Rich Harang

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:

1. 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?
2. 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?
3. 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?
4. How can we improve the mixing time of samplers within some subspace of permissible graphs, with either exact or approximate sampling?

I'm currently exploring applications of wavelets to time series and stochastic processes, particularly in application to biological systems which exhibit quasi-periodic behavior. In many cases, we see what looks like a clear rhythm emerge from the system we're interested in studying, but this apparent rhythm is actually not (using the narrow, mathematical sense of the word) periodic, emerging from a complicated network of underlying stochastic processes operating on timescales several orders of magnitude faster, in some cases as little as ten-thousandths of a second. Using something like Fourier analysis, with its very strong assumptions of periodicity and stationarity, yields estimators and analyses that are at best fragile and at worse profoundly and unpredictably biased. Wavelets and similar compactly-supported bases allow us to evaluate more local behavior at the cost of frequency resolution.

Ongoing questions my work addresses are:

1. Can we find stochastic analogues of metrics that are used for deterministic system, such as the phase response curve?
2. How can we use these results to classify oscillators, and infer network topologies for systems of coupled oscillators?
3. How well do stochastic diffusions model the behavior of biological systems?  Do networks of these oscillators have the same properties as networks of SCN neurons?
4. What are the limits of inference that can be performed on the unobserved states of a biological oscillator given the limited outputs we can observe?