Outline Mechanisms Mechanisms Mechanisms for Generating Random - PowerPoint PPT Presentation

Power-Law Power-Law Outline Mechanisms Mechanisms Mechanisms for Generating Random Walks Random Walks Power-Law Distributions Random Walks The First Return Problem The First Return Problem Examples Examples The First Return Problem Principles of Complex Systems Variable Variable transformation transformation Examples Course 300, Fall, 2008 Basics Basics Holtsmark’s Distribution Holtsmark’s Distribution PLIPLO PLIPLO Variable transformation Growth Growth Mechanisms Mechanisms Prof. Peter Dodds Basics Random Copying Random Copying Words, Cities, and the Web Holtsmark’s Distribution Words, Cities, and the Web References References Department of Mathematics & Statistics PLIPLO University of Vermont Growth Mechanisms Random Copying Words, Cities, and the Web References Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/87 Frame 2/87 Power-Law Power-Law Mechanisms Random walks Mechanisms Mechanisms Random Walks Random Walks The First Return Problem The First Return Problem Examples Examples Variable Variable The essential random walk: transformation transformation Basics Basics ◮ One spatial dimension. Holtsmark’s Distribution Holtsmark’s Distribution PLIPLO PLIPLO A powerful theme in complex systems: ◮ Time and space are discrete Growth Growth Mechanisms Mechanisms ◮ Random walker (e.g., a drunk) starts at origin x = 0. Random Copying Random Copying ◮ structure arises out of randomness. Words, Cities, and the Web Words, Cities, and the Web ◮ Step at time t is ǫ t : References References ◮ Exhibit A: Random walks... ( ⊞ ) � + 1 with probability 1/2 ǫ t = − 1 with probability 1/2 Frame 3/87 Frame 4/87

Power-Law Power-Law Random walks Random walks Mechanisms Mechanisms Random Walks Random Walks The First Return Problem The First Return Problem Examples Examples Displacement after t steps: Variable Variable Variances sum: ( ⊞ ) ∗ transformation transformation Basics Basics t Holtsmark’s Distribution Holtsmark’s Distribution � � � t PLIPLO PLIPLO � x t = ǫ i Var ( x t ) = Var ǫ i Growth Growth Mechanisms Mechanisms i = 1 i = 1 Random Copying Random Copying Words, Cities, and the Web Words, Cities, and the Web t t References References � � Expected displacement: = Var ( ǫ i ) = 1 = t i = 1 i = 1 � � t t � � ∗ Sum rule = a good reason for using the variance to measure � x t � = ǫ i = � ǫ i � = 0 i = 1 i = 1 spread Frame 5/87 Frame 6/87 Power-Law Power-Law Random walks Random walks Mechanisms Mechanisms Random Walks Random Walks The First Return Problem The First Return Problem Examples Examples Random walks are weirder than you might think... Variable Variable transformation transformation Basics Basics For example: Holtsmark’s Distribution Holtsmark’s Distribution So typical displacement from the origin scales as PLIPLO PLIPLO ◮ ξ r , t = the probability that by time step t , a random Growth Growth Mechanisms Mechanisms σ = t 1 / 2 walk has crossed the origin r times. Random Copying Random Copying Words, Cities, and the Web Words, Cities, and the Web ◮ Think of a coin flip game with ten thousand tosses. References References ⇒ A non-trivial power-law arises out of ◮ If you are behind early on, what are the chances you additive aggregation or accumulation. will make a comeback? ◮ The most likely number of lead changes is... 0. See Feller, [3] Intro to Probability Theory, Volume I Frame 7/87 Frame 8/87

Power-Law Power-Law Random walks Random walks—some examples Mechanisms Mechanisms Random Walks Random Walks The First Return Problem The First Return Problem Examples Examples 50 Variable Variable 0 transformation transformation x −50 Basics Basics In fact: Holtsmark’s Distribution Holtsmark’s Distribution −100 PLIPLO PLIPLO 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 t Growth Growth 50 Mechanisms Mechanisms ξ 0 , t > ξ 1 , t > ξ 2 , t > · · · Random Copying Random Copying 0 Words, Cities, and the Web x Words, Cities, and the Web −50 References References −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Even crazier: t 200 The expected time between tied scores = ∞ ! 100 x 0 −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 t Frame 9/87 Frame 10/87 Power-Law Power-Law Random walks—some examples Random walks Mechanisms Mechanisms Random Walks Random Walks The First Return Problem The First Return Problem Examples Examples 50 Variable Variable 0 transformation transformation x −50 Basics Basics The problem of first return: Holtsmark’s Distribution Holtsmark’s Distribution −100 PLIPLO PLIPLO 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 t ◮ What is the probability that a random walker in one Growth Growth 200 Mechanisms Mechanisms dimension returns to the origin for the first time after t Random Copying Random Copying 100 x Words, Cities, and the Web Words, Cities, and the Web steps? 0 References References −100 ◮ Will our drunkard always return to the origin? 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 t 200 ◮ What about higher dimensions? 100 x 0 −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 t Frame 11/87 Frame 12/87

