TY - JOUR
T1 - Transitions from trees to cycles in adaptive flow networks
AU - Martens, Erik A.
AU - Klemm, Konstantin
N1 - Funding Information:
EM would like to thank J. C. Brings Jacobsen for helpful discussions on circulatory physiology and E. Katifori on adaptive networks. We acknowledge travel funding from Action CA15109, European Cooperation for Statistics of Network Data Science (COSTNET). Research conducted by EM is supported by the Dynamical Systems Interdisciplinary Network, University of Copenhagen. KK acknowledges funding from MINECO through the Ram?n y Cajal program and through project SPASIMM, FIS2016-80067-P (AEI/FEDER, EU).
PY - 2017/11/28
Y1 - 2017/11/28
N2 - 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.
AB - 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.
KW - Adaptive networks
KW - Cycles
KW - Flow networks
KW - Heterogeneous network structures
KW - Loops
KW - Transcritical bifurcation
KW - Transport networks
KW - Tree-like structures
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U2 - 10.3389/fphy.2017.00062
DO - 10.3389/fphy.2017.00062
M3 - Article
AN - SCOPUS:85043718620
VL - 5
JO - Frontiers in Physics
JF - Frontiers in Physics
SN - 2296-424X
IS - NOV
M1 - 62
ER -