Changing the tune: mixtures of network models that vary in time.

Many of the complex systems we study in their representation as networks are growing objects, evolving by the addition of nodes and links over time. The rules governing this growth are attributed to mechanisms such as preferential attachment and triangle closure. We demonstrate a method for estimating the relative roles of these mechanisms, and further, investigating how they change as the network evolves. We show that a rich class of network evolution models can be built from a weighted mixture of these model mechanisms. Using a likelihood based formulation we show how to calculate the optimal mixture for a given set of observations of network data, and show that this framework can be used to distinguish competing models that are indistinguishable by their summary statistics. Using real data from Facebook user interactions, we show that we can improve the ability of a model to reproduce network statistics using tuned model mixtures. We further investigate the idea that the underlying model of a network can change in time, for example, a technology based network might respond to changes in the underlying technology or a financial network might respond to economic shocks. Using artificial data we show that we can recapture the time at which a known change occurred. We use the Enron email dataset to show that we can estimate how mixtures of models change over time.