Fluctuations and Redundancy in Optimal Transport Networks

The structure of networks that provide optimal transport properties has been investigated in a variety of contexts. While many different formulations of this problem have been considered, it is recurrently found that optimal networks are trees. It is shown here that this result is contingent on the assumption of a stationary flow through the network. When time variations or fluctuations are allowed for, a different class of optimal structures is found, which share the hierarchical organization of trees yet contain loops. The transitions between different network topologies as the parameters of the problem vary are examined. These results may have strong implications for the structure and formation of natural networks, as is illustrated by the example of leaf venation networks.

Figure 1

Network structures obtained with fluctuating sources for different values of the parameter $\gamma$ (system size $n=16$, fluctuation amplitude $\sigma =1$). In each network, the sink is at the lower left corner. The width of each edge is proportional to the square root of its conductance (corresponding to the diameter of electrical wires or porous pipes). (a) Treelike network ($\gamma =0.25$). (b) Hierarchical network with loops ($\gamma =0.75$). (c) Network with many loops and no hierarchical organization ($\gamma =1.25$; notice that the network is not perfectly uniform, conductances being higher near the sink).

Figure 2

Evolution of network structure vs $\gamma$ with constant (crosses) and fluctuating (lines) sources ($n=8$, $\sigma =1$, averages over 100 realizations). (a) Loop density $\widehat{L}=L/L_{\max }$. For $\gamma >1$, all conductances are nonzero and $L=(n-1{)}^2$. (b) Normalized redundancy $\widehat{R}=R/R_{\infty }$, where $R_{\infty }$ is the limit of $R$ as $\gamma \to +\infty$.

Figure 3

Normalized redundancy $\widehat{R}=R/R_{\infty }$ vs $\gamma$ for different fluctuation amplitudes ($n=8$).

Figure 4

Dependence of network structure on system size ($\sigma =1$). (a) Loop density $\widehat{L}=L/L_{\max }$. (b) Normalized redundancy $\widehat{R}=R/R_{\infty }$.