Network approximation

To approximate a time-varying network affected by missing entries, we adopt a version of the NTF, as in [168], which takes into account the presence of missing values in the tensor to be factorized. Given a tensor X , representing a time-varying network (as in Sec. 4.1.1) with some missing entries, the NTF framework shown in Eq. (4.1) is modified by including a binary tensor mask W of the same size of X whose entries are defined as

wijk=⎧⎪⎪⎪

⎨⎪⎪⎪⎩

1 if xijk is known , 0 otherwise .

To build the tensor mask W, we assume to be aware of the nodes that have missing information in the network. More precisely, we assume that all the information relative to the nodes that are not missing and all the non-zero entries related to a missing node are meaningful. We apply the adapted version of the NTF by minimizing an optimization function of the form:

fw(A, B, C) = ∥X ∗ W − W ∗Jλ; A, B, CK∥

2

F , (5.1)

To this aim, we generate a first set of factor matrices A, B, C, we factorize a tensor ˜X whose values are given by the combination of values of X and of Jλ; A, B, CK. In particular, the tensorX˜ has the same size and the same values of X in the positions that are known, and unknown values are replaced by

5.1 Missing data recovery 113

the approximation in Jλ; A, B, CK. Thus, the tensor that we approximate is updated at each iteration by:

X˜ = X ∗ W + (1 − W)Jλ; A, B, CK . (5.2)

Joint Non-negative Tensor Factorization

When studying time-varying networks, we might have access to some external information, characterizing the nodes or links of the network. Thus, instead of just using the correlated activity patterns in a time-varying network to recover its missing values, an alternative is to use the external information.

This is possible if the new data source is correlated in some dimension with the time-varying network studied, to reconstruct the missing values.

To integrate and analyse multiple data sources at a time, each represented as a tensor, we can use a generalized NTF framework that allows to decompose multiple tensor at once. This technique is called Joint Non-negative Tensor Factorization (JNTF) [169], and allows to study the correlated activity patterns in multiple dimension by coupling different tensors.

Formally, given S different data sources, each represented as a tensor Xs

with s= 1, . . . , S, we can adapt the optimization problem in Eq. (4.1) as:

min1 2

S

∑s=1∥X^s−Jλ^s; As, Bs, CsK∥

2 F +1

2

S

∑s=1αs∥λ^s∥2 (5.3) s.t. λs, As, Bs, Cs≥ 0 .

This generalization of the NTF problem allows to integrate data from several sources by approximating together the tensors Xs. Moreover, we can envisage different couplings between tensors by imposing that some of the factor ma-trices As, Bs, Cs will be common for certain s values, and we can add some regularization terms, which ensure sparsity.

The use of the JNTF can be illustrated by the following case study. Let us consider the time-varying social network of the LSCH dataset, and suppose to have access to the position of the nodes of the network during the same time period (see Chapter 3 for details). We can represent the contact network and the location network, providing the position of the nodes at each time, by the

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Interplay between Time-varying Networks and Dynamical Processes: impact of the mesoscale structures tensors X1 and X2, whose dimensions will be respectively node-node-time and node-location-time. As a consequence, the tensors X1 and X2 are coupled in two dimensions, as they share the nodes and the snapshots in time, thus the Eq. (5.3) becomes: provide the membership of the nodes to the components, the matrix C gives their temporal activity and the matrix B2 represents the mapping between the components and the locations. It is worth noting that in this case we can have different λ vectors. Thus, this framework has the advantage of being less constrained than solving the NTF on a tensor having 4 dimensions (node-node-time-location).

The adaptation to the missing data problem in the JNTF is now straight-forward, as we can substitute to the term ,corresponding to the tensor affected by the missing information, the function in Eq. (5.1). Following the aforemen-tioned example, let us now suppose that some elements in X1 are missing. We want to recover the missing entries by using the external information provided in X2, and consequently we need to solve the following problem:

min1

To solve this minimization problem, we modified both the NCG algorithm and the KKT conditions in the ANLS method. We provide the details of the gradient computation as well as the modification of the conditions for the ANLS method in the following sections. By solving the problem, we can use the results of the raw approximation to see what characteristics of the original network we can recover.

5.1 Missing data recovery 115

NCG adaptation

The computation with the NCG for both the Joint factorization and the case of incomplete data was provided by Acar et al. [169], whose results are found by using the Poblano Toolbox [170]. However, the provided package does not include the computation of the joint factorization with the non-negativity constraints. Here, we provide the computation of the gradient, which is necessary to solve the JNTF problem. To this aim, we include the non-negativity constraints through the Hadamard product of the factor matrices, as shown in Section 2.4.6.

