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Pattern formation and control in networks of bistable elements

Nikos E. Kouvaris, PhD:

 

Traveling fronts and stationary localized patterns in bistable reaction-di usion systems have been broadly studied for classical continuous media and regular lattices. Analogs of such non-equilibrium patterns are also possible in networks. Here, we consider traveling and stationary patterns in bistable one-component systems on random Erdös-Renyi, scale-free and hierarchical tree networks. As revealed through numerical simulations, traveling fronts exist in network-organized systems. They represent waves of transition from one stable state into another, spreading over the entire network. The fronts can furthermore be pinned, thus forming stationary structures. While pinning of fronts has previously been considered for chains of di ffusively coupled bistable elements, the network architecture brings about signi cant di erences. An important role is played by the degree (the number of connections) of a node. For regular trees with a xed branching factor, the pinning conditions are analytically determined [2]. Furthermore, effects of feedbacks on self-organization phenomena in bistable networks are investigated [2]. For regular trees, an approximate analytical theory for localized stationary patterns under application of global feedbacks is constructed. Using it, properties of such patterns in diff erent parts of the parameter space are discussed. Numerical investigations are performed for large random Erdös-Renyi and scale-free networks. In both kinds of systems, localized stationary activation patterns have been observed. The active nodes in such a pattern form a subnetwork, whose size decreases as the feedback intensity is increased. For strong feedbacks, active subnetworks are organized as trees. Additionally, local feedbacks affecting only the nodes with high degrees (i.e. hubs) or the periphery nodes are considered.

 

[1] N. E. Kouvaris, H. Kori and A. S. Mikhailov, Traveling and pinned fronts in bistable reaction-di ffusion systems on networks, PLoS ONE 7(9): e45029 (2012).

[2] N. E. Kouvaris and A. S. Mikhailov, Feedback-induced stationary localized patterns in networks of diff usively coupled bistable elements, Europhysics Letters 103, 16003 (2013).

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