Understanding how transport networks, such as river systems, form and evolve is crucial to optimising their stability and resilience. Some networks branch into a treelike pattern, while other show a looplike topology. Researchers from the Faculty of Physics of the University of Warsaw, and the University of Arkansas sought to answer the question what conditions favour the formation of loops.

Transport networks, like blood vessels or river systems, are essential for many natural and human-made systems. Even seemingly similar flow systems, like river deltas, may include different morphologies. The outflow from Wax Lake in Louisiana seems to branch into a treelike pattern with smaller river outlets reaching the Atlantic Ocean. The Ganges-Brahmaputra River Delta in Bangladesh, on the other hand, shows a looplike topology, with numerous channels interconnecting primary branches. What makes these two systems different is the magnitude of flow fluctuations, driven by an interplay of river discharge and tidal flows.


The question of what environmental conditions might promote the formation of loops over treelike structures inspired a collaboration of scientists from the Faculty of Physics of the University of Warsaw, and the Department of Geosciences of the University of Arkansas, the USA, to investigate the stability of looplike topologies in flow networks.


Diversity of networks

The results of the research, published in Physical Review Letters, show that networks tend to remain looplike when the flow fluctuations remain tuned in a particular way.


“Simple growth rules can often lead to fascinating patterns. Tree-like structures are effective for transport, but networks that contain loops are more resilient to damage. Understanding the necessary conditions for the emergence of loops in evolving networks is our long-term goal,” said Prof. Piotr Szymczak from the Faculty of Physics of the University of Warsaw, a co-contributing author of the study.


“River networks can look remarkably different depending on the river and the sea – geospatial data provides us with visual evidence on the changing morphologies of river deltas and with new data being collected on flow characteristics we are trying to learn more about the dynamics of their evolution, particularly in the time of rapid climate change,” underlined Prof. John Shaw from the University of Arkansas, who spent his sabbatical at the UW thanks to the Fulbright Research Award. “This publication was born from the fusion of geological observations, equations of sedimentology and mathematical methods of physics.”


Not only rivers

“Our collaboration started with the focus on rivers, but the observations generalise to a remarkably large class of transport networks,” said Radost Waszkiewicz, the leading co-author of the paper and a doctoral candidate at the UW Faculty of Physics.


The scientists discovered that the stability of loops in these networks depends on the interplay between geometric constraints and fluctuations in flow. They found that loops require fluctuations in the relative size of flow between nodes, not just temporal variations in flow at a single node and that the loops are more stable when the fluctuations are neither too small nor too large relative to the constant flow component.


“Should the fluctuation pattern change because of external factors such as human intervention or climate change, new loops inside transport networks might appear or disappear, transforming the network’s form,” concluded Prof. Maciej Lisicki, also from the Faculty of Physics, adding: “We hope that this observation will entice more precise measurements in natural systems and will take us one step further in understanding the dynamic remodelling of transport networks.”

Details of the publication

Radost Waszkiewicz, John Burnham Shaw, Maciej Lisicki, Piotr Szymczak, Goldilocks Fluctuations: Dynamic Constraints on Loop Formation in Scale-Free Transport Networks, Physical Review Letters 132, 137401 – (2024), DOI:10.1103/PhysRevLett.132.137401