Let us consider the cost function for the JNTF in Eq. (5.4), where the tensors X1 and X2 are coupled in the first and third dimension. To incorporate the non-negativity constraints we use the Hadamard product of the factors, such that the cost function, rewritten in matricized form, becomes

h(λ^1,2, A, B1,2, C) = f^{J N T F}(λ^1,2∗ λ^1,2, A∗ A, B^1,2∗ B^1,2, C∗ C)

=1

2∥X1,(1)− (A ∗ A) (Λ¹∗ Λ¹) ((C ∗ C) ⊙ (B¹∗ B¹))^T∥²_F + +1

2∥X^2,(1)− (A ∗ A) (Λ²∗ Λ²) ((C ∗ C) ⊙ (B²∗ B²))^T∥²_F =

=1

2∥δ^1,(1)∥²_F +1

2∥δ^2,(1)∥²_F (5.5)

=1

2∥X^1,(2)− (B¹∗ B¹) (Λ¹∗ Λ¹) ((C ∗ C) ⊙ (A ∗ A))^T∥²_F + +1

2∥X2,(2)− (B²∗ B²) (Λ¹∗ Λ¹) ((C ∗ C) ⊙ (A ∗ A))^T∥²_F =

=1

2∥δ1,(2)∥²_F +1

2∥δ2,(2)∥²_F (5.6)

=1

2∥X1,(3)− (C ∗ C) (Λ1∗ Λ1) ((B1∗ B1) ⊙ (A ∗ A))^T∥²_F + +1

2∥X2,(3)− (C ∗ C) (Λ²∗ Λ²) ((B²∗ B²) ⊙ (A ∗ A))^T∥²_F =

=1

2∥δ1,(3)∥²_F +1

2∥δ2,(3)∥²_F . (5.7)

To find a solution for the minimization of this cost function, we need to derive the differential dh(λ^1,2, A, B1,2, C) with respect to all variables λ^1,2, A,B1,2,

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Interplay between Time-varying Networks and Dynamical Processes: impact of the mesoscale structures C, which is given by:

dh(λ^1,2, A, B1,2, C) =

=1

2⟨∂h(λ^1,2, A, B1,2, C)

∂A , dA⟩ +1

2⟨∂h(λ^1,2, A, B1,2, C)

∂B1

, dB1⟩+

+1

2⟨∂h(λ^1,2, A, B1,2, C)

∂B2

, dB2⟩ +1

2⟨∂h(λ^1,2, A, B1,2, C)

∂C , dC⟩+

+1

2⟨∂h(λ^1,2, A, B1,2, C)

∂λ1

, dλ1⟩ +1

2⟨∂h(λ^1,2, A, B1,2, C)

∂λ2

, dλ2⟩ .

First, the derivatives with respect to the factors B1 and B2 can be found by following the steps in Section 2.4.6. Thus,

∇^B1h(λ^1,2, A, B1,2, C) = ∂h

∂B1 = 2B¹∗ ((−δ^1,(2)) [(C ∗ C) ⊙ (A ∗ A)]) ,

∇B2h(λ1,2, A, B1,2, C) = ∂h

∂B2 = 2B2∗ ((−δ2,(2)) [(C ∗ C) ⊙ (A ∗ A)]) . Second, the derivatives with respect to the factors A and C are given by the sum of the derivatives of terms in Eq. (5.5) and Eq. (5.7) respectively. Thus

∇^Ah(λ^1,2, A, B1,2, C) = ∂h

∂A = 2A ∗ ((−δ^1,(1)) [(C ∗ C) ⊙ (B1∗ B¹)]) + + 2A ∗ ((−δ2,(1)) [(C ∗ C) ⊙ (B²∗ B²)])

∇Ch(λ1,2, A, B1,2, C) = ∂h

∂C = 2C ∗ ((−δ1,(3)) [(B1∗ B1) ⊙ (A ∗ A)]) + + 2C ∗ ((−δ^2,(3)) [(B²∗ B²) ⊙ (A ∗ A)]) .

5.1 Missing data recovery 117

Finally, we have to compute the derivatives with respect to λ1 and λ2. Let us consider fJ N T F and rewrite it as follows:

fJ N T F(λ^1,2∗ λ^1,2, A∗ A, B^1,2∗ B^1,2, C∗ C) =

We compute the element-wise derivatives for all the addends in the equation above, with respect to λ1,r^′ (similarly for λ2,r^′). The terms f1, f5, f6, f7, and The remaining derivatives can be computed as follows:

∂f2

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Interplay between Time-varying Networks and Dynamical Processes: impact of the mesoscale structures

The resulting derivatives for λ1 and analogously for λ2 are then

∇^λ1,r′h(λ^1,2, A, B1,2, C) =

In the following section, we provide the adaptation of the KKT conditions for the computation of the JNTF. This adaptation can be used as an alternative to the functions provided in the Poblano Toolbox, to solve JNTF problems through the ANLS and BPP algorithms.

