Modeling the coevolution of topology and traffic on weighted technological networks

Yan-Bo Xie, Wen-Xu Wang, and Bing-Hong Wang

Phys. Rev. E 75, 026111 – Published 28 February 2007

Abstract

For many technological networks, the network structures and the traffic taking place on them mutually interact. The demands of traffic increment spur the evolution and growth of the networks to maintain their normal and efficient functioning. In parallel, a change of the network structure leads to redistribution of the traffic. In this paper, we perform an extensive numerical and analytical study, extending results of Wang et al. [Phys. Rev. Lett. 94, 188702 (2005)]. By introducing a general strength-coupling interaction driven by the traffic increment between any pair of vertices, our model generates networks of scale-free distributions of strength, weight, and degree. In particular, the obtained nonlinear correlation between vertex strength and degree, and the disassortative property demonstrate that the model is capable of characterizing weighted technological networks. Moreover, the generated graphs possess both dense clustering structures and an anticorrelation between vertex clustering and degree, which are widely observed in real-world networks. The corresponding theoretical predictions are well consistent with simulation results.

Received 30 October 2006

DOI:https://doi.org/10.1103/PhysRevE.75.026111

Authors & Affiliations

Yan-Bo Xie¹, Wen-Xu Wang^1,2, and Bing-Hong Wang¹

¹Department of Modern Physics and Nonlinear Science Center, University of Science and Technology of China, Hefei 230026, People’s Republic of China
²Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, People’s Republic of China

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Issue

Vol. 75, Iss. 2 — February 2007

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Images

Figure 1
Illustration of the evolution dynamics. A new node $n$ connects to a node $i$ with probability proportional to $s_{i} = \sum_{j} s_{j}$ . The thicknesses of nodes and links, respectively, represent the magnitudes 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.Reuse & Permissions
Figure 2
(Color online) Time evolution of vertex strength $s_{i}$ , weight $f_{i j}$ , the difference of vertex strength and degree $s_{i} - k_{i}$ , and $(s_{i} - k_{i}) ∕ s_{i}$ for different values of parameter $W$ . Here $i$ and $j$ are chosen as $4$ and $5$ , respectively. Data points are obtained by averaging over 100 network realizations. The dashed lines are the fitting of the power-law behavior. The network size is $10 000$ . The insets are the comparisons of theoretical predictions and simulation results. The continuous curve is the analytical expression, while data points are the corresponding data fitting of the distribution.Reuse & Permissions
Figure 3
(Color online) Distributions of strength (left panel) and weight (right panel) for different values of parameter $W$ . Data points are obtained by averaging over 20 network realizations. Here we log-bin the data with bin sizes increased by multiplying with a constant factor. The network size is $40 000$ . The insets are the comparisons of theoretical predictions and simulation results. The continuous curve is the analytical expression; while data points are the corresponding data fitting of the distribution.Reuse & Permissions
Figure 4
(Color online) Distributions of vertex degree (left panel) and the correlation between vertex strength and degree (right panel) for different values of $W$ . Data points are obtained by averaging over 20 network realizations. The network size is $40 000$ . For degree distributions, we log-bin the data with bin sizes increased by multiplying with a constant factor.Reuse & Permissions
Figure 5
(Color online) Left panel: strength correlation $s_{N N}$ versus $s$ . Right panel: vertex strength $s_{i}$ as a function of vertex number $i$ . The red (solid) lines are the accurate theoretical predictions and the blue (dashed) lines are the mean field theory. The parameter values are $W = 4$ and $M = 2$ . The network size $N = 1000$ . Simulation results are obtained by averaging over 500 different realizations.Reuse & Permissions
Figure 6
(Color online) Clustering coefficient $C$ as a function of network size $N$ (left panel) and the dependence of vertex clustering $C (k)$ on degree $k$ (right panel) for different values of parameter $W$ . The line is of slope $- 1$ . Data points are obtained by averaging over 100 different realizations.Reuse & Permissions

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