities, and the Web 0 2 4 6 8 10 12 14 16 0 2 t References ◮ A useful restatement: P fr ( 2 n ) = 2 · 1 2 Pr ( x t ≥ 1 , t = 1 , . . . , 2 n − 1 , and x 1 = x 2 n − 1 = 1 ) ◮ Want walks that can return many times to x = 1. ◮ (The 1 2 accounts for stepping to 2 instead of 0 at t = 2 n .) Frame 18/88

Power-Law First Returns Mechanisms Random Walks The First Return Problem Examples Variable transformation ◮ Counting problem (combinatorics/statistical Basics Holtsmark’s Distribution mechanics) PLIPLO ◮ Use a method of images Growth Mechanisms ◮ Define N ( i , j , t ) as the # of possible walks between Random Copying Words, Cities, and the Web x = i and x = j taking t steps. References ◮ Consider all paths starting at x = 1 and ending at x = 1 after t = 2 n − 2 steps. ◮ Subtract how many hit x = 0. Frame 19/88

Power-Law First Returns Mechanisms Random Walks The First Return Problem Examples Variable transformation Key observation: Basics Holtsmark’s Distribution # of t -step paths starting and ending at x = 1 PLIPLO and hitting x = 0 at least once Growth Mechanisms = # of t -step paths starting at x = − 1 and ending at x = 1 Random Copying Words, Cities, and the Web = N ( − 1 , 1 , t ) References So N first return ( 2 n ) = N ( 1 , 1 , 2 n − 2 ) − N ( − 1 , 1 , 2 n − 2 ) See this 1-1 correspondence visually... Frame 20/88

Power-Law First Returns Mechanisms Random Walks 4 The First Return Problem Examples 3 Variable transformation Basics 2 Holtsmark’s Distribution PLIPLO Growth 1 Mechanisms Random Copying 0 x Words, Cities, and the Web References −1 −2 −3 −4 0 2 4 6 8 10 12 14 16 t Frame 21/88