KKT adaptation

In this section, we show how to adapt the ANLS framework and in particular how to compute the new KKT optimality conditions to perform the joint factorization with non-negativity constraints. To this aim, we have to solve the minimization problem in Eq. (5.3). Here, we take as regularization terms the square of the l2-norm, which, as we will show below, will be useful to adapt the KKT conditions. We show the procedure to solve Eq.(5.4), in which the two data sources are coupled in two dimensions, i.e. S= 2, A = A¹= A², and

5.1 Missing data recovery 119

C= C¹ = C². The solution to this problem will be useful to find a solution for the application described in Section 5.1.3.

To solve the problem through the BPP algorithm we have to rewrite the minimization problem in the form of Eq. 2.22, and solve it for each of the factor matrices (here we include λ1,2 in the factor matrices). To this aim, we start by solving the problem for the factor matrices B1 and B2 which are respectively present in the first and third term of Eq. 5.4. With respect to these factor matrices we can rewrite the equation by using the 2-mode matricization of X1

and X2, which leads to the following approximations:

X_1,(2)≈ B¹Λ₁(C ⊙ A)^T and X2,(2)≈ B²Λ₂(C ⊙ A)^T ,

where Λ1 = diag (λ^1,1, . . . , λ1,R) and Λ²= diag (λ^2,1, . . . , λ2,R). We can rewrite the approximations as

X^T_1,(2)≈ (C ⊙ A) Λ¹B^T₁ and X^T_2,(2)≈ (C ⊙ A) Λ²B^T₂ , where Λ^T1,2= Λ^1,2, and thus

B^T₁ ≈ Λ⁻¹1 (C ⊙ A)^†X^T_1,(2) and B^T2 ≈ Λ⁻¹2 (C ⊙ A)^†X^T_2,(2) .

By using the third property of the Khatri-Rao product in Prop. 2, we can write the approximation as

Λ₁(C^TC∗ A^TA) B^T1 ≈ (C ⊙ A)^TX^T_1,(2), Λ₂(C^TC∗ A^TA) B^T2 ≈ (C ⊙ A)^TX^T_2,(2) .

The related subproblems of Eq.(5.4) for B1 and B2 are then reduced to:

minB1

1

2∥Λ¹(C^TC∗ A^TA) B^T1 − (C ⊙ A)^TX^T_1,(2)∥²_F , minB2

1

2∥Λ²(C^TC∗ A^TA) B^T2 − (C ⊙ A)^TX^T_2,(2)∥²

F .

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Interplay between Time-varying Networks and Dynamical Processes: impact of the mesoscale structures Finally, the subproblems can be respectively written in the form of Eq. 2.22 by assigning

V= Λ¹(C^TC∗ A^TA) , X = B^T1 and W= Λ²(C ⊙ A)^TX^T_1,(2), V= (C^TC∗ A^TA) , X = B^T2 and W= (C ⊙ A)^T X^T_2,(2).

The solution to the subproblems is now straightforward as illustrated in Sec-tion 2.4.5.

We now need to rewrite the subproblems for the factors A and C which are present in both the first and third term of Eq. (5.4). We show the procedure for A (which is analogous for C). First, we write the minimization problem by using the 1-mode matricization of X1 and X2 (3-mode for C):

minA By following the procedure shown above, we can write the problem in the following way: in which we can assign

V₁ = Λ¹C^TC+ B^T1B₁, X= A^T and W1= (C ⊙ B¹)^TX^T_1,(1), To solve the problem in Eq. (5.8) we need to adapt the KKT conditions, which result to be

5.1 Missing data recovery 121

whose solution is given by solving

X^TV_1,2^T − W^T1,2= 0 .

Since in Eq. (5.4) are present the regularization terms, we have to adapt the KKT conditions for the minimization problem with respect to λ1 and λ2. We show the procedure for λ1, which is analogous for λ2. We consider the cost function built from the terms in which λ1 is involved (the first and second in Eq.(5.4)): and we rewrite it through the vectorization:

fλ1 = 1

2∥vec (C ⊙ B ⊙ A) λ1− vec (X )∥²F +α1

2 ∥λ1∥²2 .

Minimizing the cost function fλ1 is equivalent to minimize f_λ^′₁, obtained by incorporating the regularization term as follows:

f_λ^′₁ =1

The solution to the minimization of f_λ^′₁ follows from Eq. (2.19), as shown in Section 2.4.5.

In document Tensor decomposition techniques for analysing time-varying networks (Page 132-141)