Belief and opinion evolution in social networks (SNs) can aid in understanding how people influence others’ decisions through social relationships as well as provide a solid foundation for many valuable social applications. As large numbers of users are involved in SNs, the complexity of traditional optimization techniques is high as they deal with the interactions between users separately. Moreover, the state variable (opinion) is high-dimensional because a person usually has opinions about many different social issues. To overcome those challenges, we formulate the opinion evolution in SNs as a high-dimensional stochastic mean field game (MFG). Numerical methods for high-dimensional MFGs are practically non-existent because of the need for grid-based spatial discretization. Thus, we propose a machine-learning based method, where we use an alternating population and agent control neural network (APAC-net), to tractably solve high-dimensional stochastic MFGs. Through APAC-net, solving MFGs can be regarded as a special case of training a generative adversarial network (GAN). To the best of our knowledge, the APAC-Net is the first model that can solve high-dimensional stochastic MFGs. The simulation results affirm the efficiency of the APAC-net.

The Wasserstein distance serves as a loss function for unsupervised learning which depends on the choice of a ground metric on sample space. We propose to use the Wasserstein distance itself as the ground metric on the sample space of images. This ground metric is known as an effective distance for image retrieval, that correlates with human perception. We derive the Wasserstein ground metric on pixel space and define a Riemannian Wasserstein gradient penalty to be used in the Wasserstein Generative Adversarial Network (WGAN) framework. The new gradient penalty is computed efficiently via convolutions on the $L^2$ gradients with negligible additional computational cost. The new formulation is more robust to the natural variability of the data and provides for a more continuous discriminator in sample space.