Reducing the dimensions of adaptive networks

Networks, such as neural connections or social interactions, encompass systems at all scales. Traditional network dynamics only consider how each node moves, whereas adaptive network dynamics consider both the dynamics of the nodes and the dynamics of each nodal connection. The resulting high dimensionality and multiple timescales of adaptive networks make them challenging to analyze.

Take a traditional network with four connections. The adaptation dynamics consider four dimensions — one dimension for each connection. Erik Martens and Christian Bick developed an analytical framework that reduces the dimensionality of an adaptive network, facilitating methodological analysis of complex network systems.

The dimension reduction is rooted in the “weight,” or coupling strength, of each connection. If the weights of each connection evolve together over time, the four dimensions of the connections can be simplified to a single dimension. If the weights naturally evolve together, the system possesses an intrinsic constraint.

Constraints can also be forced on the system at different levels. A network-level constraint imposes a constant coupling strength between each pair of nodes. A nodal-level constraint limits the coupling capacity of each node. The strengths of certain connections can also be frozen or made inactive altogether using what’s called a “subgraph constraint.”

Systematically increasing and decreasing the constraints enables researchers to identify how dynamics link to adaptivity, such as the critical point at which chaos occurs or where the system transitions from non-synchronous to synchronous behavior.

“Now is the time to develop new mathematical theories and identify homeostasis mechanisms, because then we can say, this real-world system is something that we can deal with using slow-fast dynamics because it’s effectively low-dimensional enough that more general theory can be applied,” said Bick.

Source: “Multiple timescale dynamics of network adaptation with constraints,” by Erik A. Martens and Christian Bick, Chaos (2025). The article can be accessed at https://doi.org/10.1063/5.0289706 .