Discrete and Continuous Models in the Theory of Networks
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.
Publications
- All 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)
Dynamical systems associated with adjacency matrices
D Mugnolo (2017)
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)
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 delaysW Lu, FM Atay (2016)
IEEE Transactions on Automatic Control
Schrödinger operators on graphs: symmetrization and Eulerian cyclesG Karreskog, P Kurasov, I Trygg Kupersmidt (2016)
Proc. AMS
A delayed consensus algorithm in networks of anticipatory agentsFM Atay, D Irofti (2016)
European Control Conference (ECC)
On the symmetry of the Laplacian spectra of signed graphsFM 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 MatricesP 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 NetworksD Mugnolo (2015)
Cham: Springer International Publishing
On vertex conditions for elastic systemsJC Kiik, P Kurasov, M Usman (2015)
Phys. Lett. A
RT-Symmetric Laplace Operators on Star Graphs: Real Spectrum and Self-AdjointnessM Astudillo, P Kurasov, and M Usman (2015)
Adv. Mathematical Physics
Topological damping of Aharonov-Bohm effect: quantum graphs and vertex conditionsP Kurasov, A Serio (2015)
Nanosystems: Phys., Chem., Math.
Spectral gap for complete graphs: upper and lower estimatesP 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 matricesP Kurasov, R Ogik and A Rauf (2014)
Opuscula Math., 2014
Rayleigh estimates for differential operators on graphsP Kurasov, S Naboko (2014)
J. Spectral Theory
- Preprints
Schrödinger Operators on Graphs and Geometry II. Integrable Potentials and an Ambartsumian Theorem
J Boman, P Kurasov, and R Suhr (2020)
preprint