Random walks and extreme events on complex networks :
Random walk models in physics are more than a century old. It has many applications ranging from diffusion phenomenon to stock market dynamics to even algorithms for search engines on the world wide web. The standard random walk is defined on a one-dimensional lattice of nodes. However, if we replace this one-dimensional lattice with a complex network (of nodes connected by edges) many interesting phenomena can be studied. Extreme events is one such phenomena when a large number of random walkers meet purely by chance at a particular node. We study the probabilities for the occurrence of such events and how such extreme events can disable a network. These models can ultimately lead to a better understanding of transport processes on networks, such as for instance air/rail traffic network and TCP/IP packets on IP networks.
Our first paper on this
Extreme events on complex networks provides an introduction.
Many natural and socio-economic processes, for example, earth quakes, temperature records,
stock market dynamics display long memory such that the correlations that do not decay
in finite time. Many properties of
such systems would differ from those that have short memory, e.g. , typical ensembles one
studies in equilibrium statistical physics. We are interested in various questions
related to the long range correlated systems. They are also somewhat difficult to
handle analytically. If suppose an extreme event takes place now (say, something like
an large earthquake), when is it more likely for the next such event to take place ?
What is the distribution of extreme events ?
These are our current interests in the context of long range
correlated systems. Since long range correlations appear in various garbs
in a variety of fields, these have many applications across many areas of sciences.
This involves two rather big ideas in physics ; 1/f noise and random matrix theory (RMT). As the cliched saying goes (again!), 1/f noise is ubiquitous in nature. RMT had its origins in spectroscopy of nuclear physics but later came to be extensively applied in condensed matter systems and quantum chaos. Is there 1/f noise in random matrix spectra ? It turns out that the answer is yes; provided we redefine the spectra to make it analogous a time series. We are working on the implications of this for real-life time series that display 1/f noise. Can we learn something more about such time series from RMT results ? What are its possible generalisations ? These are some of our current pursuits.
If interested, a somewhat dated listing of papers on
1/f noise and quantum chaos