General Dynamics of Topology and Traffic on Weighted Technological Networks

For most technical networks, the interplay of dynamics, traffic, and topology is assumed crucial to their evolution. In this Letter, we propose a traffic-driven evolution model of weighted technological networks. By introducing a general strength-coupling mechanism under which the traffic and topology mutually interact, the model gives power-law distributions of degree, weight, and strength, as confirmed in many real networks. Particularly, depending on a parameter $W$ that controls the total weight growth of the system, the nontrivial clustering coefficient $C$, degree assortativity coefficient $r$, and degree-strength correlation are all consistent with empirical evidence.

Figure 1

Illustration of the evolution dynamics. A new node $n$ connects to a node $i$ with probability proportional to $s_i/{\Sigma }_j{s}_j$. The thickness of nodes and links, respectively, represents the magnitude of the strength and weight. New connections (dashed lines) can be built between preexisting nodes, and the bilateral links represent the traffic growing process along links under the general mechanism of strength couplings.

Figure 2

(a) Probability distribution $P(s)$. Data are consistent with a power-law behavior $s^{-\alpha }$. In the inset, we give the value of $\alpha$ obtained by data fitting (solid circles), together with the analytical expression $\alpha =2+m/(m+2W)$ (line). The data are averaged over 20 networks of size $N=5000$. (b) Strength $s_i$ versus $k_i$ for different $W$ (log-log scale). Linear data fitting gives slopes 1.04, 1.17, 1.25, and 1.30 (from bottom to top), demonstrating the correlation of $s\sim k^{\beta }$.

Figure 3

(a) Probability distribution of the degrees $P(k)\sim k^{-\gamma }$. (b) Probability distribution of the weights $P(w)\sim w^{-\eta }$. The data are averaged over 20 networks of size $N=5000$.

Figure 4

(a) Clustering coefficient $C$ depending on the parameter $W$. In the inset, we report the evolution of clustering coefficient (or $C$ versus $N$) which converges soon. (b) Degree-degree correlation $r$ depending on $W$. In the inset, we report its evolution which converges soon.