Correlations between communicability sequence entropy and transport performance in spatially embedded networks

We investigate electric current transport performances in spatially embedded networks with total cost restriction introduced by Li et al. [Phys. Rev. Lett. 104, 018701 (2010)]. Precisely, the network is built from a $d$-dimensional regular lattice to be improved by adding long-range connections with probability $P_{ij}\sim r_{ij}^{-\alpha }$, where $r_{ij}$ is the Manhattan distance between sites $i$ and $j$, and $\alpha$ is a variable exponent, the total length of the long-range connections is restricted. In addition, each link has a local conductance given by $g_{ij}\sim r_{ij}^{-C}$, where the exponent $C$ is to measure the impact of long-range connections on network flow. By calculating mean effective conductance of the network for different exponent $\alpha$, we find that the optimal electric current transport conditions are obtained with ${\alpha }_{\text{opt}}=d+1$ for all $C$. Interestingly, the optimal transportation condition is identical to the one obtained for optimal navigation in spatially embedded networks with total cost constraint. In addition, the phenomenon can be possibly explained by the communicability sequence entropy; we find that when $\alpha =d+1$, the spatial network with total cost constraint can obtain the maximum communicability sequence entropy. The results show that the transport performance is strongly correlated with the communicability sequence entropy, which can provide an effective strategy for designing a power network with high transmission efficiency, that is, the transport performance can be optimized by improving the communicability sequence entropy of the network.

Figure 1

The network is implemented by added long-range connections (in red) to a one-dimensional nearest-neighbor coupled network (nearest neighbor number $K=2$). Each connection has a local conductance given by $g_{ij}$, a unitary global current is added between nodes $A$ and $B$, then $G_{AB}=1/\left(V_A-V_B\right)$. So the mean global conductance $〈G〉$ is obtained by averaging over different pairs of nodes and different realizations of the network.

Figure 2

Local conductance probability distribution for each connection of a one-dimensional embedded network, where the total length is limited to $\lambda =10N$, $N=512$. (a) $C=0$ and (b) $C=1$. We sampled $500N$ realizations for each $\alpha$.

Figure 3

Dependence of mean effective conductance $〈G〉$ of one-dimensional spatially embedded networks with total cost constraints on network parameters ($\alpha$ and $N$). For $C=0$, in (a) we show the relationship between $〈G〉$ and the control parameter $\alpha$, for large $N$, the maximum conductance is obtained at ${\alpha }_{\text{opt}}=2$, and in (b) we show the case of $C=1$. In (c) and (d) we show the dependence of mean effective conductance $〈G〉$ on the network size $N$ for $C=0$ and $C=1$. The average conductance always obeys a power law behavior $〈G〉\sim N^{-\beta }$ for different values of $\alpha$. The inset shows the relationship between exponent $\beta$ and $\alpha$. The total length $\lambda$ of the added long-range connections is limited to $10N$. The results are averaged $5000N$ realizations for $N=128$ and $N=256$, $500N$ realizations for $N=512$, $250N$ realizations for $N=1024$, and $50N$ realizations for $N=2048$.

Figure 4

We show the dependence on $\alpha$ of the average conductance of the network is built form $N=L\times L$ square lattices to be improved by adding long-range connections. In (a), the maximum values of the average conductance are obtained at ${\alpha }_{\text{opt}}\approx 3$ for $C=0$. In (b) the maximum values of the average conductance are obtained at ${\alpha }_{\text{opt}}\approx 3.9$ for $L=50$, and at ${\alpha }_{\text{opt}}\approx 3.5$ for $L=100$. The inset shows the dependence of ${\alpha }_{\text{opt}}$ on $L^{-1}$, the result of linear fitting is ${\alpha }_{\text{opt}}=3.086+40.228L^{-1}$.

Figure 5

We show the dependence on $\alpha$ of the mean effective conductance of the network for different values of $C$. The underlying substrate is an one-dimensional spatially embedded network with total cost constraints ($N=2048$, $\lambda =10N$). The optimal global conductance is obtained at ${\alpha }_{\text{opt}}=2$ for all $C$. The results are averaged $1000N$ realizations.

Figure 6

Communicability sequence entropy of spatially embedded networks with total cost restriction. The normalized entropy $S_N$ as a function of $\alpha$ for one- and two-dimensional spatially embedded networks, the total length $\lambda$ of the added long-range connections is limited to $10N$ for the one-dimensional lattice, and $N$ for the two-dimensional lattice. In (a) $d=1$, $C=0$ and (b) $d=1$, $C=1$, the maximum normalized entropy is obtained at ${\alpha }_{\text{opt}}=2$. For $d=2$, when $C=0$, the maximum normalized entropy is obtained at ${\alpha }_{\text{opt}}=3$. When $C=1$, the maximum values of the normalized entropy are obtained at ${\alpha }_{\text{opt}}\approx 3.8$ for $L=50$, and at ${\alpha }_{\text{opt}}\approx 3.6$ for $L=70$.

Figure 7

The normalized entropy $S_N$ as a function of $\alpha$ for two-dimensional Kleinberg networks. In (a) for $C=0$, the maximum normalized entropy is obtained at ${\alpha }_{\text{opt}}=0$. In (b) for $C=1$, the maximum normalized entropy is obtained at ${\alpha }_{\text{opt}}=2$.