The spotlight of my current research are Population Algorithms (this is my own term), i.e.

What computational problems can be solved "incrementally" in scale-free systems?

Populations are essentially graphs, and anything that can be computed on these graphs is intimately related to the nature of diffusion processes on these graphs. Problems which can be computed incrementally are the ones whose solution is related to a rapidly-mixing diffusion process. Almost by definition, the solutions to such problems can be maintained under dynamic changes of their input parameters.

Perhaps a good example of a problem in this area is:

Can a population self-assemble a routing mechanism in a rapid and robust manner?

Examples of other problems are information classification and page ranking of data sets stored in a population graph, and so on.

Pleasingly, these types of questions call for techniques from pretty areas of Mathematics ...

• Metric Embeddings and Spectral Geometry
• Analytical Combinatorics (a.k.a. Harmonic Analysis), and
• Matrix Computation and Approximation

... all seen through the discrete eye of a Computer Scientist!