Temporal Latent Space Models for Social Networks

• In Event: 46.028 - Quantitative Methods for the Analysis of Social Network Data in Education

Abstract

We introduce the Temporal Latent Space Model, an extension of the Latent Space Model for static social networks, to study the generative process of networks observed over several time points. The motivation for developing this model is to understand how the structure of the network evolves over time. A lot of real world networks are observed over several time points, and there is value in modeling how these networks evolve to understand changes in interaction pattern, to detect formation and/or dissolution of groups, and to investigate many other interesting aspects of the networks.

The Latent Space Model (LSM) assumes that each individual in the network occupies a position in a latent social space. The pairwise distances between these positions predict the probability of tie. Thus, a network with a fixed number of nodes can evolve in time by either expanding the latent positions which makes the network sparser, by shrinking of latent positions which makes the network denser, or by not changing which implies no substantive change in the structure of the network. Our framework for Temporal Latent Space Models allows us to model these changes. Moreover, similar to static latent space model, this model also accounts for basic network properties like transitivity and reciprocity at a time point, and further, along different time points.

Given an observed network at time, t, and assume that some of the nodes remain fixed. Then the probability of forming an edge at that time depends on the network at time t-1 only through the pairwise distances of the latent positions such than the latent space positions at time t are distributed with a multivariate normal distribution centered at the latent space positions at time t-1. We present both simulated and real-world empirical examples to illustrate the utility of our model.

Authors

• Samrachana Adhikari, Carnegie Mellon University

• Brian W. Junker, Carnegie Mellon University