Calibración de los modelos hidrodinámicos

1{W >n−1}. Then

P (v = w) = P

v_{dW e}^(j) = w

P (W ≤ n − 1) + P W − (n − 1)

δ_in



= w



P (W > n − 1)

= D_in⁽ⁿ⁻¹⁾(w) n − 1

n − 1

n − 1 + N (n − 1)δ_in

+ 1

N (n − 1)

N (n − 1)δ_in n − 1 + N (n − 1)δ_in

= D_in⁽ⁿ⁻¹⁾(w) + δ_in n − 1 + N (n − 1)δ_in,

which corresponds to the desired selection probability (4.1).

4.3 Parameter estimation: MLE based on the full network history

In this section, we estimate the preferential attachment parameter vector θ = (α, β, δ_in, δ_out) under two assumptions about what data is available. In the first scenario, the full evolution of the network is observed, from which the likelihood function can be computed. The resulting MLE is strongly consistent and asymptotically normal. For the second scenario, the data only consist of one snapshot of the network with n edges, without the knowledge of the network history that produced these edges. For this scenario we give an estimation approach through approximating the score function and moment matching, which produces parameter estimators that are also strongly consistent but less efficient than those based on the full evolution of the network. In both cases, the estimators are uniquely determined.

4.3.1 Likelihood calculation

Assume the network begins with the graph G(n₀) (consisting of n₀ edges) and then evolves according to the description in Section 4.2.1 with parameters (α, β, δ_in, δ_out), where δ_in, δ_out >

0 and α, β are non-negative probabilities. The γ is implicitly defined by γ = 1 − α − β. To avoid trivial cases, we will also assume α, β, γ < 1 for the rest of the chapter. For MLE estimation we restrict the parameter space for δ_in, δ_out to be [, K], for some sufficiently small  > 0 and large K. In particular, the true value of δ_in, δ_out is assumed to be contained in (, K). Let e_t = (v_t⁽¹⁾, v_t⁽²⁾) be the newly created edge when the random graph evolves from G(t − 1) to G(t). We sometimes refer to t as the time rather than the number of edges.

Assume we observe the initial graph G(n₀) and the edges {e_t}ⁿ_t=n0+1 in the order of their formation. For t = n₀+ 1, . . . , n, the values of the following variables are known:

• N (t), the number of nodes in graph G(t);

• D_in^(t−1)(v), D_out^(t−1)(v), the in- and out-degree of node v in G(t − 1), for all v ∈ V (t − 1);

• J_t, the scenario under which e_t is created.

Then the likelihood function is L(α, β, δ_in, δ_out| G(n₀), (e_t)ⁿ_t=n

0+1)

=

n

Y

t=n0+1

α D^(t−1)_in (v_t⁽²⁾) + δ_in t − 1 + δ_inN (t − 1)

!1_{Jt=1}

×

n

Y

t=n0+1

β D_in^(t−1)(v⁽²⁾_t ) + δ_in t − 1 + δinN (t − 1)

 D^(t−1)_out (v_t⁽¹⁾) + δ_out t − 1 + δoutN (t − 1)



!1_{Jt=2}

×

n

Y

t=n0+1

(1 − α − β) D^(t−1)_out (v_t⁽¹⁾) + δout

t − 1 + δ_outN (t − 1)

!1_{Jt=3}

(4.7) and the log likelihood function is

log L(α, β, δ_in, δ_out| G(n₀), (e_t)ⁿ_t=n0+1) (4.8)

= log α

n

X

t=n0+1

1{Jt=1}+ log β

n

X

t=n0+1

1{Jt=2}+ log(1 − α − β)

n

X

t=n0+1

1{Jt=3}

+

n

X

t=n0+1

log

D_in^(t−1)(v_t⁽²⁾) + δ_in

1{Jt∈{1,2}}

+

n

X

t=n0+1

log

D^(t−1)_out (v⁽¹⁾_t ) + δ_out

1{Jt∈{2,3}}

−

n

X

t=n0+1

log(t − 1 + δ_inN (t − 1))1_{Jt∈{1,2}}

−

n

X

t=n0+1

log(t − 1 + δ_outN (t − 1))1{Jt∈{2,3}}.

