Graph Theory Cambridge

[Mellissa Sprock]

Thereare really no prerequisites for this course. Occasionally, it would be useful to have met some terminology from graph theory. Any course in graph theory will contain such terminology, and if you have not done any graph theory course then that is also fine, as the terminology will be minimal and very easy to pick up.

There are really no prerequisites for this course. If you have already met Ramsey's theorem then that is a help, but the course is designed with the assumption that the audience have not met Ramsey's theorem at all.

This is a second course in Graph Theory. Pretty much any first course in Graph Theory will be sufficient, as long as it has some theorems in it and is not just a catalogue of definitions. For example, any course that contains Turan's theorem will be fine. But if you have not done any Graph Theory course at all then you might find this course is not followable.

The Cambridge Graph Theory Conference, held at Trinity College from 11 to 13 March 1981, brought together top ranking workers from diverse areas of the subject. The papers presented were by invitation only. This volume contains most of the contniutions, suitably refereed and revised.

For many years now, graph theory has been developing at a great pace and in many directions. In order to emphasize the variety of questions and to preserve the freshness of research, the theme of the meeting was not restricted. Consequently, the papers in this volume deal with many aspects of graph theory, including colouring, connectivity, cycles, Ramsey theory, random graphs, flows, simplicial decompositions and directed graphs. A number of other papers are concerned with related areas, including hypergraphs, designs, algorithms, games and social models.This wealth of topics should enhance the attractiveness of the volume.

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There is increasing interest in the potential of whole-brain computational models to provide mechanistic insights into resting-state brain networks. It is therefore important to determine the degree to which computational models reproduce the topological features of empirical functional brain networks. We used empirical connectivity data derived from diffusion spectrum and resting-state functional magnetic resonance imaging data from healthy individuals. Empirical and simulated functional networks, constrained by structural connectivity, were defined based on 66 brain anatomical regions (nodes). Simulated functional data were generated using the Kuramoto model in which each anatomical region acts as a phase oscillator. Network topology was studied using graph theory in the empirical and simulated data. The difference (relative error) between graph theory measures derived from empirical and simulated data was then estimated. We found that simulated data can be used with confidence to model graph measures of global network organization at different dynamic states and highlight the sensitive dependence of the solutions obtained in simulated data on the specified connection densities. This study provides a method for the quantitative evaluation and external validation of graph theory metrics derived from simulated data that can be used to inform future study designs.

I am interested in understanding human brain network organization from neuroimaging data in health and disease. My recent methodological work has focused on graph theory to measure aspects of brain network topology. I am also interested in better neuroscientific understanding and treatment of psychiatric disorders. I am currently leading a consortium funded by the Wellcome Trust and pharmaceutical companies (GSK, Janssen, Lundbeck) - the Neuroimmunology of Mood Disorders and Alzheimer's Disease (NIMA) consortium - which is exploring immune mechanisms and therapeutics for depression and dementia.

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A week after the start of Lent Term 2022 at Cambridge (also known as Hilary Term to Oxon), I jumped onto a commuter train from Cambridge to London King's Cross. Sitting at the hard seat on a typical commuter train, I hardly get to have a nap, though the view outside is charming, highlighting the classic rural field within Cambridgeshire and later Hertfordshire. Somehow I started to appreciate recurrently how exciting it is to teach students from a different university on statistical matters that are cutting edge and in active research. The deep mathematical contents being earned as a second year PhD student at Cambridge Mathematics Faculty enables me to appreciate the rewarding nature of many of the Machine Learning (ML) techniques in a theoretical level, yet it is always a challenge transferring the knowledge to students from different academic and industrial backgrounds. From Cambridge to London, I need to resemble my equations, figures, and codes, drawing a tale between these two universities. From London to Cambridge, I remind myself how exciting these teachings are, the exchange of intellectuals with those students, collecting these tales into a blog that portraits me the enjoyable days in London, skipping Cambridge lectures to teach my fellow students at LSE.

The course is titled as ST456 Deep Learning offered by the Statistics Department at LSE, with students from a variety of Master programmes taking this course. In the first week, we had a helicopter view on the course and codes for implementation, with the second week getting to the meat: Architecture of Neural Networks.

Feedforward Neural Networks (NN) are introduced in lectures and practiced in classes for codings. NN builds the fundamentals of modern ML, especially Deep Learning, and understanding the basic concepts such as layers and activation functions are crucial. A historically motivating example is the XoR problem, which gets covered in greater details both mathematically and in codings. Later on, we also tried a few functions in tf.Sequential groups in the TensorFlow package to run experiments as a starting taste of the state of the art: in the next few weeks, more to come on those learning algorithms, followed by more complicated yet practical applications to images (CNN) and time-based data (RNN) such as text analytics and time series.

As a side note raised when answering question from the students: it is common to introduce NN to students using neurons and / or vertices. Though, I found it interesting that the notion of graph theory is being exploited here --- the idea of (V, E) denoting vertices and edges. In Cambridge, many pure mathematicians embark on combinatorics and graph theory, cooking up results that are profound and applicable, for some, to ML. When a student asked me about what's the difference between neurons and vertices, I highlighted this remark of linguistic difference between ML practitioners and mathematicians.

In the next two weeks, I have designed more practical NN approximation problems students need to work on --- related to option pricing --- surprise surprise. This will engage with modern learning algorithms as well as interesting mathematical questions raised from option pricing theory. I will also update this blog on a weekly basis as I sit on the returning train from London to Cambridge.

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We propose a graph-theoretical formalism to study generic circuit quantum electrodynamics systems consisting of a two level qubit coupled with a single-mode resonator in arbitrary coupling strength regimes beyond rotating-wave approximation. We define colored-weighted graphs, and introduce different products between them to investigate the dynamics of superconducting qubits in transverse, longitudinal, and bidirectional coupling schemes. The intuitive and predictive picture provided by this method, and the simplicity of the mathematical construction, are demonstrated with some numerical studies of the multiphoton resonance processes and quantum interference phenomena for the superconducting qubit systems driven by intense ac fields.

Dressed atom formalism was developed in 1969 by Cohen-Tannoudji and Haroche1 to explain the behavior of atoms exposed to radio-frequency fields described in terms of photons2. In fact, the Floquet quasienergy diagram is equivalent to the fully quantized dressed-atom picture in the limit of strong fields3. Generalization of the Floquet theory for non-perturbative treatment of infinite-level systems, including both bound and continuum states, was first introduced by Chu and Reinhardt4 in 1977. Dressed superconducting qubits5,6, have been theoretically studied7, and experimentally demonstrated8,9. For strongly driven superconducting qubits, the Floquet formalism can describe the ac Stark level shift and power broadening for multiphoton resonance processes, which appear beyond rotating-wave approximation (RWA)10. Also, the RWA is not applicable in the ultrastrong-coupling (USC) regime11,12. This new regime of cavity quantum electrodynamics (QED), where the coupling rate becomes an appreciable fraction of the unperturbed frequency of the bare systems, has been experimentally reached in a variety of solid state systems13,14,15,16. In RWA the excitation-number-nonconserving processes or virtual transitions have been excluded in calculations. Therefore the Jaynes-Cummings model cannot describe higher order resonant transitions via intermediate states connected by counter-rotating terms17,18.

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