sign in sign up
<<
Home , Wiener process

Scheidegger Networks

Scheidegger Networks—A Bonus First return

Calculation References Complex Networks, Course 295A, Spring, 2008

Prof. Peter Dodds

Department of & University of Vermont

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. Frame 1/11 References

Scheidegger Outline Networks

First return random walk

References

First return random walk

Frame 2/11 Scheidegger Outline Networks

First return random walk

References

First return random walk

References

Frame 2/11 I 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

I We solved this with a counting argument for the discrete random walk the preceding Complex Systems course

I For fun and the constitution, let’s work on the continuous time Wiener process version

I A classic, delightful problem

Scheidegger Random walks Networks

First return random walk

References I We’ve seen that Scheidegger networks have random walk boundaries [1, 2]

Frame 3/11 I We solved this with a counting argument for the discrete random walk the preceding Complex Systems course

I For fun and the constitution, let’s work on the continuous time Wiener process version

I A classic, delightful problem

Scheidegger Random walks Networks

First return random walk

References I We’ve seen that Scheidegger networks have random walk boundaries [1, 2]

I 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

Frame 3/11 I For fun and the constitution, let’s work on the continuous time Wiener process version

I A classic, delightful problem

Scheidegger Random walks Networks

First return random walk

References I We’ve seen that Scheidegger networks have random walk boundaries [1, 2]

I 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

I We solved this with a counting argument for the discrete random walk the preceding Complex Systems course

Frame 3/11 I A classic, delightful problem

Scheidegger Random walks Networks

First return random walk

References I We’ve seen that Scheidegger networks have random walk boundaries [1, 2]

I 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

I We solved this with a counting argument for the discrete random walk the preceding Complex Systems course

I For fun and the constitution, let’s work on the continuous time Wiener process version

Frame 3/11 Scheidegger Random walks Networks

First return random walk

References I We’ve seen that Scheidegger networks have random walk boundaries [1, 2]

I 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

I We solved this with a counting argument for the discrete random walk the preceding Complex Systems course

I For fun and the constitution, let’s work on the continuous time Wiener process version

I A classic, delightful problem

Frame 3/11 Scheidegger Random walks Networks

First return random walk

References

The Wiener process ()

Frame 4/11 Scheidegger Random walking on a sphere... Networks

First return random walk

References

The Wiener process () Frame 5/11 I x(t2) − x(t1) ∼ N (0, t2 − t1) where 1 2 N (x, t) = √ e−x /2t 2πt

I Continuous but nowhere differentiable

Scheidegger Random walks Networks

First return random walk

References

I Wiener process =

Frame 6/11 I Continuous but nowhere differentiable

Scheidegger Random walks Networks

First return random walk

References

I Wiener process = Brownian motion

I x(t2) − x(t1) ∼ N (0, t2 − t1) where 1 2 N (x, t) = √ e−x /2t 2πt

Frame 6/11 Scheidegger Random walks Networks

First return random walk

References

I Wiener process = Brownian motion

I x(t2) − x(t1) ∼ N (0, t2 − t1) where 1 2 N (x, t) = √ e−x /2t 2πt

I Continuous but nowhere differentiable

Frame 6/11 I Use what we know: the probability density for a return (not necessarily the first) at time t is

1 1 f (t) = √ e−0/2t = √ 2πt 2πt

I Observe that f and g are connected like this:

Z t f (t) = f (τ)g(t − τ)dτ + δ(t) τ=0 |{z}

I In words: Probability of returning at time t equals the of the probability of returning at time τ and then not returning until exactly t − τ time units later.

Scheidegger First return Networks

First return I Objective: find g(t), the probability that Wiener random walk

process first returns to the origin at time t. References

Frame 7/11 I Observe that f and g are connected like this:

Z t f (t) = f (τ)g(t − τ)dτ + δ(t) τ=0 |{z} Dirac delta function

I 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.

Scheidegger First return Networks

First return I Objective: find g(t), the probability that Wiener random walk

process first returns to the origin at time t. References

I Use what we know: the probability density for a return (not necessarily the first) at time t is

1 1 f (t) = √ e−0/2t = √ 2πt 2πt

Frame 7/11 I 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.

Scheidegger First return Networks

First return I Objective: find g(t), the probability that Wiener random walk

process first returns to the origin at time t. References

I Use what we know: the probability density for a return (not necessarily the first) at time t is

1 1 f (t) = √ e−0/2t = √ 2πt 2πt

I Observe that f and g are connected like this:

Z t f (t) = f (τ)g(t − τ)dτ + δ(t) τ=0 |{z} Dirac delta function

Frame 7/11 Scheidegger First return Networks

First return I Objective: find g(t), the probability that Wiener random walk

process first returns to the origin at time t. References

I Use what we know: the probability density for a return (not necessarily the first) at time t is

1 1 f (t) = √ e−0/2t = √ 2πt 2πt

I Observe that f and g are connected like this:

Z t f (t) = f (τ)g(t − τ)dτ + δ(t) τ=0 |{z} Dirac delta function

I 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 I So we take the Laplace transform: Z ∞ L[f (t)] = F(s) = f (t)e−st dt t=0−

I and obtain F(s) = F(s)G(s) + 1

I Rearrange: G(s) = 1 − 1/F(s)

Scheidegger First return Networks

First return random walk

I Next see that right hand side of References R t f (t) = τ=0 f (τ)g(t − τ)dτ + δ(t) is a juicy convolution.

