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
Jenna
Birch (U of Liverpool)
Will Cunningham (Northeastern U)
Or Eisenberg (Northeastern U)
Alfredo
GarbuñoInigo (U of Liverpool)
Stephen
Wu (Caltech, now at ETH)


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

Research
Areas
I
am a mathematician with broad research interests. I like to branch
out, learn new areas of math and applied science, and jump on
new problems. I have written two completely independent Ph.D.
dissertations and published papers
being affiliated with departments of mathematics, physics, and
civil engineering. So, 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 and has a strong geometric
flavor. In particular, I am interested in complex networks, Markov
chain Monte Carlo algorithms, rare event estimation, Bayesian
inference, and integrable Hamiltonian systems.

 Somewhere in Hong Kong

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.
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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.
Markov
Chain Monte Carlo
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.
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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.
Rare Event Estimation
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.
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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 estimating
of small probabilities or rare events t is currently a hot topic
in applied probability and uncertainty quantification.
Bayesian
Inference
A Bayesian approach to statistics
views probability as a measure of the plausibility of a proposition
when incomplete information does not allow us to establish its truth
or falsehood with certainty. Bayesian probability theory was, in fact,
originally developed by Laplace for statistical analysis of astronomical
observations.
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Moreover, Laplace proved
the famous Bayes’ theorem in full generality, while Bayes did
it only for a special case. From mathematical point of view, Bayesian
methods are often reduced to the following challenging problems: to
sample from the posterior distribution (which is difficult to sample
from); and to estimate its normalizing constant, i.e. to calculate
a highdimensional integral that often cannot be evaluated analytically
nor numerically by straightforward quadrature. This makes MCMC algorithms
very important computational tool for Bayesian Inference. Over the
last few decades it has been shown that Bayesian methods can be used
with a great success in a variety of situations, with applications
ranging from astrophysics to the social sciences.
Integrable
Hamiltonian Systems
A Hamiltonian system is a system
of differential equations which can be written in a very “nice”
(Hamiltonian) form. Liouville integrability means that a system has
sufficiently many first integrals in involution. In this case one
can already make a strong qualitative conclusions on the trajectories’
behavior.
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Although integrable systems
are highly exceptional among all Hamiltonian systems (Poincaré
showed that a general Hamiltonian system is not integrable), they
play a fundamental role in mathematical physics, since many physical
interesting systems were found to be wellenough described by integrable
systems. Having originated in classical mechanics, the theory of integrable
systems plays a unifying role in mathematics combining algebra, geometry
and analysis.
