Latent Space Models

In this section, we first give a detailed introduction of the inner-product model (4.1) and conditions for its identifiability. In addition, we introduce a more general class of latent space models that includes the inner-product model as a special case. The methods we propose later will be motivated by the inner-product model and can also be applied to the more general class.

4.2.1. Inner-product models

Recall the inner-product model defined in (4.1), i.e., for any observed A and X and any i < j,

Aij =Ajiind.∼ Bernoulli(Pij), with logit(Pij) = Θij =αi+αj +βXij+zi>zj.

Fixing all other parameters, if we increaseαi, then nodeihas higher chances of connecting

with other nodes. Therefore, the αi’s model degree heterogeneity of nodes and we call

them degree heterogeneity parameters. Next, the regression coefficient β moderates the contribution of covariate to edge formation. For instance, if Xij indicates whether nodesi

and j share some common attribute such as gender, then a positive β value implies that nodes that share common feature are more likely to connect. Such a phenomenon is called homophily in the social network literaute. Last but not least, the latent variables {zi}ni=1

enter the model through their inner-productz_i>zj, and hence is the name of the model. We

impose no additional structural/distributional assumptions on the latent variables for the sake of modeling flexibility.

We note that model (4.1) also allows the latent variables to enter the second equation in the form ofg(zi, zj) =−1₂kzi−zjk2. To see this, note thatg(zi, zj) =−1₂kzik2−1₂kzjk2+z_i>zj,

and we may re-parameterize by setting ˜αi=αi−₂1kzik2 for all i. Then we have

Θij =αi+αj+βXij −

1

An important implication of this observation is that the function g(zi, zj) =−1₂kzi−zjk2

directly models transitivity, i.e., nodes with common neighbors are more likely to connect since their latent variables are more likely to be close to each other in the latent space. In view of the foregoing discussion, the inner-product model (4.1) also enjoys this nice modeling capacity.

In matrix form, we have

Θ =α1n>+ 1nα>+βX+G (4.2)

where 1n is the all one vector in Rn and G=ZZ> with Z = (z1,· · ·, zn)> ∈Rn×k. Since

there is no self-edge and Θ is symmetric, only the upper diagonal elements of Θ are well defined, which we denote by Θu. Nonetheless we define the diagonal element of Θ as in (4.2) since it is inconsequential. To ensure identifiability of model parameters in (4.1), we assume the latent variables are centered, that is

J Z =Z where J =In−

1 n1n1n

>_{. (4.3)}

Note that this condition uniquely identifies Z up to an orthogonal transformation of the rows while G=ZZ> is now directly identifiable.

4.2.2. A more general class of latent space models

Model (4.1) is a special case of a more general class of latent space models, which can be defined by Aij =Aji ind. ∼ Bernoulli(Pij), with logit(Pij) = Θij = ˜αi+ ˜αj+βXij+h(zi, zj) (4.4)

where h(·,·) is a smooth symmetric function on Rk×Rk. We shall impose an additional

constraint on h following the discussion below. In matrix form, for ˜α = ( ˜α1, . . . ,α˜n)0 and

To better connect with (4.2), let

G=J HJ, and α1n>+ 1nα>= ˜α1n>+ 1nα˜>+H−J HJ. (4.5)

Note that the second equation in the last display holds since the expression on its right hand side is symmetric and of rank at most two. Then we can rewrite the second last display as

Θ =α1n>+ 1nα>+βX+G (4.6)

which reduces to (4.2) and G satisfies J G = G. Our additional constraint on h is the following Schoenberg Condition:

For any positive integern≥2 and anyz1, . . . , zn∈Rk,

G=J HJ is positive semi-definite forH = (h(zi, zj)) andJ =In−_n11n1n>.

(4.7)

Condition (4.7) may seem abstract, while the following lemma provides two important sets of symmetric functions for which it is satisfied.

Lemma 4.1. Condition (4.7) is satisfied in the following cases:

1. h is a positive semi-definite kernel function on Rk×Rk;

2. h(x, y) =−kx−ykqp for some p ∈(0,2]and q ∈(0, p] where k · kp is the p-norm (or

p-seminorm when p <1) on Rk.

The first claim of Lemma 4.1 is a direct consequence of the definition of positive semi-definite kernel function which ensures that the matrix H itself is positive semi-definite and so is G=J HJ sinceJ is also positive semi-definite. The second claim is a direct consequence of the famous Hilbert space embedding result by Schoenberg (Schoenberg, 1937, 1938). See, for instance, Theorems 1 and 2 of Schoenberg (1937).