The score functions for α, β, δ_in, δ_out are calculated as follows:

∂

∂αlog L(α, β, δ_in, δ_out| G(n₀), (e_t)ⁿ_t=n

0+1)

= 1 α

n

X

t=n0+1

1{J_t=1}− 1 1 − α − β

n

X

t=n0+1

1{J_t=3}, (4.9)

∂

∂βlog L(α, β, δin, δout| G(n0), (et)ⁿ_t=n0+1)

= 1 β

n

X

t=n0+1

1{Jt=2}− 1 1 − α − β

n

X

t=n0+1

1{Jt=3}, (4.10)

∂

∂δ_inlog L(α, β, δ_in, δ_out| G(n₀), (e_t)ⁿ_t=n

0+1)

=

n

X

t=n0+1

1

D_in^(t−1)(v⁽²⁾_t ) + δ_in1{Jt∈{1,2}}

−

n

X

t=n0+1

N (t − 1)

t − 1 + δ_inN (t − 1)1{J_t∈{1,2}}, (4.11)

∂

∂δ_outlog L(α, β, δ_in, δ_out| G(n₀), (e_t)ⁿ_t=n0+1)

=

n

X

t=n0+1

1

D_out^(t−1)(v⁽¹⁾_t ) + δ_out1{J_t∈{2,3}}

−

n

X

t=n0+1

N (t − 1)

t − 1 + δ_outN (t − 1)1{Jt∈{2,3}}.

Note that the score functions (4.9), (4.10) for α and β do not depend on δ_in and δ_out. One can show that the Hessian matrix of the log-likelihood for (α, β) is positive definite.

Setting (4.9) and (4.10) to zero gives the unique MLE estimates for α and β, ˆ

α^{M LE} = 1 n − n₀

n

X

t=n0+1

1{Jt=1}, (4.12)

βˆ^{M LE} = 1 n − n₀

n

X

t=n0+1

1{Jt=2}. (4.13)

These estimates are strongly consistent by applying the strong law of large numbers for the {J_t} sequence.

Next, consider the first term of the score function for δ_in in (4.11), and we have

n

X

t=n0+1

1

D_in^(t−1)(v⁽²⁾_t ) + δin

1_{Jt∈{1,2}}

=

∞

X

i=0

1 i + δ_in

n

X

t=n0+1

1ⁿ

D^(t−1)_in (v_t⁽²⁾)=i,Jt∈{1,2}o. Observe that n

D_in^(t−1)(v_t⁽²⁾) = i, J_t∈ {1, 2}o

describes the event that the in-degree of node v_t⁽²⁾∈ V (t − 1) is i at time t − 1 and is augmented to i + 1 at time t. For each i ≥ 1, such an event happens at some stage t ∈ {n₀+ 1, n₀+ 2, . . . , n} only for those nodes with in-degree

≤ i at time n₀ and in-degree > i at time n. Let N_ij(n) denote the number of nodes with in-degree i and out-degree j at time n, and N_iⁱⁿ(n) and N_>iⁱⁿ(n) to be the number of nodes with in-degree equal to i and greater than i, respectively, i.e.,

N_iⁱⁿ(n) =

∞

X

j=0

N_ij(n), N_>iⁱⁿ(n) = X

k>i

N_kⁱⁿ(n).

Then

n

X

t=n0+1

1ⁿ

D^(t−1)_in (v_t⁽²⁾)=i,Jt∈{1,2}o = N_>iⁱⁿ(n) − N_>iⁱⁿ(n₀), i ≥ 1.

On the other hand, when i = 0, n

D_in^(t−1)(v⁽²⁾_t ) = 0, J_t∈ {1, 2}o

occurs for some t if and only if all of the following three events happen:

(i) v_t⁽²⁾ has in-degree > 0 at time n;

(ii) v_t⁽²⁾ does not have in-degree > 0 at time n₀;

(iii) v_t⁽²⁾ was not created under the γ-scheme (otherwise it would have been born with in-degree 1).

