Social Network Mediation Analysis: A Latent Space Approach

Abstract

A social network comprises both actors and the social connections among them. Such connections reflect the dependence among social actors, which is essential for individuals’ mental health and social development. In this article, we propose a mediation model with a social network as a mediator to investigate the potential mediation role of a social network. In the model, the dependence among actors is accounted for by a few mutually orthogonal latent dimensions which form a social space. The individuals’ positions in such a latent social space are directly involved in the mediation process between an independent and dependent variable. After showing that all the latent dimensions are equivalent in terms of their relationship to the social network and the meaning of each dimension is arbitrary, we propose to measure the whole mediation effect of a network. Although individuals’ positions in the latent space are not unique, we rigorously articulate that the proposed network mediation effect is still well defined. We use a Bayesian estimation method to estimate the model and evaluate its performance through an extensive simulation study under representative conditions. The usefulness of the network mediation model is demonstrated through an application to a college friendship network.

Appendix

1.1 Gibbs Sampler

Because the posterior distribution has no closed form, we thus use the Markov chain Monte Carlo method to draw samples of parameters from their posterior distributions. The steps of Gibbs sampler are provided below. Given desired length T and initials ($\varvec{i}_{1}^{0},i_{2}^{0},\varvec{a}^{0},\varvec{b}^{0}$, $\alpha ^{0},c'^{0},(\sigma _{1,d}^{2},d=1,2,\ldots ,D)^{0}$,$(\sigma _{2}^{2})^{0}$),

• 1. In the $k'$th iteration, draw $\mathbf{z}_{i}^{k}$ from its conditional posterior distribution $P(\mathbf{z}_{i}|\varvec{i}_{1}^{k-1},i_{2}^{k-1},\varvec{a}^{k-1},\varvec{b}^{k-1},\alpha ^{k-1},(c'){}^{k-1},(\sigma _{1,d}^{2},d=1,2,\ldots ,D)^{k-1},(\sigma _{2}^{2})^{k-1},X, Y,{} \mathbf{M})$, for all actors i in the network;

• 2. Draw $\varvec{i}_{1}^{k}$ from its conditional posterior distribution $P(\varvec{i}_{1}|i_{2}^{k-1}{,}\varvec{a}^{k-1},\varvec{b}^{k-1},\alpha ^{k-1},(c')^{k-1},(\sigma _{1,d}^{2},d{=}1,2,{\ldots },D)^{k{-}1},(\sigma _{2}^{2})^{k-1},\mathbf{z}_{i}^{k},X,Y,\mathbf{M})$ with updated $\mathbf{z}_{i}^{k};$

• 3. Draw $i_{2}^{k}$ from its conditional posterior distribution $P(i_{2}|\varvec{i}_{1}^{k},\varvec{a}^{k-1},\varvec{b}^{k-1},\alpha ^{k-1},c'^{k-1},(\sigma _{1,d}^{2},d=1,2,\ldots ,D)^{k-1},(\sigma _{2}^{2}){}^{k-1},\mathbf{z}_{i}^{k},X, Y,\mathbf{M})$

• 4. Draw $\alpha ^{k}$ from its conditional posterior distribution $P(\alpha |\varvec{i}_{1}^{k},i_{2}^{k},\varvec{a}^{k-1},\varvec{b}^{k-1},(c'){}^{k-1},(\sigma _{1,d}^{2},d=1,2,\ldots ,D)^{k-1},(\sigma _{2}^{2})^{k-1},{\mathbf{z}}_{i}^{k},X, Y,\mathbf{M}$);

• 5. Draw $\varvec{a}^{k}$from its conditional posterior distribution P($\varvec{a}|\varvec{i}_{1}^{k},i_{2}^{k},\varvec{b}^{k-1},\alpha ^{k},(c'){}^{k-1},(\sigma _{1,d}^{2},d=1,2,\ldots ,D)^{k-1},(\sigma _{2}^{2})^{k-1}$,$\mathbf{z}_{i}^{k},X, Y$, $\mathbf{M}$);

• 6. Draw $\varvec{b}^{k}$ from its conditional posterior distribution P($\varvec{b}|\varvec{i}_{1}^{k},i_{2}^{k},\varvec{a}^{k},\alpha ^{k},(c')^{k-1},(\sigma _{1,d}^{2},d=1,2,\ldots ,D)^{k-1},(\sigma _{2}^{2})^{k-1}$,$\mathbf{z}_{i}^{k},X,Y$, $\mathbf{M}$);;

• 7. Draw $c'^{k}$ from its conditional posterior distribution P($c'|\varvec{i}_{1}^{k},i_{2}^{k},\varvec{a}^{k}, {\varvec{b}^{k}},\alpha ^{k},(\sigma _{1,d}^{2},d=1,2,\ldots ,D)^{k-1},(\sigma _{2}^{2})^{k-1}$,$\mathbf{z}_{i}^{k},X,Y$,$\mathbf{M}$);;

• 8. Draw $(\sigma _{1,d}^{2},d=1,2,\ldots ,D)^{k}$ from its conditional posterior distribution P($\sigma _{1,d}^{2},d=1,2,\ldots ,D|\varvec{i}_{1}^{k},i_{2}^{k},\varvec{a}^{k}, {\varvec{b}^{k}},\alpha ^{k},c'^{k},(\sigma _{2}^{2})^{k-1}$,$\mathbf{z}_{i}^{k},X$, $\mathbf{M}$);

• 9. Draw $(\sigma _{2}^{2})^{k}$ from its conditional posterior distribution P($\sigma _{2,}^{2}|\varvec{i}_{1}^{k},i_{2}^{k},\varvec{a}^{k}, {\varvec{b}^{k}},\alpha ^{k}$,$(\sigma _{1,d}^{2},d=1,2,\ldots ,D)^{k},X,Y$,$\mathbf{M}$).

Repeat steps 1–9 until the chain reaches convergence and has sufficient posterior samples.

1.2 OpenBUG Code

The following is the OpenBUG code used for empirical data analysis with three latent dimensions.