Impact of Complex Network Topology on Synchronization Dynamics
Der Einfluß komplexer Netzwerktopologie auf die Synchronisationsdynamik
by Carsten Grabow
Date of Examination:2012-01-27
Date of issue:2012-12-20
Advisor:Prof. Dr. Marc Timme
Referee:Prof. Dr. Annette Zippelius
Referee:Prof. Dr. Stephan Herminghaus
Persistent Address:
http://dx.doi.org/10.53846/goediss-2596
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Abstract
English
Synchrony is one of the most common dynamical states emerging on networks. The speed of convergence towards synchrony provides a fundamental collective time scale for synchronizing systems. Here we study the asymptotic synchronization times for directed networks with topologies ranging from completely ordered, grid-like, to completely disordered, random topoplogies - including intermediate, partially disordered, the so-called small-world, topologies. These small-world topologies simultaneously exhibit large local clustering as well as short average path length and have therefore been suggested to support network synchronization. We investigate the asymptotic speed of synchronization of coupled oscillators in dependence on the degree of randomness of their interaction topology in generalized Watts-Strogatz ensembles. We find that networks with fixed in-degree synchronize faster the more random they are, with small worlds just appearing as an intermediate case. For any generic network ensemble, if synchronization speed is at all extremal at intermediate randomness, it is slowest in the small-world regime. This phenomenon occurs for various types of oscillators, intrinsic dynamics and coupling schemes. Randomly rewiring real-world neural, social and transport networks confirms this picture. We extend the approach of master stability functions to quantify synchronization times. Synchronization dynamics on small-world networks emerge in a broad range of systems with their spectra characterizing fundamental asymptotic features. We derive analytic mean-field predictions for the spectra of small-world models. These theoretical predictions agree well with the actual spectra (obtained by numerical diagonalization) for undirected and directed networks and from fully regular to strongly random topologies. These results may provide analytical insights to empirically found features of dynamics on small-world networks from various research fields, including biology, physics, engineering, and social science. Another network architecture, the scale-free topology, is characterized by a heavy-tailed distribution of degree per node with no characteristic scale. For neural circuits this means, although most nerve cells (neurons) display local connectivity, a small number of hub neurons, characterized by long-range connections linking large numbers of cells, can confer synchronicity on the network. Thus, the presence of hub neurons, which act as super-connected nodes, has been postulated as a substrate for widespread neural synchronization. Intriguingly, by stimulating single hub neurons one may remove the synchronous collective oscillations completely. When the stimulation is switched off, the synchronicity , but if a non-hub unit driven in the same way, the oscillation stays almost identically as in the undriven state. Such synchronous oscillations constitute one of the most dominant collective dynamics of complex networks. They occur not only in circuits of neurons, but in a large range of systems, ranging from metabolic and gene regulatory networks within cells to food webs of crossfeeding species or even to oscillations in the global climate system. Thus, understanding the functional role of hubs – not only in neuronal circuits – is a task of paramount importance and has recently attracted widespread attention. However, the mechanisms underlying the suppression of global oscillations in a neurobiological system have not yet been understood. We set up a detailed theoretical framework and numerically investigate different neuronal network models to reveal potential mechanisms that underlie the experimentally discovered phenomena.
Keywords: Synchronization. complex networks; gekoppelte Oszillatoren
Other Languages
Synchronisation ist einer der bekanntesten
dynamischen Zust
Schlagwörter: Synchronisation; komplexe Netzwerke; gekoppelte Oszillatoren