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Discrete and Continuous Models in the Theory of Networks

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Universität Bielefeld/P. Ottendörfer
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Convenors

Fatihcan M. Atay (Bilkent University Ankara, TUR)

Prof. Dr. Delio Mugnolo (University of Hagen, GER)

Prof. Dr. Pavel Kurasov (Stockholm University, SWE)

Coordinator at ZiF

Mo Tschache

Discrete and Continuous Models in the Theory of Networks

October 2012 - September 2017

The idea of representing a complex interacting systems of human and/or non-human actors as a network is very old. Already Aristotle in his Politicsconsiders the society as a community of different individuals clustered in several subgroups and is aware of the founding dichotomy of networks: the individual nodes that pursue their own interests, as opposed to the health of the net as an impersonal whole. Networks and webs appear in a manifold of areas of human activities. It is therefore no surprise that they have played a role - for decades already - in many research areas in natural sciences, originally as a convenient representation form and more recently as a deeper paradigm, too. Let us mention the invention of Feynman diagrams, developed in 1948 by Nobel Prize-winning physicist Richard Feynman to allow for a compact representation of interactions between sub-atomic particles in quantum field theory, which have ever since mutated into a source of extremely efficient computational algorithms. The medium is (part of) the message, in this and in many further network-based schemes that arise in natural sciences.

Today network methods are being adopted at an ever-faster rate, to provide a convenient description of a model, or perhaps to perform computations making use of a combinatorial approach, or finally to convey information among subsystems in a more efficient way. Now they are crucial theoretical tools in fields as far away as biology (ecological networks, neuroscience), physics (lattice field theories), computer sciences (decision trees), civil engineering (traffic flows on road networks), just to name a few.

Seemingly, the sole common point to all these studies is the use of a network formalism. To introduce these systems, which describe completely different objects/actors but usually have similar qualitative properties, completely different methods are often used - at a first glance. Indeed, many applied scientists make sometimes use (often in a naive, enthusiastic, and rewarding way) of mathematical tools and notions. Far from being only an elegant human construct out of any touch with reality, modern mathematics - combinatorics and analysis in the first place - is a set of key technologies that are particularly mighty in the analysis of network-shaped systems. Not despite but rather exactly because of their abstraction, mathematical methods are applicable to a manifold of different fields. This research project will bring together researchers whose research areas benefit from, or feature approaches based on, or are directly involved with, the analysis of graphs and networks. We are going to concentrate our investigations on evolutionary networks describing systems where not only the individual nodes, but also the interactions between the nodes are subject to time evolution, possibly along with changes of the underlying network topology, with a special focus on applications in physical and natural sciences.

Our main bet will be that mathematics can (and in fact, following Galileo Galilei's opinion on the language of the universe, ought to) be the common language of this loose network of network-conscious scientists. Our aim will be to initiate interdisciplinary collaborations and mutual interactions, even with pure mathematicians who already have experience with accurate description of complicated systems.

Members

Fatihcan M. Atay

Mathematics

Bilkent University Ankara (TUR)

Prof. Dr. Pavel Kurasov

Mathematics

Stockholms Universitet (SWE)

Prof. Dr. Delio Mugnolo

Mathematics and Computer Science

Hagen University (GER)

Prof. Dr. Ginestra Bianconi

Mathematics

Queen Mary University of London (UK)

Prof. Dr. Till Becker

Production and Logistics

University of Bremen (GER)

Prof. Dr. Jonathan Breuer

Mathematics

Hebrew University of Jerusalem (ISR)

Radu Cascaval

Mathematics

University of Colorado (USA)

J.-Prof. Mareike Fischer

Biomathematics

University of Greifswald (GER)

Júlia Gallinaro

Computational Neuroscience

Bernstein Center Freiburg (GER)

Dr. Jiao Gu

Computer Science

Leipzig University (GER)

Bobo Hua

Mathematics

Leipzig University

Stojan Jovanovic

Mathematics

University of Belgrade (SER)

Christopher Kaiser-Bunbury

Biosciences

University of Exeter (UK)

Kosmas Kosmidis

Computational Physics

University of Thessaloniki (GR)

Annick Lesne

Theoretical Physics

Sorbonne University (FRA)

Prof. Dr. Wenlian Lu

Mathematics

Fudan University (CHI)

Gabriela Malenova

Mathematics

Ulm University (GER)

Benjamin Mauroy

Mathematics/Biology

Nice University (FRA)

Fumito Mori

Chemics

Max Planck Society (GER)

Gabor Pete

Mathematics

Budapest University of Technology and Economics (HUN)

Mats-Erik Pistol

Physics/Mathematics

Lund University (SWE)

Patricia Alonso Ruiz

Mathematics

Ulm University (GER)

