Nonlocal evolution of weighted scale-free networks

We introduce the notion of globally updating evolution for a class of weighted networks, in which the weight of a link is characterized by the amount of data packet transport flowing through it. By noting that the packet transport over the network is determined nonlocally, this approach can explain the generic nonlinear scaling between the strength and the degree of a node. We demonstrate by a simple model that the strength-driven evolution scheme recently introduced can be generalized to a nonlinear preferential attachment rule, generating the power-law behaviors in degree and in strength simultaneously.

Figure 1

(a) Link load ${\ell }_{ij}$ vs the product of the two degrees $k_i{k}_j$ of the nodes at each end of the link in the BA model with the system size $N={10}^3$. Data points are logarithmically binned to reduce fluctuation. The straight line with slope 0.63 is drawn for reference; (b) relation between the node strength (the sum of the link loads attached to the node) and the load of the node. As indicated by the guideline with slope 1, they scale linearly for large $\ell$.

Figure 2

Cumulative distributions for the degree (a) and the strength (b) of the packet transport-driven network growth with Eq. (7). The open circles denote the data for $\alpha =1$ and the filled circles for $\alpha =0.6$. The guidelines have slopes $-2.0$ (a) and $-1.2$ (b). Note, the humps for large degrees and large strengths in the case of $\alpha =1$, indicating the breakdown of the SF nature. The local clustering function $C\left(k\right)$ (c) and the average neighbor degree function $k_{nn}\left(k\right)$ (d) for the network with $\alpha =0.6$. The filled circles denote the weighted version of the corresponding quantities introduced in Ref. 12 and the cross symbols the original binary version of them.

Figure 3

Histogram of the links having the link load ${\ell }_{ij}$ and the link weight $w_{ij}$ in the coauthorship network of cond-mat subset of arXiv.org [31]. The gray level denotes the number of links in each bin, in logarithm with base 10.

Figure 4

Strength $s\left(k\right)$ based on the link load (◻), the load $\ell \left(k\right)$ based on the vertex load (▵), and the BBV strength $s^{\left(\mathrm{BBV}\right)}\left(k\right)$ (엯) vs the degree $k$ for the BBV model network with system size $N={10}^4$, the number of links emanating from a new node $m=1$, and the weight increment $w_0=\Delta =1.0$. The slopes of the guide lines are 1.33 (dotted) and 1.0 (dashed), respectively.