Transitions from trees to cycles in adaptive flow networks

TL;DR

This study models adaptive flow networks to understand how load fluctuations induce the emergence of cyclic and tree-like structures, revealing a bifurcation that partitions the network based on fluctuation amplitude.

Contribution

It introduces a dynamical systems approach to analyze how load fluctuations lead to the formation of cyclic and tree-like structures in adaptive networks.

Findings

Cyclic structures emerge via bifurcation as load fluctuations increase.

Networks partition into cyclic and tree-like regions based on fluctuation amplitude.

The model explains the structural variation in natural transport networks.

Abstract

Transport networks are crucial to the functioning of natural and technological systems. Nature features transport networks that are adaptive over a vast range of parameters, thus providing an impressive level of robustness in supply. Theoretical and experimental studies have found that real-world transport networks exhibit both tree-like motifs and cycles. When the network is subject to load fluctuations, the presence of cyclic motifs may help to reduce flow fluctuations and, thus, render supply in the network more robust. While previous studies considered network topology via optimization principles, here, we take a dynamical systems approach and study a simple model of a flow network with dynamically adapting weights (conductances). We assume a spatially non-uniform distribution of rapidly fluctuating loads in the sinks and investigate what network configurations are dynamically stable. The network converges to a spatially non-uniform stable configuration composed of both cyclic and tree-like structures. Cyclic structures emerge locally in a transcritical bifurcation as the amplitude of the load fluctuations is increased. The resulting adaptive dynamics thus partitions the network into two distinct regions with cyclic and tree-like structures. The location of the boundary between these two regions is determined by the amplitude of the fluctuations. These findings may explain why natural transport networks display cyclic structures in the micro-vascular regions near terminal nodes, but tree-like features in the regions with larger veins.