Outline Scheidegger Networks Networks Scheidegger NetworksA Bonus - PowerPoint PPT Presentation

Scheidegger Outline Scheidegger Networks Networks Scheidegger Networks—A Bonus First return First return random walk random walk Calculation References References Complex Networks, Course 295A, Spring, 2008 First return random walk Prof. Peter Dodds Department of Mathematics & Statistics University of Vermont References Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/11 Frame 2/11 Random walks Scheidegger Random walks Scheidegger Networks Networks First return First return random walk random walk ◮ We’ve seen that Scheidegger networks have random References References walk boundaries [1, 2] ◮ Determining expected shape of a ‘basin’ becomes a problem of finding the probability that a 1-d random walk returns to the origin after t time steps ◮ We solved this with a counting argument for the discrete random walk the preceding Complex Systems course ◮ For fun and the constitution, let’s work on the continuous time Wiener process version ◮ A classic, delightful problem The Wiener process ( ⊞ ) Frame 3/11 Frame 4/11

Random walking on a sphere... Scheidegger Random walks Scheidegger Networks Networks First return First return random walk random walk References References ◮ Wiener process = Brownian motion ◮ x ( t 2 ) − x ( t 1 ) ∼ N ( 0 , t 2 − t 1 ) where 1 e − x 2 / 2 t √ N ( x , t ) = 2 π t ◮ Continuous but nowhere differentiable The Wiener process ( ⊞ ) Frame 5/11 Frame 6/11 First return Scheidegger First return Scheidegger Networks Networks ◮ Objective: find g ( t ) , the probability that Wiener First return First return random walk random walk ◮ Next see that right hand side of process first returns to the origin at time t . References � t References f ( t ) = τ = 0 f ( τ ) g ( t − τ ) d τ + δ ( t ) is a juicy ◮ Use what we know: the probability density for a convolution. return (not necessarily the first) at time t is ◮ So we take the Laplace transform: 1 1 e − 0 / 2 t = f ( t ) = √ √ � ∞ 2 π t 2 π t t = 0 − f ( t ) e − st d t L [ f ( t )] = F ( s ) = ◮ Observe that f and g are connected like this: ◮ and obtain � t F ( s ) = F ( s ) G ( s ) + 1 f ( t ) = f ( τ ) g ( t − τ ) d τ + δ ( t ) τ = 0 ���� ◮ Rearrange: Dirac delta function G ( s ) = 1 − 1 / F ( s ) ◮ In words: Probability of returning at time t equals the integral of the probability of returning at time τ and then not returning until exactly t − τ time units later. Frame 7/11 Frame 8/11

First return Scheidegger First return Scheidegger Networks Networks First return First return random walk random walk References References Groovy aspects of g ( t ) ∼ t − 3 / 2 : ◮ We are here: G ( s ) = 1 − 1 / F ( s ) ◮ Variance is infinite (weird but okay...) ◮ Now we want to invert G ( s ) to find g ( t ) ◮ Mean is also infinite (just plain crazy...) ◮ Use calculation that F ( s ) = ( 2 s ) − 1 / 2 ◮ Distribution is normalizable so process always ◮ G ( s ) = 1 − ( 2 s ) 1 / 2 ≃ e − ( 2 s ) 1 / 2 returns to 0. ◮ For river networks: P ( ℓ ) ∼ ℓ − γ so γ = 3 / 2 for Scheidegger networks. Frame 9/11 Frame 10/11 References I Scheidegger Networks First return random walk References A. E. Scheidegger. A stochastic model for drainage patterns into an intramontane trench. Bull. Int. Assoc. Sci. Hydrol. , 12(1):15–20, 1967. . A. E. Scheidegger. Theoretical Geomorphology . Springer-Verlag, New York, third edition, 1991. . Frame 11/11