This implies:

n

X

t=n0+1

1ⁿ_D(t−1)

in (v_t⁽²⁾) = 0,Jt∈{1,2}o = N_>0ⁱⁿ(n) − N_>0ⁱⁿ(n₀) −

n

X

t=n0+1

1{Jt=3},

since there are, in total, Pn

t=n0+11_{Jt=3} nodes created under the γ-scheme. Therefore,

n

X

t=n0+1

1

D_in^(t−1)(v⁽²⁾_t ) + δ_in1{Jt∈{1,2}} =

∞

X

i=0

1 i + δ_in

n

X

t=n0+1

1ⁿ

D^(t−1)_in (v_t⁽²⁾)=i,Jt∈{1,2}o

=

∞

X

i=0

N_>iⁱⁿ(n) − N_>iⁱⁿ(n₀) i + δ_in −

Pn

t=n0+11{Jt=3}

δ_in .(4.14) Setting the score function (4.11) for δin to 0 and dividing both sides by n − n0 leads to

1 n − n₀

∞

X

i=0

N_>iⁱⁿ(n) − N_>iⁱⁿ(n₀)

i + δ_in − 1

δ_in(n − n₀)

n

X

t=n0+1

1{J_t=3}

− 1

n − n₀

n

X

t=n0+1

N (t − 1)

t − 1 + δ_inN (t − 1)1_{Jt∈{1,2}} = 0, (4.15) where the only unknown parameter is δ_in. In Section 4.3.2, we show that the solution to (4.15) actually maximizes the likelihood function in δ_in. Similarly, the MLE for δ_out can be solved from

1 n − n₀

∞

X

j=0

N_>j^out(n) − N_>j^out(n₀) j + δ_out −

1 n−n0

Pn

t=n0+11{J_t=1}

δ_out

− 1

n − n₀

n

X

t=n0+1

N (t − 1)

t − 1 + δ_outN (t − 1)1{J_t∈{2,3}} = 0, where N_>j^out(n) is defined in the same fashion as N_>iⁱⁿ(n).

Remark 4.3.1. The arguments leading to (4.14) allow us to rewrite the likelihood function (4.7):

L(α, β, δ_in, δ_out| G(n₀), (e_t)ⁿ_t=n0+1)

= α

Pn

t=n0+11_{Jt=1}

β

Pn

t=n0+11_{Jt=2}

(1 − α − β)

Pn

t=n0+11_{Jt=3}

×

n

Y

t=n0+1

(t − 1 + δinN (t − 1))^{−1{Jt∈{1,2}}} (t − 1 + δoutN (t − 1))^{−1{Jt∈{2,3}}}

×

n

Y

t=n0+1

" _∞ Y

i=0

(i + δ_in)

1{^D(t−1)in (v(2)

t )=i,Jt∈{1,2}}

∞

Y

j=0

(j + δ_out)

1{^D(t−1)out (v(1)

t )=j,Jt∈{2,3}}

#

= α^{Pnt=n0+11{Jt=1}} β^{Pnt=n0+11{Jt=2}} (1 − α − β)^{Pnt=n0+11{Jt=3}}

×

n

Y

t=n0+1

(t − 1 + δ_inN (t − 1))^{−1{Jt∈{1,2}}} (t − 1 + δ_outN (t − 1))^{−1{Jt∈{2,3}}}

δ_in^−1{Jt=3} δ_out^−1{Jt=1}i

×

∞

Y

i=0

(i + δ_in)^{N>iin(n)−N>iin(n0)}

∞

Y

j=0

(j + δ_out)^{N>jout(n)−N>jout(n0)}.

Hence by the factorization theorem, N (n0), (Jt)ⁿ_t=n0+1, (N_>iⁱⁿ(n) − N_>iⁱⁿ(n0))i≥0, (N_>j^out(n) − N_>j^out(n₀))_j≥0 are sufficient statistics for (α, β, δ_in, δ_out).