Sadra Sadeh

Computational Neuoscience

Bernstein Center Freiburg (GER)

Ruben Sanchez-Garcia

Mathematics

University of Southampton (UK)

Jonathan Schiefer

Mathematics/ Computational Neuroscience

Bernstein Center Freiburg (GER)

Liu Shiping

Mathematics

Max Planck Institute Leipzig (GER)

Leszek Sirko

Physics

Polish Academy of Sciences (POL)

Dimitri Volchenkov

Theoretical Physics

Bielefeld University, Center of Excellence Cognitive Interaction Technology (GER)

Verena Wolf

Mathematics

Saarland University (GER)

Publications

Discrete and Continuous Models in the Theory of Networks

Fatihcan M. Atay, Pavel B. Kurasov, Delio Mugnolo (eds.) (2020)

Basel: Birkhäuser 2020 (Operator Theory: Advances and Applications 281)

Quantum graphs: PT-symmetry and reflection symmetry of the spectrum

P Kurasov, B Majidzadeh Garjani (2017)

J. Math. Physics

Edge connectivity and the spectral gap of combinatorial and quantum graphs

G Berkolaiko, JB Kennedy, P Kurasov, D Mugnolo (2017)

Surgery of graphs: M-function and spectral gap

P Kurasov (2017)

Acta Physica Polonica A 2017

On the spectral gap of a quantum graph

JB Kennedy, P Kurasov, G Malenová, D Mugnolo (2016)

Ann. H. Poincaré

Local pinning of networks of multi-agent systems with transmission and pinning delays

W Lu, FM Atay (2016)

IEEE Transactions on Automatic Control

Schrödinger operators on graphs: symmetrization and Eulerian cycles

G Karreskog, P Kurasov, I Trygg Kupersmidt (2016)

Proc. AMS

A delayed consensus algorithm in networks of anticipatory agents

FM Atay, D Irofti (2016)

European Control Conference (ECC)

On the symmetry of the Laplacian spectra of signed graphs

FM Atay, B Hua (2016)

Linear Alg. Appl.

On the delay margin for consensus in directed networks of anticipatory agents.

D Irofti, FM Atay (2016)

IFAC-PapersOnLine

On Equi-transmitting Matrices

P Kurasov and R Ogik (2016)

Reports on Math. Phys.

Time regularity and long-time behavior of parabolic p-Laplace equations on infinite graphs

B Hua, D Mugnolo (2015)

J. Diff. Eq.

Mathematical Technology of Networks

D Mugnolo (2015)

Cham: Springer International Publishing

On vertex conditions for elastic systems

JC Kiik, P Kurasov, M Usman (2015)

Phys. Lett. A

RT-Symmetric Laplace Operators on Star Graphs: Real Spectrum and Self-Adjointness

M Astudillo, P Kurasov, and M Usman (2015)

Adv. Mathematical Physics

Topological damping of Aharonov-Bohm effect: quantum graphs and vertex conditions

P Kurasov, A Serio (2015)

Nanosystems: Phys., Chem., Math.

Spectral gap for complete graphs: upper and lower estimates

P Kurasov (2015)

in: D Mugnolo (Ed.) Mathematical technology of networks, Springer, 2015.

Laplacians on quantum hypergraphs

D Mugnolo (2014)

PAMM, 2014

On the spectrum of the normalized Laplacian for signed graphs: Interlacing, contraction, and replication.

FM Atay, H. Tuncel (2014)

Linear Algebra and its Applications 442:165?177, 2014

On reflectionless equi-transmitting matrices

P Kurasov, R Ogik and A Rauf (2014)

Opuscula Math., 2014

Rayleigh estimates for differential operators on graphs

P Kurasov, S Naboko (2014)

J. Spectral Theory

Spectral gap for quantum graphs and their edge connectivity

P Kurasov, G Malenová, S Naboko (2013)

Inverse scattering for lasso graph

P Kurasov (2013)

J. Math. Phys.

On the Spectral Gap for Laplacians on Metric Graphs

P Kurasov (2013)

Acta Physica Polonica A, 2013

Inverse problems for quantum graphs: recent developments and perspectives.

P Kurasov (2012)

Acta Physica Polonica A, 2012

Schrödinger Operators on Graphs and Geometry II. Integrable Potentials and an Ambartsumian Theorem

J Boman, P Kurasov, and R Suhr (2020)

preprint

On the sharpness of spectral estimate for graph Laplacians

P Kurasov and A Serio (2020)

preprint

Schrödinger operators on graphs and geometry III. Non-standard conditions and a geometric version of the Ambartsumian theorem

P Kurasov and R Suhr (2020)

preprint

Surgery of graphs and spectral gap: Titchmarsh-Weyl operator-function approach

P Kurasov and S Naboko (2020)

preprint

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