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Real data from networks often consists of severely limited information, both in terms of node observations and edge observations, often across several loosely-related networks. Many common estimation techniques produce point estimates that have loose (or no) bounds on the errors of those estimates, and often have difficulty either dealing with missing information, or including 'extra' information to improve inference. Introducing a time-dependent element to these problems further complicates matters, particularly when the networks are large enough that the sampling time is on the same order as the time to change in a network. Bayesian methods allow for very general network models to be constructed, and flexible and robust estimates to be obtained in many cases, but the space of possible graphs is often too large to ensure adequate sampling through most MCMC methods.
Ongoing questions of interest to me include:
- How can we use time-series information to produce meaningful estimates for the topology of a graph? How can we generate meaningful error bounds on those estimates, and what form should those error bounds take?
- What sort of network statistics can we collect from a dynamic network, and how can we either estimate or correct for the bias that is introduced by the time dependency?
- Can we generalize the notions of smoothness that exist for functions in normal topologies to the topology of a graph, and use this notion to guide our inference about edges in the graph?
- How can we improve the mixing time of samplers within some subspace of permissible graphs, with either exact or approximate sampling?
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