4.3.2 Consistency of MLE

We remarked after (4.12) and (4.13) that ˆα^{M LE} and ˆβ^{M LE} converge almost surely to α and β. We now prove that the MLE of (δin, δout) is also strongly consistent. Note that if we initiate the network with G(n₀) (for both n₀ and N (n₀) finite), then almost surely for all i, j ≥ 0,

N_>iⁱⁿ(n₀)

n ≤ N (n₀)

n → 0, N_>j^out(n₀)

n ≤ N (n₀)

n → 0, as n → ∞,

and (n − n₀)/n → 1. In other words, n₀, N_>iⁱⁿ(n₀), N_>j^out(n₀) are all o(n). So for simplicity, we assume that the graph is initiated with finitely many nodes and no edges, that is, n₀ = 0 and N (0) ≥ 1. In particular, these assumptions imply the sum of the in-degrees at time n is equal to n.

Let Ψ_n(·), Φ_n(·) be the functional forms of the terms in the log-likelihood function (4.8) involving δ_in and δ_out respectively, normalized by 1/n, i.e.,

Ψ_n(λ) :=

∞

X

i=0

N_>iⁱⁿ(n)

n log(i + λ) − log λ n

n

X

t=1

1{J_t=3}

− 1 n

n

X

t=1

log (t − 1 + λN (t − 1)) 1{Jt∈{1,2}},

Φ_n(µ) :=

∞

X

j=0

N_>j^out(n)

n log(j + µ) − log µ n

n

X

t=1

1{Jt=1}

− 1 n

n

X

t=1

log (t − 1 + µN (t − 1)) 1{Jt∈{2,3}}. The following theorem gives the consistency of the MLE of δ_in and δ_out. Theorem 4.3.2. Suppose δ_in, δ_out ∈ (, K) ⊂ (0, ∞). Define

ˆδ_in^{M LE} = ˆδ_in^{M LE}(n) := argmax

≤λ≤K

Ψ_n(λ), δˆ_out^{M LE} = ˆδ_out^{M LE}(n) := argmax

≤µ≤K

Φ_n(µ).

Then these are the MLE estimators of δ_in, δ_out and they are strongly consistent; that is, δˆ^{M LE}_in −→ δ^a.s. _in, ˆδ_out^{M LE} −→ δ^a.s. _out, n → ∞.

Proof of Theorem 4.3.2. We only verify the consistency of ˆδ_in^{M LE} since similar arguments apply to ˆδ^{M LE}_out . Define

ψ_n(λ) := Ψ⁰_n(λ) =

∞

X

i=0

N_>iⁱⁿ(n)/n i + λ −

1 n

Pn

t=11{J_t=3}

λ − 1

n

n

X

t=1

N (t − 1)

t − 1 + λN (t − 1)1_{Jt∈{1,2}}. Let us consider a limit version of ψn:

ψ(λ) :=

∞

X

i=0

pⁱⁿ_>i(δ_in) i + λ − γ

λ − (1 − β)a₁(λ), (4.16)

where pⁱⁿ_>i(δ_in) := P

k>ipⁱⁿ_k(δ_in) with pⁱⁿ_k(δ_in) := pⁱⁿ_k as defined in (4.4), and a₁(λ) := α + β

1 + λ(1 − β), λ > 0.

Here we write pⁱⁿ_i (δ_in) to emphasize the dependence on δ_in. In Lemmas 4.7.1 and 4.7.2, provided in Section 4.7, it is shown that ψ(·) has a unique zero at δ_in, where ψ(λ) > 0 when λ < δin and ψ(λ) < 0 when λ > δin, and

sup

λ≥

|ψ_n(λ) − ψ(λ)| → 0. (4.17)

Since ψ is continuous, for any κ > 0 arbitrarily small, there exists ε_κ > 0 such that ψ(λ) > ε_κ for λ ∈ [, δ_in− κ] and ψ(λ) < −ε_κ for λ ∈ [δ_in+ κ, K]. From (4.17),

P ∃N_κ s.t. sup

n>Nκ

sup

λ∈[,K]

|ψ_n(λ) − ψ(λ)| < ε_κ/2

!

