Find the value of mu

Question

This is an equation used in a synthetic network generator 

 

⟨k⟩ = ((2*π^(D/2)*δ*μ*I₁) / Γ(D/2) ) * ⟨κ⟩^2

 

where ⟨k⟩ is the desired average degree of the nodes 

and  ⟨κ⟩ is the analytical prediction of the degree of the nodes

I1 is the integral of the connection probability between nodes (it can be any integral function) in my case is the fermi dirac distribution.

and Γ(D/2) is the gamma distribution.

D = 1 is the dimensions of the geometric space where the network will be generated.

δ = N/2*π*cosh(α*(R-1)) is the density of the nodes in the hyperbolic space disk (space im using)

 

So the point of this equation is to to find the value of μ such that ⟨k⟩ = ⟨κ⟩ (the observed average degree will match the prediction) 

 

1) I have to first set ⟨κ⟩ =  ((2*π^(D/2)*δ*μ*I1) / Γ(D/2) ) * ⟨κ⟩^2  ?

 

2) therefore μ = Γ(D/2) / (<κ>*2*π^(D/2)*δ*I₁) ?

 

now I₁ has a closed form solution which is (β/π * sin(π/β))^-1 

by replacing the values of D, I1 and δ in the equation we have:

 

μ = (β*sin(π/β)*cosh(α*(R-1)))/(⟨κ⟩*N) is this correct ?

 

 

 

Answer 1

Antonio,

 

Could you explain the context of this question? What do the quantities k, D, δ, μ, κ, and l₁ represent? Is Γ the gamma function?

 

No one will be able to completely answer this question unless you provide more detail about where it originated.

 

Thanks for posting.