Prospective
Students
If you
are interested in working with me in one of the research areas described
on the right, or have your own project in mind that you would like
to discuss, please send me an email to kostia [at] caltech [dot]
edu.


Students
Coauthors
Erdös Number
My Erdös
number is (at most) 4:
KZ→Bianconi→Severini→Cameron→Erdös
KZ→Beer→Kreinovich→Blass→Erdös
Citations
Google
Scholar:

Research
Areas
My research is
interdisciplinary. I like to branch out, learn new areas of
math, engineering, and science, and jump on new problems.
I have written
two independent Ph.D. dissertations and
published papers being affiliated
with departments of civil
engineering, mathematics, physics,
and computer science. According
to Dyson's "birdsandfrogs"
classification, I am a frog which occasionally makes very long
jumps. Most of my research falls under the umbrella of applied
probability & statistics with applicatins to engineering
problems and has a strong geometric flavor.

 Somewhere in Hong Kong

My research spans topics in three interdisciplinary
areas of Engineering, Applied Mathematics, and Statistics.
Reliability
Engineering: Rare Event Simulation, Estimation of Small Failure
Probabilities, Structural Reliability, Risk Analysis, Reliability
and Resilience of Critical Infrastructure Networks, Cascading Failures.
>
continue reading about rare
events
An
event is rare if its occurrence is unlikely. Getting ten heads in
ten flips of a fair coin, for example, is a rare event. In probabilistic
terms, this means that the event occupies a tiny portion of the
underlying sample space. If the occurrence of a rare event does
not cause substantial consequences (say, you lose a dollar in the
coin example), the event can be safely ignored, since it is both
unimportant and statistically negligible. In many applications,
however, there are events which, although rare, may lead to catastrophes.
For example, a building collapse,
a stock market crash, or you lose a million dollars in the coin
example. In these cases, accurate estimation of the small probabilities
of such extreme events is very important. The probability of a rare
event can often be written as an integral over a highdimensional
parameter space with an expensivetocompute integrand. This makes
its estimation quite challenging, since neither numerical integration
nor the Monte Carlo method would work. As a result, one needs to
develop more advanced stochastic simulation methods, based, for
example, on Markov chain Monte Carlo algorithms. Accurate estimation
of small probabilities or rare events is one of the most fundamental
and challenging problems in reliability engineering and uncertainty
quantification.
Network Science: Complex Network Models, Hyperbolic Geometry
of Network Data, Network Dynamics, Exponential Random Graphs, Simplicial
Complexes, Multilayer Networks, Infrastructure Networks.
>
continue reading about complex
networks
Networks are everywhere. Examples
include networks of roads, railways, airline routes, electric power
grids, the Internet, the World Wide Web, networks of friendship
between individuals, business relations between companies, citations
between papers, protein interaction networks, food webs, etc. Networks
are different, but many have some structural properties in common.
One of the fundamental problems in the study of complex networks
is to identify evolution mechanisms that shape the structure and
dynamics of large real networks.
The study of networks goes back
to Euler and his famous solution of the Königsberg bridge problem
which is often regarded as the starting point of graph theory, the
first mathematical study of networks. Recent study of networks has
been driven mostly by observations of the properties of realworld
networks. As more data became available, it has been recognized
that the topology and evolution of real networks are not completely
random, instead, they are governed by robust organizing principles:
the smallworld effect, transitivity, powerlaw degree distribution,
etc. Thanks to their giant size (millions or even billions of vertices),
real networks cannot be analyzed by just using combinatorial tools
from graph theory; rather, modern statistical methods should be
used. Thus, the modern theory of complex networks has emerged. This
is a new field of research that studies the statistical properties
of large networks, develops their mathematical models, and aims
to predict the behavior of networks based on their internal structure.
I am especially interested in infrastructure networks.
Computational
Statistics: Markov Chain Monte Carlo, Subset Simulation,
Importance Sampling, Simulated Annealing, Bootstrap, Permutation Tests,
Stochastic Geometry and Optimization, Bayesian Inference.
>
continue reading about MCMC
Markov chain Monte Carlo (MCMC)
is a class of algorithms for sampling from complex probability distributions.
These methods are based on constructing a Markov chain whose stationary
distribution is the target distribution of interest. The most common
application of MCMC is numerical calculation of multidimensional
integrals, which is known to be a very important, yet difficult
problem in applied mathematics.
MCMC originated in works of N. Metropolis,
who used it for sampling from the Boltzmann distribution (and implemented
it on the first MANIAC!),
and W.K. Hastings, who generalized the method to what now is known
as the famous MetropolisHastings algorithm. Since then great progress
has been made in both theory and applications of MCMC algorithms,
and recent years have witnessed the socalled MCMC
revolution in applied mathematics. Moreover, there is some
evidence that solving any problem is a “matter of cooking
up an appropriate Markov chain.” Having applications in physics,
biology, scientific computing, and in other fields, MCMC is considered
to be one of the top ten most important algorithms ever.