= 1. (4.18)

Note sup_λ∈[,K]|ψ_n(λ) − ψ(λ)| < ε_κ/2 implies ψn(λ) ≥ ψ(λ) − sup

λ∈[,K]

|ψn(λ) − ψ(λ)| ≥ εκ− εκ/2 > 0, λ ∈ [, δin− κ), and

ψ_n(λ) ≤ ψ(λ) + sup

λ∈[,K]

|ψ_n(λ) − ψ(λ)| ≤ −ε_κ + ε_κ/2 < 0, λ ∈ (δ_in+ κ, K].

These jointly indicate that δ_in− κ ≤ ˆδ_in^{M LE} ≤ δ_in+ κ. Hence (4.18) implies P



n→∞lim |ˆδ_in^{M LE} − δ_in| ≤ κ

= 1, for arbitrary κ > 0. That is, ˆδ_in^{M LE} −→ δ^a.s. _in.

4.3.3 Asymptotic normality of MLE

In the following theorem, we establish the asymptotic normality for the MLE estimator θˆ_n^{M LE} = ( ˆα^{M LE}, ˆβ^{M LE}, ˆδ_in^{M LE}, ˆδ^{M LE}_out ).

Theorem 4.3.3. Let ˆθ_n^{M LE} be the MLE estimator for θ, the parameter vector of the pref-erential attachment model. Then

√n( ˆθ^MLE_n − θ) → N (0, Σ(θ)) ,^d

where

Σ⁻¹(θ) = I(θ) :=



















1−β α(1−α−β)

1

1−α−β 0 0

1 1−α−β

1−α

β(1−α−β) 0 0

0 0 Iin 0

0 0 0 I_out



















, (4.19)

with

I_in :=

∞

X

i=0

pⁱⁿ_>i

(i + δ_in)² − γ

δ²_in − (α + β)(1 − β)²

(1 + δ_in(1 − β))², (4.20) Iout :=

∞

X

j=0

p^out_>j

(j + δ_out)² − α

δ²_out − (γ + β)(1 − β)² (1 + δ_out(1 − β))².

In particular, I(θ) is the asymptotic Fisher information matrix for the parameters, and hence the MLE estimator is efficient.

Remark 4.3.4. From Theorem 4.3.3, the estimators ( ˆα^{M LE}, ˆβ^{M LE}), ˆδ_in^{M LE}, and ˆδ_out^{M LE} are asymptotically independent.

Proof of Theorem 4.3.3. We first show the limiting distributions for the MLE’s, i.e. ( ˆα^{M LE}, ˆβ^{M LE}), δˆ^{M LE}_in and ˆδ_out^{M LE}. From (4.12) and (4.13),

( ˆα^{M LE}, ˆβ^{M LE}) = 1 n

n

X

t=1

1{Jt=1}, 1{Jt=2}
,

where {Jt} is a sequence of iid random variables. Hence the limiting distribution of the pair

 ˆ

α^{M LE}, ˆβ^{M LE}

follows directly from standard central limit theorem for sums of independent random variables.

Next we show the asymptotic normality for ˆδ_in^{M LE}; the argument for ˆδ_out^{M LE} is similar.

Recall from (4.11) that the score function for δ_in can be written as

∂

∂δ_in log L(α, β, δin, δout)    δ

=:

n

X

t=1

ut(δ), where u_t is defined by

u_t(δ) := 1

D^(t−1)_in (v_t⁽²⁾) + δ

1_{Jt∈{1,2}}− N (t − 1)

t − 1 + δN (t − 1)1_{Jt∈{1,2}}. (4.21) The MLE estimator ˆδ_in^{M LE} can be obtained by solvingPn

t=1u_t(δ) = 0. By a Taylor expansion of Pn

t=1u_t(δ),

0 =

n

X

t=1

u_t(ˆδ_in^{M LE}) =

n

X

t=1

u_t(δ_in) + (ˆδ^{M LE}_in − δ_in)

n

X

t=1

˙u_t(ˆδ^∗_in), (4.22)

where ˙u_t denotes the derivative of u_t and ˆδ_in^∗ = δ_in+ ξ(ˆδ_in^{M LE} − δ_in) for some ξ ∈ [0, 1]. An elementary transformation of (4.22) gives

n^1/2(ˆδ^{M LE}_in − δ_in) = − 1 n⁻¹Pn

t=1 ˙u_t(ˆδ_in^∗)

