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Synchronization and clustering in complex quadratic networks

In continuation of prior work, we investigate ties between a network's connectivity and ensemble dynamics. This relationship is notoriously difficult to approach mathematically in natural, complex networks. In our work, we aim to understand it in a canonical framework, using complex quadratic node dynamics, coupled in networks which we call complex quadratic networks (CQNs). After previously defining extensions of the Mandelbrot and Julia sets for networks, we currently focus on the behavior of the node-wise projections of these sets, and on defining and analyzing the phenomena of node clustering and synchronization. We investigate the mechanisms that lead to nodes exhibiting identical or different Mandelbrot set. We propose that clustering is strongly determined by the network connectivity patterns, with the geometry of these clusters further controlled by the connection weights. We then illustrate the concept of synchronization in an existing set of whole brain, tractography-based networks obtained from 197 human subjects using diffusion tensor imaging. Synchronization and clustering are well-studied in the context of networks of oscillators, such as neural networks. Understanding the similarities to how these concepts apply to CQNs contributes to our understanding of universal principles in dynamic networks, and may help extend theoretical results to natural, complex systems.