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Generating Random Networks and Graphs$

Ton Coolen, Alessia Annibale, and Ekaterina Roberts

Print publication date: 2017

Print ISBN-13: 9780198709893

Published to Oxford Scholarship Online: May 2017

DOI: 10.1093/oso/9780198709893.001.0001

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(p.229) Appendix A The delta distribution

(p.229) Appendix A The delta distribution

Generating Random Networks and Graphs

A.C.C. Coolen

A. Annibale

E.S. Roberts

Oxford University Press

Definition. We define the δ-distribution as the probability distribution δ(x) corresponding to a zero-average random variable x in the limit where the randomness in the variable vanishes. So

dx f(x)δ(x)=f(0)      for any function f

By the same token, the expression δ(xa) will then represent the distribution for a random variable x with average a, in the limit where the randomness vanishes, since

dx f(x)δ(xa)=dx f(x+a)δ(x)=f(a)      for any function f

Formulae for the δ-distribution. A problem arises when we want to write down a formula for δ(x). Intuitively one could propose taking a zero-average normal distribution and setting its width to zero:

δ(x)=limσ0pσ(x)        pσ(x)=1σ2π ex2/2σ2

This is not a true function in a mathematical sense: δ(x) is zero for x0 and δ(0)=. However, we realize that δ(x) only serves to calculate averages; it only has a meaning inside an integration. If we adopt the convention that one should set σ0 in (A.1) only after performing the integration, we can use (A.1) to derive the following properties (for sufficiently well-behaved functions f):

dx δ(x)f(x)=limσ0dx pσ(x)f(x)=limσ0dx2π ex2/2f(σx)=f(0)dx δ(x)f(x)=limσ0dxddx(pσ(x)f(x))pσ(x)f(x)=limσ0pσ(x)f(x)f(0)=f(0)

The following relation links the δ-distribution to the step function:

δ(x)=ddxθ(x)θ(x)=1if x>00if x<0

(p.230) This is proved by showing that both sides of the equation have the same effect inside an integration:

dx(δ(x)ddxθ(x))f(x)=f(0)limϵ0ϵϵdxddx(θ(x)f(x))f(x)θ(x)=f(0)limϵ0f(ϵ)0+limϵ00ϵdx f(x)=0

Finally one can use the definitions of Fourier transforms and inverse Fourier transforms to obtain the following integral representation of the δ-distribution:

δ(x)=dk2π eikx