Power-Law Power-Law First returns Random Walks Mechanisms Mechanisms Random Walks Random Walks The First Return Problem The First Return Problem Examples Examples Variable Variable 50 transformation transformation 0 Reasons for caring: Basics Basics x Holtsmark’s Distribution Holtsmark’s Distribution −50 PLIPLO PLIPLO 1. We will find a power-law size distribution with an −100 Growth 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Growth t Mechanisms Mechanisms interesting exponent 200 Random Copying Random Copying Words, Cities, and the Web Words, Cities, and the Web 2. Some physical structures may result from random 100 References x References 0 walks −100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 3. We’ll start to see how different scalings relate to t each other Again: expected time between ties = ∞ ... Let’s find out why... [3] Frame 13/87 Frame 15/87 Power-Law Power-Law First Returns First Returns Mechanisms Mechanisms Random Walks Random Walks The First Return Problem The First Return Problem Examples Examples For random walks in 1-d: Variable Variable transformation transformation 4 Basics Basics ◮ Return can only happen when t = 2 n . Holtsmark’s Distribution Holtsmark’s Distribution PLIPLO PLIPLO 2 ◮ Call P first return ( 2 n ) = P fr ( 2 n ) probability of first return Growth Growth Mechanisms at t = 2 n . Mechanisms x 0 Random Copying Random Copying Words, Cities, and the Web ◮ Assume drunkard first lurches to x = 1. Words, Cities, and the Web References References −2 ◮ The problem −4 0 5 10 15 20 P fr ( 2 n ) = Pr ( x t ≥ 1 , t = 1 , . . . , 2 n − 1 , and x 2 n = 0 ) t Frame 16/87 Frame 17/87

Power-Law Power-Law First Returns First Returns Mechanisms Mechanisms Random Walks Random Walks 4 4 The First Return Problem The First Return Problem Examples Examples 3 3 Variable Variable transformation transformation ◮ Counting problem (combinatorics/statistical Basics Basics 2 2 x x Holtsmark’s Distribution Holtsmark’s Distribution mechanics) PLIPLO PLIPLO 1 1 ◮ Use a method of images Growth Growth Mechanisms Mechanisms Random Copying ◮ Define N ( i , j , t ) as the # of possible walks between Random Copying 0 0 Words, Cities, and the Web Words, Cities, and the Web 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 x = i and x = j taking t steps. t References t References ◮ Consider all paths starting at x = 1 and ending at ◮ A useful restatement: P fr ( 2 n ) = x = 1 after t = 2 n − 2 steps. 1 2 Pr ( x t ≥ 1 , t = 1 , . . . , 2 n − 1 , and x 1 = x 2 n − 1 = 1 ) ◮ Subtract how many hit x = 0. ◮ Want walks that can return many times to x = 1. ◮ (The 1 2 accounts for stepping to 2 or -2 instead of 0 at t = 2 n .) Frame 18/87 Frame 19/87 Power-Law Power-Law First Returns First Returns Mechanisms Mechanisms Random Walks Random Walks 4 The First Return Problem The First Return Problem Examples Examples Variable 3 Variable transformation transformation Key observation: Basics Basics 2 Holtsmark’s Distribution Holtsmark’s Distribution # of t -step paths starting and ending at x = 1 PLIPLO PLIPLO and hitting x = 0 at least once Growth Growth 1 Mechanisms Mechanisms = # of t -step paths starting at x = − 1 and ending at x = 1 Random Copying Random Copying 0 x Words, Cities, and the Web Words, Cities, and the Web = N ( − 1 , 1 , t ) References References −1 So N first return ( 2 n ) = N ( 1 , 1 , 2 n − 2 ) − N ( − 1 , 1 , 2 n − 2 ) −2 See this 1-1 correspondence visually... −3 −4 0 2 4 6 8 10 12 14 16 t Frame 20/87 Frame 21/87