Network Dynamics

Many real-world systems can be seen as networks of dynamical nodes. Examples range from power grid networks to synchronously firing neural oscillators in the brain. The dynamics – and therefore the function – of such networks are determined by the individual dynamics of each node as well as the interaction between nodes.

Crucially, in many cases the interactions between two nodes are not static: They may be influenced by the state of other nodes or change over time, e.g., via external forcing, intrinsic adaptation such plastic connections. In this case the network is adaptive or co-evolutionary. That adaptive effects are crucial for the network dynamics has been demonstrated, for example, in ecological networks and networks of neuronal oscillators. Despite this, there are only realtively limited mathematical insights into the dynamics of networks with state-dependent and adaptive interactions: Many commonly used models either assume static connections, or they assume static node dynamics. Even if there is co-evolutionary dynamics on and of the network, direct numerical simulation or formal moment approximations are some of the predominant approaches. Hence, there is still a very serious challenge to understand the mathematical fundamentals of adaptive networks. One key goal of the Focus Group "Network Dynamics" is to push the boundaries on the mathematical understanding on network dynamical systems with state-dependent and adaptive interactions.

Furthermore, robust function of networks may be thought of as an emergent dynamic property of the network resulting from the intrinsic dynamics within nodes, the structure and dynamics of connections between nodes, and the interplay of the network with the external environment. Consequently important transient and excitable behaviour is widespread in such networks, e.g., in biomedical applications. Given perturbations different transient and emergent dynamics arise in networks. The significance of such transient effects has been demonstrated in avalanches of gene activation in gene regulatory pathways to drive cell differentiation, development and cancer as well as cell fate in biofilm formation, just to name a few. However, our mathematical understanding of transient network dynamics is in its infancy in spite of some recents efforts. Accordingly, the Focus Group "Network Dynamics"– consisting of Prof. Dr. Krasimira Tsaneva-Atanasova (Hans Fischer Senior Fellow)Dr. Christian Bick (Hans Fischer Fellow) and Prof. Dr. Christian Kühn (TUM-Professor for Multiscale and Stochastic Dynamics) – aims at developing the mathematical understanding on transient emergent network dynamics.

TUM-IAS funded doctoral candidates:

  • Tobias Böhle
  • Maria Elena Gonzalez

Selected Publications by the Focus Group

Salman, M., Bick, C., & Krischer, K. (2020). Collective oscillations of globally coupled bistable, nonresonant components. Physical Review Research, 2(4), 043125. doi:10.1103/PhysRevResearch.2.043125

"On the reliable and efficient numerical integration of the Kuramoto model and related dynamical systems on graphs"
T. BöhleC. Kuehn, and M. Thalhammer

"Coupled hypergraph maps and chaotic cluster synchronization"
T. BöhleC. Kuehn, R. Mulas and J. Jost

"Multi-population phase oscillator networks with higher-order interactions"
C. BickT. Böhle and C. Kuehn

"Reduced models of cardiomyocytes excitability: comparing Karma and FitzHugh-Nagumo"
M.E. Gonzalez HerreroC. Kuehn and K. Tsaneva-Atanasova

"A geometric analysis of the SIRS epidemiological model on a homogeneous network"
H. Jardon Kojakhmetov,C. KuehnA. Pugliese and M. Sensi

"Balancing quarantine and self-distancing measures in adaptive epidemic networks"
L. Horstmeyer,C. Kuehn and S. Thurner

"Graphop mean-field limits for Kuramoto-type models"
M.-A. Gkogkas and C. Kuehn

"A universal route to explosive phenomena"
C. Kuehn and C. Bick
Science Advances, accepted / to appear

"Coupled dynamics on hypergraphs: master stability of steady states and synchronization" 
R. MulasC. Kuehn and J. Jost
Physical Review E, Vol. 101, No. 6, 062313, 2020

"On fast-slow consensus networks with a dynamic weight" 
H. Jardon Kojakhmetov and C. Kuehn
Journal of Nonlinear Science, Vol. 30, pp. 2737-2786, 2020

"Network dynamics on graphops"
C. Kuehn
New Journal of Physics, Vol. 22, 053030, 2020

"Adaptive voter model on simplicial complexes"
L. Horstmeyer and C. Kuehn
Physical Review E, Vol. 101, No.2, 022305, 2020

"Mathematical analysis of nonlocal  PDEs for network generation"
T. Böhle and C. Kuehn
Mathematical Modelling of Natural Phenomena, Vol. 14, No. 5, 506, 2019

"Power network dynamics on graphons"
C. Kuehn and S. Throm
SIAM Journal on Applied Mathematics, Vol. 79, No. 4, pp. 1271-1292, 2019

"Multiscale dynamics of an adaptive catalytic network"
C. Kuehn
Mathematical Modelling of Natural Phenomena, Vol. 14, No. 4, 402, 2019