/
MFIsingModelAnalytic.jl
167 lines (151 loc) · 5.94 KB
/
MFIsingModelAnalytic.jl
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
module MFIsingAnalytic
export AIM, prob, SSprob
using Plots, Parameters, LinearAlgebra, Distributions, StatsBase
"""
Define the Analytic Ising Model struct
"""
@with_kw struct AIM
N::Int64 = 100 # number of agents
γ::Float64 = 1.0 # timescale
h::Float64
J::Float64
z::Float64 = 1.0 # could generalise to be 2*d for NN.
pars::Vector{Float64} = [γ, h, J, z]
A::Matrix{Float64} = make_TRM(pars, N)
λ::Array{Complex{BigFloat}} = GetEigVals(A)
q_arr::Vector{Vector{Complex{BigFloat}}} = [GetOrthoQ(pars,N,λ[j]) for j in 1:N+1]
p_arr::Vector{Vector{Complex{BigFloat}}} = [GetOrthoP(pars,N,λ[j]) for j in 1:N+1]
den_prod::Vector{Complex{BigFloat}} = [prod([λ[i]-λ[j] for j in filter!(e->e≠i,[j for j in 1:N+1])]) for i in 1:N+1]
As::Vector{BigFloat} = [a(pars,N,j) for j in 1:N]
Bs::Vector{BigFloat} = [b(pars,N,j) for j in 0:N-1]
end
"""
Define rate function for N₊ prod
"""
function r1(pars::Vector{Float64}, N::Int64, n::Int64)
γ, h, J, z = pars
return convert(BigFloat,γ*(N-n)*(1+exp(-2*(h+J*z*((2*n-N)/N))))^-1)::BigFloat
end
# rescaled definition in solution
a(pars,N,n) = r1(pars,N,n-1);
"""
Define rate function for N₋ prod
"""
function r2(pars::Vector{Float64}, N::Int64, n::Int64)
γ, h, J, z = pars
return convert(BigFloat,γ*n*(1+exp(2*(h+J*z*((2*n-N)/N))))^-1)::BigFloat
end
# rescaled definition in solution
b(pars,N,n) = r2(pars,N,n+1);
"""
Make the transition rate matrix
"""
function make_TRM(pars::Vector{Float64}, N::Int64)
A = zeros(N+1, N+1);
for i in 1:size(A)[1]
for j in 1:size(A)[2]
if i == 1 && j == 1
A[1,1] = - a(pars,N,1)
elseif i == j && i>1
A[i,i] = -(a(pars,N,i)+b(pars,N,i-2))
elseif i == j+1
A[i,j] = a(pars,N,j)
elseif i == j-1
A[i,j] = b(pars,N,i-1)
else
continue
end
end
end
return A::Matrix{Float64}
end
"""
Get eigenvalues
"""
function GetEigVals(A::Matrix{Float64})
λ = convert(Array{Complex{BigFloat}}, reverse(eigvals(A)));
λ[1] = 0.0 # explicitly code in the zero for the s.s.
return λ::Array{Complex{BigFloat}}
end
"""
Get the p orthogonal polynomials
"""
function GetOrthoP(pars::Vector{Float64}, N::Int64, λᵢ::Complex{BigFloat})
p = Array{Complex{BigFloat},1}(undef, N+1);
p[1] = 1.0; p[2] = λᵢ+a(pars,N,1);
for i in 3:N+1
p[i] = (λᵢ+a(pars,N,i-1)+b(pars,N,i-3))*p[i-1] - b(pars,N,i-3)*a(pars,N,i-2)*p[i-2]
end
return p::Vector{Complex{BigFloat}}
end
"""
Get the q orthogonal polynomials
"""
function GetOrthoQ(pars::Vector{Float64}, N::Int64, λᵢ::Complex{BigFloat})
q = Array{Complex{BigFloat},1}(undef, N+3);
q[N+3] = 1.0; q[N+2] = λᵢ + b(pars,N,N-1);
for i in reverse([j for j in 3:N+1])
q[i] = (λᵢ + a(pars,N,i-1)+b(pars,N,i-3))*q[i+1] - b(pars,N,i-2)*a(pars,N,i-1)*q[i+2]
end
return q::Vector{Complex{BigFloat}}
end
"""
Define the sum of the elements from the solution
"""
function sum_elems(λᵢ::Complex{BigFloat}, t::Float64, m::Int64, m₀::Int64, pars::Array{Float64,1}, N::Int64, p_arrᵢ::Vector{Complex{BigFloat}}, q_arrᵢ::Vector{Complex{BigFloat}}, den_prodᵢ::Complex{BigFloat})
return exp(λᵢ*t)*p_arrᵢ[m+1]*q_arrᵢ[m₀+3]/den_prodᵢ::Complex{BigFloat}
end
"""
Define P(m,t|m₀)
"""
function pm(t::Float64, IM::AIM, m::Int64, m₀::Int64)
@unpack λ, pars, N, As, Bs, q_arr, p_arr, den_prod = IM
if m<m₀
return prod(Bs[m+1:m₀])*sum([sum_elems(λ[i], t, m, m₀, pars, N, p_arr[i], q_arr[i], den_prod[i]) for i in 1:N+1])::Complex{BigFloat}
elseif m==m₀
return sum([sum_elems(λ[i], t, m, m, pars, N, p_arr[i], q_arr[i], den_prod[i]) for i in 1:N+1])::Complex{BigFloat}
else
return prod(As[m₀+1:m])*sum([sum_elems(λ[i], t, m₀, m, pars, N, p_arr[i], q_arr[i], den_prod[i]) for i in 1:N+1])::Complex{BigFloat}
end
end
"""
Define the probability distribution return function from a initial distribution.
"""
function prob(IM::AIM, t::Float64, q_init_D::Distribution{Univariate, Discrete})
@unpack N = IM
q_init = pdf(q_init_D)
pmt = Array{Complex{BigFloat}}(undef,N+1)
for i in 1:N+1 # loop over the m0's
pmt[i] = sum([q_init[n+1]*pm(t, IM, i-1, n) for n in 0:N])
end
return (LinRange(0.0,N,N+1),real(pmt))::Tuple{LinRange{Float64}, Vector{BigFloat}}
end
"""
Define the probability distribution return function from a precise value of m₀.
"""
function prob(IM::AIM, t::Float64, m₀::Int64)
@unpack N = IM
return (LinRange(0.0,N,N+1),real([pm(t, IM, m, m₀) for m in 0:N]))::Tuple{LinRange{Float64}, Vector{BigFloat}}
end
"""
Define the probability distribution return function for N/2 value of m₀.
"""
function prob(IM::AIM, t::Float64)
@unpack N = IM
m₀ = round(Int64,N/2)
return (LinRange(0.0,N,N+1),real([pm(t, IM, m, m₀) for m in 0:N]))::Tuple{LinRange{Float64}, Vector{BigFloat}}
end
"""
Return the steady state distribution
"""
function SSprob(IM::AIM)
@unpack As, Bs, N = IM
ps = Vector{BigFloat}(undef, N+1)
for m in 2:N # use the product rule
ps[m] = prod(As[1:m-1])*prod(Bs[m:N])
end
ps[1] = prod(Bs[1:N]) # do product for the B's
ps[N+1] = prod(As[1:N]) # do product for the A's
return (LinRange(0.0,N,N+1),ps/sum(ps))::Tuple{LinRange{Float64}, Vector{BigFloat}}
end
end # module end