A two-way heterogeneity model for dynamic networks

Abstract

Analysis of networks that evolve dynamically requires the joint modelling of individual snapshots and time dynamics. In spite of the existence of a large body of literature on dynamic networks, network models that capture both the network dynamics and the degree heterogeneity are still underdeveloped. Furthermore, a more interesting yet much more challenging question is to address the node heterogeneity in the network dynamics when there are very few networks. This thesis proposes a new flexible two-way heterogeneity model which directly depicts the dynamic change of the edges and degree sequences over time for dynamic network processes. The new model equips each node of the network with two heterogeneity parameters, one to characterize the propensity to form ties with other nodes statically and the other to differentiate the tendency to retain existing ties over time. With n observed networks each having p nodes, we develop a new asymptotic theory for the maximum likelihood estimation of 2p parameters when np → ∞. Instead of assuming the number of observed networks n → ∞, we mainly investigate the behaviour of the maximum likelihood estimator (MLE) under the condition n ≥ 2 and np → ∞, where p is the number of nodes, i.e. we allow n to be finite. We overcome the global non-convexity of the negative log-likelihood function by the virtue of its local convexity, and propose a novel method of moment estimator as the initial value for a simple algorithm that leads to the consistent local MLE. To establish the upper bounds for the estimation error of the MLE, we derive a new uniform deviation bound, which is of independent interest. The theory of the model and its usefulness are further supported by extensive simulation and a data analysis examining social interactions of ants.