Designing Complex Networks

Abstract

We suggest a new perspective of research towards understanding the relations between structure and dynamics of a complex network: Can we design a network, e.g. by modifying the features of units or interactions, such that it exhibits a desired dynamics? Here we present a case study where we positively answer this question analytically for networks of spiking neural oscillators. First, we present a method of finding the set of all networks (defined by all mutual coupling strengths) that exhibit an arbitrary given periodic pattern of spikes as an invariant solution. In such a pattern all the spike times of all the neurons are exactly predefined. The method is very general as it covers networks of different types of neurons, excitatory and inhibitory couplings, interaction delays that may be heterogeneously distributed, and arbitrary network connectivities. Second, we show how to design networks if further restrictions are imposed, for instance by predefining the detailed network connectivity. We illustrate the applicability of the method by examples of Erdös-Rényi and power-law random networks. Third, the method can be used to design networks that optimize network properties. To illustrate this idea, we design networks that exhibit a predefined pattern dynamics while at the same time minimizing the networks' wiring costs.

42 pages, 12 figures

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• Stable Irregular Dynamics in Complex Neural Networks
• The simplest problem in the collective dynamics of neural networks: Is synchrony stable?
• Bounding network spectra for network design
• Inferring network connectivity from event timing patterns
• Two-cluster bifurcations in systems of globally pulse-coupled oscillators
• Reduction of Interaction Delays in Networks
• Hysteretic percolation from locally optimal individual decisions
• Smooth Exact Gradient Descent Learning in Spiking Neural Networks
• Reconfigurable Computation in Spiking Neural Networks
• Dynamics and computation in mixed networks containing neurons that accelerate towards spiking
• GINA: Neural Relational Inference From Independent Snapshots