Quantum Probability and Applications to Complex Networks - lectures by Nobuaki Obata


24.03 - 23.05.2008, Mondays and Wednesdays 16:15-18:00 in WS

Credit points: 6


Quantum probability theory provides a framework of extending the measure-theoretical (Kolmogorovian) probability theory. The idea traces back to von Neumann (1932), who, aiming at the mathematical foundation for the statistical questions in quantum mechanics, initiated a parallel theory by making a selfadjoint operator and a trace play the roles of a random variable and a probability measure, respectively. During the last two decades quantum probability theory has been related to various fields of mathematical sciences beyond the original purposes. We focus in these lectures on spectral analysis of large graphs (or of growing graphs, or networks) and show how the quantum probabilistic techniques are applied, especially, for the study of asymptotics of spectral distributions in terms of quantum central limit theorem.

Part 1 (April)

Fundamentals in quantum probability

I will review fundamental concepts and results in quantum probability accrding to Chapter 1 [1] with examples from graphs and networks. These lectures are prepared also for non-experts.

Keywords are: quantum probability space, quantum random variable, state, interacting Fock space, orthogonal polynomial, continued fraction, Cauchy-Stieltjes transform, concepts of independence, central limit theorem

Reference: [1] A. Hora and N. Obata: Quantum Probability and Spectral Analysis of Graphs Series: Theoretical and Mathematical Physics

Part 2 (May)

Quantum probabilistic approach to network science

I will discuss spectral properties of graphs and networks by means of quantum probability theory.

Lecture notes