Frame 8/11 I and obtain F(s) = F(s)G(s) + 1

I Rearrange: G(s) = 1 − 1/F(s)

Scheidegger First return Networks

First return random walk

I Next see that right hand side of References R t f (t) = τ=0 f (τ)g(t − τ)dτ + δ(t) is a juicy convolution.

I So we take the Laplace transform: Z ∞ L[f (t)] = F(s) = f (t)e−st dt t=0−

Frame 8/11 I Rearrange: G(s) = 1 − 1/F(s)

Scheidegger First return Networks

First return random walk

I Next see that right hand side of References R t f (t) = τ=0 f (τ)g(t − τ)dτ + δ(t) is a juicy convolution.

I So we take the Laplace transform: Z ∞ L[f (t)] = F(s) = f (t)e−st dt t=0−

I and obtain F(s) = F(s)G(s) + 1

Frame 8/11 Scheidegger First return Networks

First return random walk

I Next see that right hand side of References R t f (t) = τ=0 f (τ)g(t − τ)dτ + δ(t) is a juicy convolution.

I So we take the Laplace transform: Z ∞ L[f (t)] = F(s) = f (t)e−st dt t=0−

I and obtain F(s) = F(s)G(s) + 1

I Rearrange: G(s) = 1 − 1/F(s)

Frame 8/11 1/2 ' e−(2s)

I Now we want to invert G(s) to find g(t) −1/2 I Use calculation that F(s) = (2s)

I G(s) = 1 − (2s)1/2

Scheidegger First return Networks

First return random walk

References

I We are here: G(s) = 1 − 1/F(s)

Frame 9/11 1/2 ' e−(2s)

−1/2 I Use calculation that F(s) = (2s)

I G(s) = 1 − (2s)1/2

Scheidegger First return Networks

First return random walk

References

I We are here: G(s) = 1 − 1/F(s)

I Now we want to invert G(s) to find g(t)

Frame 9/11 1/2 ' e−(2s)

I G(s) = 1 − (2s)1/2

Scheidegger First return Networks

First return random walk

References

I We are here: G(s) = 1 − 1/F(s)

I Now we want to invert G(s) to find g(t) −1/2 I Use calculation that F(s) = (2s)

Frame 9/11 1/2 ' e−(2s)

Scheidegger First return Networks

First return random walk

References

I We are here: G(s) = 1 − 1/F(s)

I Now we want to invert G(s) to find g(t) −1/2 I Use calculation that F(s) = (2s)

I G(s) = 1 − (2s)1/2

Frame 9/11 Scheidegger First return Networks

First return random walk

References

I We are here: G(s) = 1 − 1/F(s)

I Now we want to invert G(s) to find g(t) −1/2 I Use calculation that F(s) = (2s)

I 1/2 G(s) = 1 − (2s)1/2 ' e−(2s)

Frame 9/11 I is infinite (weird but okay...)

I Mean is also infinite (just plain crazy...)

I Distribution is normalizable so process always returns to 0. −γ I For river networks: P(`) ∼ ` so γ = 3/2 for Scheidegger networks.

Scheidegger First return Networks

First return random walk

References

Groovy aspects of g(t) ∼ t −3/2:

Frame 10/11 I Mean is also infinite (just plain crazy...)

I Distribution is normalizable so process always returns to 0. −γ I For river networks: P(`) ∼ ` so γ = 3/2 for Scheidegger networks.

Scheidegger First return Networks

First return random walk

References

Groovy aspects of g(t) ∼ t −3/2:

I Variance is infinite (weird but okay...)

Frame 10/11 I Distribution is normalizable so process always returns to 0. −γ I For river networks: P(`) ∼ ` so γ = 3/2 for Scheidegger networks.

Scheidegger First return Networks

First return random walk

References

Groovy aspects of g(t) ∼ t −3/2:

I Variance is infinite (weird but okay...)

I Mean is also infinite (just plain crazy...)

Frame 10/11 −γ I For river networks: P(`) ∼ ` so γ = 3/2 for Scheidegger networks.

Scheidegger First return Networks

First return random walk

References

Groovy aspects of g(t) ∼ t −3/2:

I Variance is infinite (weird but okay...)

I Mean is also infinite (just plain crazy...)

I Distribution is normalizable so process always returns to 0.

Frame 10/11 Scheidegger First return Networks

First return random walk

References

Groovy aspects of g(t) ∼ t −3/2:

I Variance is infinite (weird but okay...)

I Mean is also infinite (just plain crazy...)

I Distribution is normalizable so process always returns to 0. −γ I For river networks: P(`) ∼ ` so γ = 3/2 for Scheidegger networks.

Frame 10/11 Scheidegger References I Networks

First return random walk

References

A. E. Scheidegger. A 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