Load distribution in weighted complex networks

We study the load distribution in weighted networks by measuring the effective number of optimal paths passing through a given vertex. The optimal path, along which the total cost is minimum, crucially depends on the cost distribution function $p_c\left(c\right)$. In the strong disorder limit, where $p_c\left(c\right)\sim c^{-1}$, the load distribution follows a power law both in the Erdős-Rényi (ER) random graphs and in the scale-free (SF) networks, and its characteristics are determined by the structure of the minimum spanning tree. The distribution of loads at vertices with a given vertex degree also follows the SF nature similar to the whole load distribution, implying that the global transport property is not correlated to the local structural information. Finally, we measure the effect of disorder by the correlation coefficient between vertex degree and load, finding that it is larger for ER networks than for SF networks.

Figure 1

The relative optimal path length $d_{\mathrm{opt}}∕d_{\mathrm{SP}}$ (a) and the relative advantage $c_{\mathrm{SP}}∕c_{\mathrm{opt}}$ (b) as a function of the exponent $\omega$ of the cost distribution for SF network with $\gamma =3$ and $N={10}^4$. The effect of the disorder becomes maximal at $\omega =1$, which manifests itself as the peak in both quantities. The cross (triangle) symbol represents the the shortest path length of the underlying binary network (the uniform cost distribution case). The inset of (b) shows the same data in linear scale, excluding those with $c_{\mathrm{SP}}∕c_{\mathrm{opt}}⩾{10}^1$.

Figure 2

The load distribution of the weighted networks in the presence of disorder. Plotted are in each panel for the ER networks (a), the SF networks with $\gamma =4$ (b), $\gamma =3$ (c), and $\gamma =2.0$ (d). All networks are of size $N={10}^4$. Data are logarithmically binned. The symbols are for the exponential cost distribution (◇), the uniform cost distribution (◻), and the strong disorder limit (엯). The guidelines are plotted together, with the slope $-1.6$ (a), $-1.7$ (b), $-1.8$ (solid) and $-2.2$ (dotted) (c), and $-2.0$ (d).

Figure 3

The condition probability $p(\ell \mid k)$ with $k=2$ for the ER network (a) and for the SF network with $\gamma =3$ (b). Data are for the exponential cost distribution (˟), the uniform cost distribution (◻), and the strong disorder limit (엯). Plotted are the guidelines with slopes $-3$ and $-1.7$ in (a); $-2.5$ and $-2$ in (b).

Figure 4

The correlation coefficient $r_{{\ell }_w{\ell }_b}$ as a function of $\omega$ for the ER network (엯), the SF network with $\gamma =3$ (◻), and $\gamma =2$ (◇). The dotted lines interpolating the data are drawn for the eye. The plot of $r_{k{\ell }_w}$, which is not shown here, follows very similar feature, owing to the high correlation between $k$ and ${\ell }_b$.