! n^−1/2

n

X

t=1

u_t(δ_in)

! . To establish

n^1/2(ˆδ^{M LE}_in − δin) → N (0, I^d _in⁻¹),

where I_in is as defined in (4.19), it suffices to show the following two results:

(i) n^−1/2Pn

t=1u_t(δ_in) → N (0, I^d _in), (ii) n⁻¹Pn

t=1 ˙u_t(ˆδ^∗_in) → −I^p _in.

These are proved in Lemmas 4.7.3 and 4.7.4 in the Section 4.7.1, respectively.

To establish the joint asymptotic normality of the MLE estimator ˆθ_n^{M LE}, denote the joint score function vector for θ by

∂

∂θ log L(θ) =: S_n(θ) = (S_n(α), S_n(β), S_n(δ_in), S_n(δ_out))^T ,

where S_n(α), S_n(β), S_n(δ_in), S_n(δ_out) are the score functions for α, β, δ_in, δ_out, respectively. A multivariate Taylor expansion gives

0 = S_n ˆθ^{M LE}_n 

= S_n(θ) + ˙S_n ˆθ^∗_n  ˆθ^{M LE}_n − θ

, (4.23)

where ˙S_n denotes the Hessian matrix of the log-likelihood function log L(θ), and ˆθ_n^∗ = θ + ξ ◦ ˆθ_n^{M LE}− θ

for some vector ξ ∈ [0, 1]⁴, where “◦” denotes the Hadamard product.

From Remark 4.3.1, the likelihood function L(θ) can be factored into L(θ) = f₁(α, β)f₂(δ_in)f₃(δ_out).

Hence

1 n

S˙_n( ˆθ_n^∗) =



















∂²log Ln( ˆθ^∗_n)

∂α²

∂²log Ln( ˆθ_n^∗)

∂α∂β 0 0

∂²log Ln( ˆθ^∗_n)

∂β∂α

∂²log Ln( ˆθ_n^∗)

∂β² 0 0

0 0 ^{∂2log L}_∂δ2^{n( ˆθ∗n)} in

0 0 0 0 ^{∂2log L}_∂δ2^{n( ˆθ∗n)}

out



















→ I(θ)p (4.24)

as implied in the previous part of the proof, where I(θ) (defined in (4.19)) is positive semi-definite.

Note that (S_n(α), S_n(β)), S_n(δ_in), S_n(δ_out) are pairwise uncorrelated. As an example, observe that

E[Sⁿ(α)Sn(δin)] =

Z ∂ log L(θ)

∂α

∂ log L(θ)

∂δ_in L(θ)dx

=

Z ∂ log f₁(α, β)

∂α

∂ log f₂(δ_in)

∂δ_in f1(α, β)f2(δin)f3(δout)dx

=

Z ∂f₁(α, β)

∂α

∂f₂(δ_in)

∂δ_in f3(δout) dx

= ∂²

∂α∂δin

Z

L(θ)dx

= 0 = E[Sn(α)]E[Sn(δ_in)].

Using the Cram´er-Wold device, the joint convergence of Sn(θ) follows easily, i.e., n^−1/2S_n(θ) → N (0, I(θ)).^d

From here, the result of the theorem follows from (4.23) and (4.24).

In document ANEXO Nº 3. TRABAJO DE ASESORAMIENTO PARA LA REDACCIÓN DEL PROYECTO DEL EMISARIO DE GORLIZ (UNICAN) (página 20-28)